LET (Math Major) Reviewer Combined 195Pages 16mb

| Mathematics Major [1]

Philippine Normal University LICENSURE EXAMINATION FOR TEACHERS (LET) Refresher Course for Year 2015

WHAT TO EXPECT MAJORSHIP Area: Mathematics Focus: ARITHMETIC AND BUSINESS MATH LET Competencies: 1. Simplifying expressions involving series of operations 2. Solve problems involving a. GFC and LCMF b. prime and composite c. divisibility d. inverse and partitive proportions e. compound interest

PART I

– CONTENT

UPDATE

INTEGERS The set of integers (also called the signed numbers) is the union of the set of counting numbers { 1, 2, 3, 4, 5…}, – – – – – the set of their opposites { 1, 2, 3, 4, 5…} and zero {0}. It is usually denoted by Z. Thus, Z = {… –5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5…} The set of positive integers is {1, 2, 3, 4, 5…}. – – – – – The set of negative integers is {… 5, 4, 3, 2, 1}. The integer zero (0) is neither positive nor negative. OPERATIONS ON SIGNED INTEGERS Addition. 1. To add two integers with like signs, add their absolute values and affix their common sign. – – – – + + + + ( 5) + ( 9) = │ 5│ + │ 9│ ( 3) + ( 7) = │ 3│ + │ 7│ =5+9 =3+7 – = +14 = 10 2. To add two integers with unlike signs, find the difference of their absolute values and use the sign of the number with the larger absolute value. – – – – + + + + ( 15) + ( 9) = │ 15│ – │ 9│ ( 21) + ( 9) = │ 21│ – │ 9│ 9 = 15 – = 21 –9 – = +6 = 12 Subtraction. To subtract two integers, change the sign of the subtrahend and then proceed to the rules for addition. – – – – (+15) – ( 9) = │+15│ + │ +9│ ( 28) – (+23) = │ 28│ + │ 23│ = 15 + 9 = 28 + 23 – = +34 = 51

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| Mathematics Major [1] Multiplication and Division 1. To multiply/divide two integers with like signs, multiply/divide the absolute values of the given integers and affix positive sign. + + + + (–28) x (–21) = │ –28│ x │–21│ ( 5) x ( 29) = │ 5│ x │ 29│ = 5 x 29 = 28 x 21 + + = 145 = 588 – – – – (+125) ÷ (+5) = │+125│ ÷ │+5│ ( 861) ÷ ( 123) = │ 861│ ÷ │ 123│ = 125 ÷ 5 = 861 ÷ 123 + + = 25 = 7 2. To multiply/divide two integers with unlike signs, multiply/divide the absolute values of the given integers and affix the negative sign. – – – – + + + + ( 32) x ( 14) = │ 32│ x │ 14│ ( 15) x ( 31) = │ 15│ x │ 31│ = 32 x 14 = 15 x 31 –

–

= 448 – = –465 – – ( 2706)  (+11) = │ 2706│  │+11│ (+2184)  ( 12) = /+2184/  / 12/ = 2706 ÷ 11 = 2184 ÷ 12 – – = 246 = 182 Odd and Even Numbers Any integer that ends with 0, 2, 4, 6 or 8 is called an even number, and any integer that ends with 1, 3, 5, 7 or 9 is called an odd number. Example: 4, 20, 36, 772, 1958 are even numbers. 11, 223, 805, 907, 8009 are odd numbers. Three important facts about even and odd numbers a) If two integers are both even or both odd, their sum or difference is even. b) If one integer is even and the other is odd, their sum or difference is odd. c) The product of two integers is even unless both of them are odd.

+/Even – Even Odd

Odd Even Odd

x

Even Even Odd

Odd Even

Odd Even Even

Even Odd

PRIME AND COMPOSITE NUMBERS Any positive integer with exactly two distinct positive factors in the set of integers is called a prime number. Any positive integer that has more than two distinct positive factors in the set of integers is called composite number.

Example: Which of the following numbers are prime, composite, or neither? a) 53 prime b) 421 prime c) 24,638 composite d) 43,101 composite e) 1 neither

Fundamental Theorem of Arithmetic Every composite whole numbers can be expressed as the product of primes in exactly one way (the order of the factors is disregarded). Example: Find the prime factorization of 300. 300 5

60 10 5

6 2

2

3 2

2

the prime factors of 300: 5 x 5 x 2 x 2 x 3 = 5 x 2 x 3

2

| Mathematics Major [1] Factors and Multiples If a and b are whole numbers and a  0, then a is a factor of b if and only if there is a whole number c such that ac = b. If a is a factor of b, we can also say that a divides b (a b), or b is a multiple of a. Every number has a finite set of factors (or divisors) and an infinite set of multiples.

Examples: Classify each of the following as true or false. a) –3 is a factor of 12. true b) 03 c) 30 d) 2 is a multiple of 8. e) for all integers a, 1a.

false true false true

GREATEST COMMON FACTOR Let a and b be natural numbers. The greatest natural number d that divides both a and b is called their greatest common factor (divisor), and we write d = (a, b). Example: Find the greatest common factor. (180, 220) 180 = 2 x 2 x 3 x 3 x 5 220 = 2 x 2 x 5 x 11

2x2

x5

So, (180, 220) = 2 x 2 x 5 = 20

LEAST COMMON MULTIPLE Let a and b be natural numbers. The least natural number m that is a multiple of both a and b is called their least common multiple and we write m = [a, b]. Example: Find the least common multiple. [28, 44] 28 = 2 x 2 x 7 44 = 2 x 2 x 11

2 x 2 x 7 x 11 So, [28, 44] = 2 x 2 x 7 x 11 = 308 Remember: For positive integers a and b,

[a, b] =

ab (a, b)

and when (a, b) = 1, then [a, b] = a x b

DIVISIBILITY RULES Divisibility Properties a) If a number divides each of two other numbers, then it divides their sum. If ab and ac, then a(b + c). b) If a number divides one of two numbers but not the other, then it will not divide their sum. If ab and a ł c, then a ł (b + c) . c) If one number divides another number, then it will divide the product of that number with any other whole number. If ab, then abk. Divisibility A number is divisible by 2 if the number ends with an even number (0,2,4,6,8). Example: 158 and 5792 are divisible by 2 since the numbers end with 8 and 2 respectively which are both even numbers. A number is divisible by 3 if the sum of its digits is divisible by 3. Example: 2301 is divisible by 3 since the sum of its digits 2+3+0+1 = 6 is divisible by 3. A number is divisible by 4 if the last two digits of the number form a number which is divisible by 4. Example: 2900, 3136, and 745084 are divisible by 4. A number is divisible by 5 if the number ends with 0 or 5. Example: 1015, 2890, and 802525 are divisible by 5.

3


A number is divisible by 6 if the number is divisible by both 2 and 3. Example: 1350 and 201564 are divisible by 6 because they are both divisible by 2 and by 3. A number is divisible by 7 if the difference between twice the unit digit and the number formed by the remaining digits is divisible by 7. Example: 2191 is divisible by 7, since 2191  219 – 2 (twice 1) 217  21 –14 (twice 7) 7



divisible by 7

A number is divisible by 8 if the last three digits of the number form a number which is divisible by 8. Example: 413000, 6739048, and 9013816 are divisible by 8. A number is divisible by 9 if the sum of its digits is divisible by 9. Example: 19008 and 324594 are divisible by 9, since 1+9+0+0+8 = 18 and 3+2+4+5+9+4 = 27 are divisible by 9 A number is divisible by 10 if the number ends with 0. Example: 47020 and 390580 are both divisible by 10. A number is divisible by 11 if the difference between the sum of the digits in the even position and the sum of the digits in the odd position is divisible by 11. Example: 7654235050 is divisible by 11 since, (7+5+2+5+5) – (6+4+3+0+0) = 24 – 13 = 11 is divisible by 11 even position odd position A number is divisible by 12 if the number is divisible by both 3 and 4. Example: 16178904 is divisible by 12 because it is divisible by 3 (since 1+6+1+7+8+9+0+4 = 36 is divisible by 3) and by 4 (since the last 2 digits 04 form a number which is divisible by 3). A number is divisible by 13 if the sum of four times the unit digits and the number formed by the remaining digits is divisible be 13. Example: 195 is divisible by 13, since 195  19 + 20 (four times 5) 39



is divisible by 13.

RATIO AND PROPORTION A ratio is a comparison of 2 numbers a and b, with b ≠ 0, and may be expressed in the form “a to b”, “a : b” or “a/b”.

Ratio compares two quantities with same units.

“12 Math books to 15 English books” is a ratio “10 kilometers per liter” is a rate. Example: In 1993, for every woman arrested in the United States, 4 men were arrested. 1. What is the ratio of the number of men arrested to the number of women arrested? Ans.

4 1

or 4:1

2. What is the ratio of the number of women arrested to the number of men arrested? Ans.

1 4

or 1:4

For any two equal ratios a/b and c/d, a/b = c/d is called a proportion. This is also written as a:b = c:d. In this form, a and d are called extremes and b and c are called the means, and the rule states that “the product of the m eans equals the product of the extremes.” This can also be written as; if a : b = c : d, then ad = bc.

4


Example: If the ratio of teachers to students in a school is 1 to 18 and there are 360 students, how many teachers are there? Let x be the number of teachers,

x

1 18



360

or 1 : 18 = x : 360

18x = 360 x = 20 teachers PARTITIVE PROPORTION If a quantity q is to be partitioned into p1, p2, p3, . . . p n, so that the partitions are in the ratio a1 : a2 : a3 : . . . : an, then the size of the kth partition may be computed as follows:

q Pk = a1



a2



a3

 ... 

an



ak

Example: Joshua divides his day into leisure, sleep, and work. In the ratio 1 : 2 : 3. How many hours does he spend working? 24

P3 =

1 2  3

3

= 12 hours.

DIRECT PROPORTION

If the ratio of two quantities being compared is constant, then they are directly proportional. y1 = kx1 and y2 = kx2, then

y1 x1



k and

y2 x2



k , therefore

y1 x1

y2 

x2

Example: Junior paid 125php for 14 chocolate candies. How much would 25 of such chocolate candies cost? 125 

14

x 25

→ 14x = 125(25) = 223.21php

INVERSE PROPORTION If the product of two quantities being compared is constant, then they are inversely proportional

x1y1 = k and x2y2 = k, then x1y1 = x2y2 or

x1 y2



x2 y1

Example: It takes 20 men to build a house for 60 days. How many men will be needed to build it in 15 days?

x

20 15



60

→ 15x = 20(60) = 1 200 → x = 80 men

COMPUTING INTEREST Simple Interest ( Simple Interest = Prt ) Simple interest is an interest computed on the srcinal principal. The srcinal amount deposited or borrowed is called the principal. The percent used to determine the interest is called the interest rate. Interest rates are given for specific periods of time such as years, months or days. Example: Christian opens a savings account that pays simple interest at the rate of 5¼% per year. If he deposits 2 000php and makes no other deposits, find the interest and the final amount for 90 days.

I = (2 000php) ● (5¼%) ● (90/365) = (2 000php) ● 0.0525) ● (90/365) = 25.89php Final Amount = Principal + I = P(1 + rt) = 2 000php + 25.89php = 2 025.89php n Compound Interest ( Final Amount = P[ 1 + r ] ) Compound interest is different from simple interest because after the first interest calculation, the interest is added to the principal, so interest is earned on previous interest in addition to the principal. Compound Interest rates are usually given as annual (1 time a year), semiannual (2 times a year), quarterly (4 times a year), monthly (12 times a year), and daily (365 times a year). Example: If 500php is invested at 8% compounded semiannually, what will the final amount be after three years? Final Amount = P[ 1 + r ]n = 500[ 1 + (8% / 2)] 3 * 2 = 500[ 1 + 0.04 ] 6 = 500[1.27] = 635

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Philippine Normal University LICENSURE EXAMINATION FOR TEACHERS (LET) Refresher Course for year 2015

PLANE AND SOLID GEOMETRY –

PART I

I.

CONTENT UPDATE

Basic Ideas The undefined terms, point, line, and plane are geometric ideas and they are visually represented by a tiny dot, a thin wire, and a smooth flat surface, respectively. Points are labeled by means of capital letters, lines by naming any two of its points, and planes by naming at least three of its points. The subsets of a line are ray, segment, and the line itself.

A . .

B. Line

AB

A.

B.

A.

AB

B. Line

Ray

AB

Space is the set of all points.

Some postulates on points, lines and planes:  An infinite number of lines may pass through a given point.  To every pair of different points there corresponds a unique positive number.  This number is called the distance between the two points.     

For every two different points there is exactly one line that contains both points. If two points of a line lie in a plane, then the line lies in the same plane. Any three points lie in at least one plane, and any three non-collinear points lie in exactly one plane. If two planes intersect, then their intersection is a line. An infinite number of planes may pass through a given line.

Further:  Every segment has exactly one midpoint.  If a line intersects a plane not containing it, then the intersection is a point.  Given a line and a point not on the line, there is exactly one plane containing both.  Given two intersecting lines, there is exactly one plane containing both. II.

ANGLES If two rays have a common endpoint, but do not lie on the same line, then their union is an angle. Their common end point is called its vertex and the two rays are called its sides. The following are angles:

A point may be on the angle, in the interior or neither on the angle nor in its interior called the exterior. P. P. P.

The unit of measure for an angle is called a degree. An angle is measured with a protractor. Angle Measurement Postulate. To every angle there corresponds a real number between 0 and 180.

1

| Mathematics Major [2] Kinds of Angles 1. Acute Angle. An acute is an angle whose measure is less than 90. 2. Right Angle. A right angle is one that measures 90. It is usually represented by a small the vertex. Two lines that intersect and form right angles are called perpendicular (  ) 3. Obtuse Angle. This is an angle whose measure is more than 90 but less than 180.

square at lines.

Remarks:In plane Geometry, an angle is simply a set of points. This should be distinguished from the angles in trigonometry where we speak of directed angles. When we use directed angles, we a llow “zero angles” and “straight angles”. In the study of directed angles, we seldom use the degree as a unit measure. Instead, we use the radian. FACTS ABOUT PAIRS OF ANGLES 1. Vertical Angles. Two angles are vertical angles if and only if their sides form two pairs of opposite rays. When two lines intersect, they form two pairs of vertical angles. 2. Complementary Angles . Two angles whose measures total 90 . 3. Supplementary Angles. Two angles whose measures total 180 . C 4. Linear Pair.Two angles that are formed by two opposite rays and a third common ray. Two angles,  BAC and  CAD, form a linear pair if only if B, A, and D

.

.

are collinear and C is not on BA . A linear pair is a supplementary pair.

A

B

.

D

An angle bisector is a ray in the interior of the angle dividing it into two congruent parts. III. EQUIVALENCE RELATIONS Equality between two numbers has the following properties  Reflexive Property: a  a , for every a 

Symmetric Property: If



Transitive property: If

a  b , then b  a . a  b and b  c , then a  c .

IV. CONGRUENCE RELATIONS Congruence Properties between Angles (or between segments): 

Reflexive Property: A  A for every A .



Symmetric property: If A  B , then B  A . Transitive Property: If A  B and B  C , then A  C .



AB  CD implies that AB  CD and vice versa. Similarly, A  B implies that

mA  mB

and vice versa.

Equality is used for measures while congruence is used for figures. Some Theorems on Angles  If the angles in a linear pair are congruent, then each of them is a right angle.  If two angles are complementary, then they are both acute.  Any two right angles are congruent.  If two angles are both congruent and supplementary, then each is a right angle.  Supplements of congruent angles are congruent.  Complements of congruent angles are congruent.  Vertical angles are congruent.  Two perpendicular lines form four right angles.  The angles in a linear pair are supplementary.  The shortest segment from a point not on a line to the line is the perpendicular () segment. Examples: 1. Estimate the measure of an angle if it is three times the measure of its supplement. Solution: x = 3(180 – x) 4x = 3(180) x = 135

2. On the edge of a half-plane, take points, M, K, A such that A is between M and K. Take ray

AT

so that m TAK = 35. In the

same half-plane take ray AV such that m MAV = 85. What is the measure of TAV? ( Ans. 60) 3. If mA = 64, find the measures of the angles that are complementary and supplementary to A. Solutions: The measures of two complementary angles must add to 90, so the measure of the complement of A is given by 90 - 64 = 26. The measures of two supplementary angles must add to 180, so the measure of the supplement of A is given by 180 - 64 = 116. 4. In the figure, 1 is complementary to2, and m3 = 141. Find the measure of each numbered angle in the figure.

2


Solutions:  Since 3 and m6 are vertical angles, then these angles are congruent. Thus, if m3 = 141, therefore, m6 = 141 .     

5.

1

and 3 form linear pair and therefore they are supplementary. Thus, m1 = 39. 1 and 5 are vertical angles and vertical angles are congruent. Thus, m5 = 39. Since 1 is complementary to 2, then m2 = 51. Since 2 and 8 are vertical angles, then m 8 = 51. Now, m4 = m7 = 129. Why? ____________________________________

In the figure below, the measures of angles 1, 2, and 3 are in the ratio 1:2:3, respectively. Find the measure of each angle. Solution: Adding the terms of the ratio 1,2, and 3 gives 6.

12

1/6 of 90 = 15; 2/6 of 90 = 30; and 3/6 of 90 = 45 Thus, the measures of the angles are 15, 30 and 45.

6.

3

Find the measure of an angle whose measure is 40 more than the measure of its supplement.

Let x = the measure of the supplement of the angle x + 40 = the measure of the angle x + (x + 40) = 140 2x = 140 x = 70 and x + 40 = 100 Thus, the measure of the angle is 110.

H

C

G

F

IV. PARALLEL LINES Facts about parallel lines: 1. Parallel lines are coplanar lines that do not intersect 2. Skew lines are noncoplanar and nonintersecting lines. Examples:

a.)

B

A

Solution:

D

E

In the parallelepiped above, name all the lines that contain the edges parallel to AB .

b.) Name all the lines containing the edges that are skew to EF . Solutions: a.) The lines parallel to AB are FG , HC and ED b.) The lines that are skew to EF are AB , HC , BG and CD . 3. A transversal is a line that intersects two coplanar lines at two different points. Angles formed by Transversals a.) Alternate Interior Angles

4 8

b.) Alternate Exterior Angles

1 2 3

3 5

5 6 7

4

and 6; 3 and 5 are alternate interior angles

c.) Same-side Interior angles

1 and 7; 2 and 8 are alternate exterior angles

d.) Corresponding angles 3

1 2

1 2


4 and 5; 3

and 6 are angles on the same side of the transversal

1 and 5; 2 and 6; 4 and 8; 3 and 7 are corresponding angles

PRINCIPLES ON PARALLEL LINES Principle 1: Parallel – lines postulates: Through a given point P, not on a line l, exactly one

P

.

line maybe drawn parallel to line l.

l

Angles Relationships for Parallel Lines that are C ut by a Transversal

Principle 2: If parallel lines are cut by a transversal, then the alternate interior angles are congruent. Principle 3: If parallel lines are cut buy a transversal, then the corresponding angles are congruent. Principle 4: If parallel lines are cut by a transversal, then the alternate exterior angles are congruent. Principle 5: If parallel lines are cut by a transversal, then the same-side interior angles are supplementary. Principles on Proving Lines to be Parallel

Principle 6: If two lines are cut by a transversal so that a pair of alternate interior angles are congruent, then the lines are parallel. Principle 7: If two lines are cut by a transversal so that a pair of corresponding angles are congruent, then the lines are parallel. Principle 8: If two lines are cut by a transversal so that a pair of alternate exterior angles are congruent, then two lines are parallel. Principle 9: If two lines are cut by a transversal so that a pair of same-side interior angles are supplementary then two lines are parallel. Principle 10: Lines are parallel if they are parallel to the same line. Examples: A. Use the given information to decide which lines are parallel. Justify your answers with a principle for parallel lines.

1. 2. 3. 4. 5. 6.

6  9

1 2 4 3

4  8

m3 + m8 = 180 m5 + m7 = 180  10  11  2   10

a

12 5

10

Solutions:

9 8 d

6 7 11

b c

e

1. d ║e by Congruent Corresponding Angles (Principle 7) 2. a ║ c by Alternate Interior Angles Postulate (Principle 6) 3. a║c by Same-Side Interior Angles are Supplementary (Principle 9) 4. b ║c by Same-Side Interior Angles are Supplementary (Principle 9) 5. d ║e by Congruent Corresponding Angles (Principle 7) 6. a ║c by Congruent Alternate Exterior Angle (Principle 8)

B. In the figure at the right, find the value of x given s║t cut by a transversal l 4

| Mathematics Major [2] 1. m 2 = 2x, m3 = 4x

l

2. m1 = 2x, m6 = 136

1 2

3. m1 = 3x, m5 = 60 4.

s

3 5 4 6

m1 = 6x, m3 = 120

t

Solutions:

1. Since s║t, then 2 is supplementary to 3, (by Principle 5) Thus, 2x + 4x = 180 6x = 180 x = 30 2. Since s║t, then 1  6 (Principle 4) Thus, 2x = 136 x = 68 3. Since s║t, then 2  5 (Principle 2) Thus, 2 = 60 But 1and 2 are supplementary (Def. of linear pair) Therefore, from the given 1 = 3x and 2 = 60 Then 3x + 60 = 180 3x = 120 x = 40 10. Since s║t, then 1  3 (Principle 3) Thus, 6x = 120 x = 20 V.

TRIANGLES

A triangle is the union of the segments determined by three non-collinear points. Triangles may be classified according to the congruence or noncongruence of their sides. 1. 2. 3.

Scalene triangle. A scalene triangle is a triangle having no congruent sides. Isosceles triangle. An isosceles triangle is a triangle having at least two congruent sides Equilateral triangle. An equilateral triangle is a triangle having three congruent sides.

Scalene

Isosceles

Equilateral

Triangles may also be classified according to the kinds of angles they have. 1. Right triangle. A triangle having one right angle. 2. 3.

Obtuse triangle. A triangle having one obtuse angle. Acute triangle. A triangle having three acute angles.

Right

Acute

Obtuse

Some Theorems about Triangles

1. 2. 3.

Isosceles Triangle Theorem. If two sides of a triangle are congruent, then the angles opposite these sides are congruent. Conversely, if two angles of a triangle are congruent, then the sides opposite them are congruent Every equilateral triangle is equiangular and conversely. The sum of the measures of the angles of a triangle is 180.

Special Lines in a Triangle 5

| Mathematics Major [2] 1.

Angle bisector of a triangle . An angle bisector of a triangle is a segment or ray that bisects an angle and extends to the opposite side.

2. Median of a triangle . A median of a triangle is a segment from a vertex to the midpoint of the opposite side. 3. Perpendicular bisector of a side . A perpendicular bisector of a side of a triangle is a line and is perpendicular to a side.

that bisects

4. Altitude to a side of a triangle . An altitude of a triangle is segment from a vertex perpendicular to the opposite side. 5. Altitude of obtuse triangle . In an obtuse triangle, the two altitudes fall outside the 6. The altitudes to the congruent sides of an isosceles triangle are congruent. 7. The altitudes of an equilateral triangle are congruent.

triangle.

Theorems on Right Triangles 1. In a 30-60-90 triangle, a. the hypotenuse is twice as long as the shorter leg (the leg opposite the 30 angle), and

b.

the longer leg is

60

2a

a

3 times as long as the shorter leg.

30 a 3

2.

In a 45-45-90 triangle, the hypotenuse is leg.

2 times as long as either

45

a 2

a

45

a 3. Pythagorean Theorem In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

c

a b

a2 + b2 = c2 4. Angles Outside the Triangle The measure of an exterior angle of a triangle is equal to the sum of the remote interior angles.

2

1

4

3

m4 = m1 + m2 5. Exterior Angle Inequality Theorem The measure of the exterior angle of a triangle is greater than the measure of either remote interior angle.

2

1

4

3 m4 > m1 m4 > m2

6. Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

c

a

b a+b>c

Examples:

6


Find the lengths of the unknown sides in the adjoining figure a

b ____

c ____

b. ____ ____

6 2

a.

3

c

a

b a=b

Solutions:

By Theorem 1, a) b = 3 and c = 3 2 2.

b) a = 6 and b = 6


b

a. ____

4 3

____

b. ____

____

10

60

c

a

c

30 b

Solutions:

By Theorem 2, a) a = 4 and c = 8 3.

b) a = 5 and b = 5 3 .

A pole is braced up by wires tied to its top portion from pegs on the ground each 6 ft from the foot of the pole. If a wire is 16 ft long, how tall is the pole? Solution:

c2

a2  b2 2 2 16  6  b 2 2 2 b  16  6 b 2  256  36 b 2  220  14.83 

2

4.

Find the measures of the labeled angles in the marked figure at the right. Solutions:

mx  60; my  120; mz  30; mw  30

TRIANGLE CONGRUENCE POSTULATES

Given two triangles. There are four ways to show that they are congruent using only three pairs of corresponding congruent parts: SAS Congruence Postulate . If two sides and the included angle of one triangle are congruent respectively to the

corresponding two sides and the included angle of another triangle, then the two triangles are congruent. ASA Congruence Postulate . If two angles and the included side of one triangle are congruent respectively to the corresponding two angles and the included side of another triangle, then the two triangles are congruent. SSS Congruence Postulate . If the three sides of one triangle are congruent respectively to the corresponding three sides of another triangle, then the two triangles are congruent. SAA Congruence Theorem. . If a side and two angles adjacent angles of one triangle are congruent respectively to the corresponding side and two adjacent angles of another triangle, then the two triangles are congruent. Corresponding Parts Principle. If two triangles are congruent by SAS, ASA, SSS, or SAA, then their remaining corresponding parts are also congruent

Examples:

7

| Mathematics Major [2] Each pair of marked triangles are congruent by the indicated congruence postulate.

60

60 32

32 SAS

ASA

65 95

95 65 SAA

SSS

Triangle Congruence for Right Triangles

From the triangle congruence postulates, any two right triangles may be congruent by any of the following principles: LL Congruence.Two right triangles are congruent if the two legs of one are congruent, respectively, to the corresponding two legs of the other. (By SAS) LA Congruence.Two right triangles are congruent if a leg and an adjacent acute angle of one are congruent, respectively, to the corresponding leg and an adjacent acute angle of the other. (By ASA; by SAA if the acute angles are not adjacent) HL Congruence.Two right triangles are congruent if the hypotenuse and a leg of one are congruent, respectively, to the corresponding hypotenuse and a leg of the other. (By Transitivity)

40 40

First Minimum Theorem

The shortest segment joining a point to a line is the perpendicular segment. Thus, the distance between a line and an external point is the length of the perpendicular segment from the point to the line. Similar Triangles

Two triangles are said to be similar if a. their corresponding angles are congruent, and b. their corresponding sides are proportional. Examples of similar triangles:

Similarity Postulates: 1. AA Similarity.If two angles of one triangle are congruent to two corresponding angles of another triangle, then the triangles are similar. 2. SAS Similarity.If an angle of one triangle is congruent to a corresponding angle of another triangle and the sides that include these angles are proportional, then the triangles are similar. 3. SSS Similarity.If all the three sides of one triangle are proportional to the lengths of the corresponding sides of another triangle, then the triangles are similar.

8


Midsegment Theorem for Triangles.

A

A segment whose endpoints are the midpoints of two sides of a triangle is

D

parallel to the third side, and half the length of the third side.

a. b.

E

B

C and DE

DE // BC

5.

Side-Splitting Theorem If a line parallel to a side of a triangle intersects the other two sides in distinct points, then it cuts off segments which are proportional to these sides.

1 

A

D

E

B

C AB AD

6. 7.

BC

2

AC 

AE

Similarity in a right triangle. The altitude to the hypotenuse of a right triangle forms two triangles that are each similar to the srcinal triangle and to each other. Given a right triangle and the altitude to the hypotenuse. (a) The altitude to the hypotenuse is the geometric mean of the segments into which it separates the hypotenuse. (b) Each leg is the geometric mean of the hypotenuse and the segment adjacent to the leg.

C

Thus, in the marked adjoining figure, ABC  ACD  CBD. Thus,

AD

CD



CD Hence,

2

CD

Hence,

B

D 

AD

Moreover,

A

BD

AD  BD



AC

AC AB

Finally,

BD

.

BC

AC2  AD DC.

And



BC BA

BC2  BD BA

Examples:

1. Two angles of ABC have measures, 45 and 15, while two angles of DEF have measures 120 and 45. Are the triangles similar? By what Similarity theorem or definition? Solution: YES, by the AAA Similarity Theorem_ 2. One angle of PQR measures 40 and the sides that include the angle measures 5 each. Another triangle has an angle that measures 70 and the sides that include these angle measures 8 each. Are the triangles similar?

Solution: YES, by the SAS Similarity Theorem 3. Given the figure as marked. Find PS. Solution:

S x

Q x

15 P

9 R

15 PS

12 T

9 

12

PS  (15)(12) 9  20

C

4. In the marked figure at the right, AD = 8 and DB = 18. Find CD. Solution:

(CD)2



(CD)2

 8  18

CD = 12

AD  BD

A

B

D 9

| Mathematics Major [2] QUADRILATERALS

A quadrilateral is a four-sided figure.

   

If all four angles of a quadrilateral are right angles, then the quadrilateral is a rectangle. If all four angles of a quadrilateral are right angles, and all four sides are congruent, then the quadrilateral is a square. If both pairs of opposite sides are parallel, then the quadrilateral is a parallelogram. If one and only one pair of opposite sides are parallel, then the quadrilateral is a trapezoid.

THEOREMS ON QUADRILATERALS

1. 2.

Each diagonal separates a parallelogram into two congruent triangles. In a parallelogram, any two opposite sides are congruent. Corollary: If two lines are parallel, then all points of each line are equidistant from the other line. Recall: The distance between a line and an external point is the length of the perpendicular segment from the point to the line.

3.

The distance between any two parallel lines is the distance from any point of one to the other. In a parallelogram, any two opposite angles are congruent.

4. 5.

In a parallelogram, any two consecutive angles are supplementary. The diagonals of a parallelogram bisect each other.

Application: 6. The segment joining the midpoints of two sides of a triangle is parallel to the third side and half as long. 7. A rhombus is a parallelogram all of whose sides are congruent. 8. A rectangle is a parallelogram all of whose angles are congruent. 9. A square is a rectangle all of whose sides are congruent. 10. If a parallelogram has one right angle, then it is a rectangle. 11. In a rhombus, the diagonals are perpendicular to each other. 12. If the diagonals of a quadrilateral bisect each other and are perpendicular, then quadrilateral is a rhombus.

the

CIRCLES Definition Let P be a point in a given plane, and r be a positive number. The circle with center P and radius is the set of all points of the plane whose distance from P is r. Definition Let P be a point, and let r be a positive number. The sphere with center P and radius r is the set of all points of space whose distance from P is r.

Basic Terms on Circles and Spheres   

  

Two or more spheres or two or more circles with t he same center are called concentric. A chord of a circle is a segment whose endpoints lie on the circle. A line which intersects a circle in two points is called a secant of the

circle.

A chord of a sphere is a segment whose endpoints lie on the A diameter of a circle or sphere is a chord containing the center.

sphere.

A radius of a circle or a sphere is a segment from the center to a the sphere.

point of

10


The interior of a circle is the set of all points of the plane whose distance from the center is less than the radius. The exterior of a circle is the set of all points of the plane whose distance from the center is greater than the radius.

 

Definition

A tangent to a circle is a line (in the same plane) which intersects the circle in one and only one point. This point is called the point of tangency. Theorems Circles and Spheres 1. The intersection of a sphere with a plane through its center is a circle with the same center and the same radius. 2. The intersection of a sphere with a plane through its center is called a great circle of the sphere. 3. A line perpendicular to a radius at its outer end is tangent to the circle.

4. Every tangent to a circle is perpendicular to the radius drawn to the point of tangency. 5. The perpendicular from the center of a circle to a chord bisects the chord. 6. The segment from the center of a circle to the midpoint of a chord which is not a diameter is perpendicular to the chord. 7. In the plane of a circle, the perpendicular bisector of a chord passes through the center. 8. In the same circle or in congruent circles, chords equidistant from the center are congruent. 9. In the same circle or in congruent circles, any two congruent chords are equidistant from the center. 10. If the line and the circle are coplanar, and line intersects the interior of the circle, then it intersects the circle in two and only two points. Definition

Two circles are tangent if they are tangent to the same line at the same point. If two tangent circles are coplanar, and their centers are on the same side of their common tangent, then their internally tangent. If two tangent circles if two tangents are coplanar, and their canters are on opposite sides of their common tangent, then their externally tangent

ARCS OF CIRCLES

In the adjoining circle at the right, P is the center. The set of points (darkened) on the circle in the interior of APB is the minor arc AB . The remaining set of points on the circle is the major arc AB . A and B are the endpoints

of the arcs. Definition

A central angle of a circle is an angle whose vertex is the center of the circle.

In the figure,

APB is a central angle.

Definition

Let C be a is the AB

circle, and let A and B be the endpoints of a diameter. A semicircle union of A, B, and the points of C that lie in a given half-plane with

as edge. The points A and B are the end points of the semicircle.

11

| Mathematics Major [2] Definition

1. 2.

The degree measure of an arc is the measure of the corresponding central angle. The degree measure of a semicircle is 180.

INSCRIBED ANGLES AND INTERCEPTED ARCS Definition

An angle is inscribed in an arc if 1. The sides of the angle contain the end points of the arc and 2. The vertex of the angle is a point, but not an end point, of the arc.

Definition

An angle intercepts an arc if 1. The end points of the arc lie on the angle, 2. All other points of the arc are in the interior of the angle, and 3. Each side of the angle contains an end point of the arc.

Theorems on Inscribed Angles and Intercepted Arcs

1. The measure of an inscribed angle is half the measure of the intercepted arc. 2. An angle inscribed in a semicircle is a right angle. 3. Every two angles inscribed in the same arc are congruent. Definitions A triangle is inscribed in a circle if the vertices of the triangle lie on the circle. If each side of the triangle is tangent to the circle, then the quadrilateral is circumscribed about the circle.

SOLIDS AND THEIR VOLUMES

12


Theorems

1. 2. 3. 4.

The volume of the prism is the product of the altitude and the area of the base. The volume of a triangular pyramid is one-third the product of its altitude and its base area. The volume of a pyramid is one-third the product of its altitude and its base area. The volume of a circular cylinder is the product of its and the area of its base. The volume of a circular cone is one-third the of its altitude and the area of its base.

5.

altitude product

TRANSFORMATIONS Starting point

These triangles are congruent.

If you cut out ΔDEF and place it onto ΔABC, they should have the same size and shape. The movement may be a slide, a flip, or a turn.

Look at the kite at the right. If we fold the kite over BD , there is to-one correspondence between the points of the kite. A  C, points along

a oneAB

and CB correspond, and points along AD and CD correspond.

Transformation

A transformation is a one-to-one correspondence between points in the plane such that each point P is associated with a '

unique point P , called the imageof P . Transformations that preserve the size and shape of geometric figures are called isometries(iso means “same” and metry means “measure”) orrigid motions . Types of isometries: translation, rotation and reflection. Translation Translation is a transformation that acts like a “slide”. 

Example:

Describe a transformation that will move ABC to

' ' ' ABC



coincide with Solution:

Slide the triangle so that A moves to A’. Since B’ and C’

are

the same distance and direction from B and C, respectively, as A’ is from point A, point B’ is the image of B and point C’ is the image

' ' '

of C. Thus, ABCmoves to A B C . Trace ABC and slide it using the arrow from A to A’. To define translation , we need the concept of directed line segment. Informally, a line segment AB can be directed in two ways: (1) pointing from A to B denoted as AB or (2) pointing from B to A denoted as BA . Further, two line segments are said to be equivalentif they are parallel, have the same length and point in the same direction.

13

| Mathematics Major [2] Definition Translation

Suppose that A and B are points in the plane. The translationassociated with directed line segment AB , denoted TAB , is '

PP' is equivalent to

the transformation that maps each point P to the point P such that

Directed segment


AB so that

AB .

'

PP ' ║ AB and P P = AB. Thus quadrilateral

'

P P BA is a parallelogram, since it has a pair of opposite sides that are parallel and congruent. We can imagine that P is “slid” by the translation TAB in the direction from A to B for a distance equal to AB. 

Rotation

This is an isometry that corresponds to turning the plane around a fixed point. Example '

'

'

Describe a transformation that will move ABC to coincide with A B C .

Solution: '

'

'

'

We can turn ABC 180º around point P, the midpoint of segment BB to coincide with A B C .

To define rotation, we need the concept of a directed angle. An angle ABC is said to be a directed angle if it satisfies the following properties:

mABC  0 , then the measure of the directed angle is 0º.

1.

If

2. 3.

If ABC is a straight angle, then the measure of the directed angle is 180º. In the adjoining figure, a)

Let BA be turned about B through the smallest possible angle so that the ray BA coincides with

b)

image of

BC .

If the direction of the turn is counterclockwise, the measure of the directed angle is the positive number mABC. If the direction is clockwise, the measure is the negative number  mABC. The directed angle ABC is denoted by ABC.

For the directed angle by ABC, ray BA is called the initial side and ray BC is called the terminal side. Definition Rotation

The rotationwith center O and angle with measure a, denoted RO,a , is the '

transformation that maps each point P other than O to the point P such that '

1.

The measure of directed angle POP is a, and

2.

OP  OP'

Point O is mapped to itself by RO,a . 14

| Mathematics Major [2] 

Reflection This isometry corresponds to flipping the plane over a fixed line.

Example: '

'

'

Describe a transformation that will move ABC to coincide with A B C

Solution: '

Flip ABCover the perpendicular bisector of AA .

'

'

'

'

'

'

Then point A moves to point A , point B to B , and C to C . Hence, ABC moves to coincide with A B C 

Reflection

A transformation that “flips” the plane over a fixed line is called a reflection. Definition

Suppose that a line l is a line in the plane. The reflection in line l, denoted by

Mt ,

is the

transformation that maps points as follows: '

1.

Each point P not on line l is mapped to the point P such that l is the

2.

perpendicular bisector of segment PP . Each point Q on line l is mapped to itself.

'

Examples of transformations

15



WHAT TO EXPECT MAJORSHIP: MATHEMATICS FOCUS: Basic Algebra LET COMPETENCIES: 1. Perform operations on Algebraic Expressions 2. Simplify a given algebraic expression with series of operations 3. 4. 5. 6. 7. 8. 9. 10.

Apply the Laws of Exponents in Multiplying and Dividing Algebraic Expressions Factor polynomials Use factoring in simplifying rational expressions Perform operations on Rational Expressions Perform operations on Radical Numbers Identify the domain and/or the range of a given function Identify/ describe the graph of a function Solve problems on a) Linear equations b) Systems of linear equations 11. Compute the value of a function f(n), where n is a counting number

PART I – CONTENT UPDATE

I. Algebraic Expressions A mathematical phrase that contains a variable is an open phrase. A number phrase is an expression that does not contain a variable. It is also referred to as a numerical expression. The English phrase ‘a certain number added to 5’ may be translated to the open phrase ‘n + 5’ where n stands for a certain number. The English phrase ‘seven added to 5’ may be translated to the number phrase ‘7 + 5’. Expressions like 8 + 2, 12 – 2, 5 x 2, and 20 ÷ 2 are some number phrases for the number 10. Expressions like

x

5 , - 2a, 2n + 8, 2(l + w) are examples of open phrases. Another name for open phrase is algebraic expression. An expression composed of constants, variables, grouping symbols, and operation symbols, is called an algebraic expression. It is the result of adding, subtracting, multiplying, dividing (except by 0), or taking roots on any combination of constants and variables. 2 x3 1 1 1  4x2  2x  5 Example a) 2 x 2  3x b. 2 x  x  c) d)  2 3 x x A. Polynomials A term of a polynomial is a constant, a variable or the product of a constant and one or more variables raised to whole number exponents. The constant preceding the variable in each term is called coefficient of 3 2 3 2 the variable. In 3x  15x  2 , the coefficient of x is 3, and the coefficient of x is –15. In algebra, a number is 3 2 constant 3 x  15 x  2 frequently referred to as a , and so the last term –2 in is called the constant term. A polynomial is a term or a finite sum of terms, with only non-negative integer exponents permitted on the variables. If the terms of a polynomial contain only the variable x, then the polynomial is called polynomial in x.

St. Louis Review Center, Inc – San Pablo City Branch http://www.slreviewcenter.com| https://www.facebook.com/SLRCspc (049)562-2239 | 0917-584-8106

1


Examples: 2

a)  8r 2 s 3 d) 8 x  3 y

2

b) x – 2xy – 3y

3

e)  4r s  2 2

3

c)

1

2 x3 3

 4x2  2x  5

f) x  8 x  11x  5 4

2

B. Degree of a Polynomial The degree of a polynomial in one variable is the highest exponent of the variable in the polynomial. A term containing more than one variable has degree equal to the sum of all the exponents appearing on the variables in the term. The degree of a polynomial in more than one variable is the highest degree of all the terms appearing in the polynomial. If a monomial consists of a constant term then the expression is called constant polynomial . The degree of a constant polynomial is zero. Examples: 1) The degree of the polynomial 6 x 3 y 2  3 xy is 5. 2) The degree of the polynomial  4 x 2  5 x  3 is 2. 2 2 3) The degree of the polynomial 2 x  5 x y  7 y is 3. C. Like Terms If two terms contain the same powers of the same variables, they are called like terms or similar 2 2 2 3 terms. For example, 3x and  5 x are like terms, whereas 3x and  2 x are not like terms. Other examples 2 2  7 x x  y . are 4 xy and  5 yx ,  3x y and 5 yx , and  4 x x  y and D. Kinds of Polynomials Certain polynomials have special names depending on the number of terms they have. 1. Monomial is a polynomial that has only one term. a)  8r 2 s 3

Examples:

b) 

5x 2 y 3 4

2. Binomial is a polynomial that has two terms.

6x 2 y

 3xy 5 3. Trinomial is a polynomial that has three terms. 3 a) 8 x  3 y

Examples:

b)

2x

2

3  5x 2 y  3 5 4. Multinomial is a polynomial that has four or more terms

Examples:

a)  4 x 2  5 x  3

b)

Examples:

a) 4 x 3  5 x 2  x  7

b)

2a 4 b 3



a 3b 2

5

4

 2a 2 b  5a

E. Evaluating Algebraic Expressions In Algebra, we replace a variable with a number. This is called substituting the variable. To evaluate an algebraic expression, substitute the variable by a number and then calculate. Evaluating an algebraic expression means obtaining or computing the value of the expression where value/s of the variable/s is/are assigned. Examples: Evaluate. x y x y  1) for x = 12 & y = 8 4 4

=

=

12  8



4

20 4

2)

12  8

4 4

=



3y

x

2

=

4

= 5 + 1

yx

42



2

6



for x = 2 & y = 4

3( 4) 2

12

2

2

=3+6

18 =6 = =9 4 2 F. Operations on Algebraic Expressions H. Simplifying Algebraic Expressions Involving Grouping Symbols =

24


2


II. Laws of Exponents A. Product Law If m and n are integers and a

0, then

a m  a n  a m n Examples: 1) x 2  x 3  x 5

2) 32  34  36

B. Power of a Power Law If m and n are integers and a

a 

m n

0, then

 a mn

Examples:

  

2 4

1) x 3

 x 6   x 24 4

 

3

2) 32

C. Power of a Product Law If m is an integer and a 0 and b

 36

0, then

abm  a mb m Examples:

 

2

1) 2 x 3

 22 x 6  4x 6





2) 32 x 4 y 2

   3

D. Quotient Law If m and n are integers and m > n, and a

3 3

2

x12 y 6   36 x 36 y 18 3

0, then

am  a mn an If m and n are integers and m < n, and a am 1

0, then



an anm If m and n are integers and m = n, and a am  amn  a0  1 an Examples: 1)

29 2

4

 29  4  25  32

2)

E. Power of a Quotient Law If n, a, and b are integers, and b n

x 4 x

2

0, then

1

 x

 2  ( 4 )

1

 x

2 4



1

x2

0, then

n

a a    n b b Examples: 3

x x3   3 y  y

1) 

4

24 16 2 2)    4  3 81 3

III. Special Products and Factoring A. Special Products A. Product of the Sum and Difference of Two Terms The product of the sum and difference of t wo terms is obtained by subtracting the product of the last terms from the product of th e first terms, and is called the difference of two squares (DTS).

Examples: b a  b  a 2  b 2 a) a  

5b  3a  5b  9a 2  25b 2 b)  3a  


3


Square of a Binomial The square of a binomial is the sum of the square of the first term, twice the product of the two terms, and the square of the last term.

a  b 2  a 2 2 a b  b 2  a 2  2ab  b 2 a  b 2  a 2 2 a  b  b 2  a 2  2ab  b 2 Product of Two Binomials of the Form ax by  and cx  dy The product of the binomials ax by  and cx  dy where a, b, c, & d are real numbers, is

 cx dy by cx  dy . equal to ax



Product of a Binomial and Trinomial of the Forma  b  and a  ab  b



2

2





2 2 The product of a binomial a  b  and a trinomial a  ab  b where the first term of the trinomial is

the square of the first term of the binomial, the middle term of the trinomial is the negative of the product of the two terms of the binomial, and the third term of the trinomial is the square of the second term of the binomial, is equal to the product of the first terms plus or minus the product of the last terms of the binomial and trinomial factors. 1)

2)

a  ba 2  ab  b 2   a 3  b 3 a  b a 2  ab  b 2   a 3  b 3

(Sum of Two Cubes) (Difference of Two Cubes)

Cube of a Binomial The cube of a binomial a  b  is equal to a  b  a  b  a  b  or a  b  . 3

a  b3

= a  b2 a  b  = a 2  ab  b 2 a  b  = a 3  3a 2 b  3ab 2  b 3





B. Factoring

Factoring is the reverse of multiplying. To factor an expression means to write an equivalent expression that is a product of two or more expressions. 1) Common Monomial Factoring 1 Get the GCF of the terms in the polynomial. 2 Divide the polynomial by the GCF of the terms in the polynomial. 3 Write as factors the GCF and the quotient. Examples: Factor

a)

5 x 4  20 x 3 = 5 x  x  4 3

2 2 4 2 2 2 2 b) 16 x y  24x y  48x y = 8 x y 2 x  3  6 y 4

2

2





2) Factoring the Difference of Two Squares (DTS) For a binomial to be a difference of two squares, two conditions must hold. 1. The given binomial is a difference of two terms. 2. The two terms must be perfect squares. Examples: Factor 8 6 b) 36 x  25 y

2 a) 4 x  25,

Procedure: 2

a   

b2



a



b



a



b



2 a) 4 x 2  25 = (2 x)2 - (5) = (2 x + 5) (2 x - 5) 2

a  

2

b  

a



b



a



b



b) 36 x  25 y = (6 x ) - (5 y 3 ) 2 = (6 x 4 + 5 y 3 ) (6 x 4 - 5 y 3 ) 8

6


4 2

4


3) Factoring the Perfect Square Trinomial (PST) The square of a binomial is often called the perfect square trinomial. Use the following to help recognize a perfect square trinomial: st

rd

a) Two of the terms (1 & 3 ) must be squares, a 2 and b 2 , and are both positive. b) If we multiply a and b and double the result, we get the middle term, 2ab, or its additive inverse, - 2ab. To factor perfect square trinomial, use the following relationships: a) a 2  2ab  b 2  a b

   a  b 2 a b a b   a  b  a  b 2 2 a 2  2ab  b 2  a b    a  b  a b a b   a  b  a  b 2

b) Remember to factor out a common factor first, if there is any. 4) Factoring the Quadratic Trinomial 2

bx  c , recall that c is the constant term. If that c is not a perfect square, the In the polynomial x trinomial cannot be factoredusing perfect square trinomial type. It may, however, be possible to factor it as the product of two different binomials. Some points to consider in factoring x 2  bx  c , where b and c are constants. a) If the sign of the constant term is positive, look for its factors whose sum is the numerical coefficient of the middle. The signs of these factors must be the same. The sign of the middle term becomes the signs of the factors of the constant term. Some points to consider in factoring ax 2  bx  c , where a, b and c are constants.

To factor ax 2  bx  c , we look for binomials in the form __ x  __ __ x  __ where the products of constants in the blanks are as follows. a) The constants in the first blanks of the binomials have product a. b) The constants in the last blanks of the binomials have product c. c) The product of the constants in the extremes and the product of the constants in the means have a sum of b. 5) Factoring the Sum or Difference of Two Cubes If we divide a 3  b 3 by a  b , we get the quotient a 2  ab  b 2 and no remainder. So a  b 2 2 and a  ab  b are factors of a 3  b 3 . Similarly, if we divide a 3  b 3 by a  b , we get the quotient a 2  ab  b 2 and no remainder. So a  b and a 2  ab  b 2 are factors of a 3  b 3 . Factoring a Difference or a Sum of Two Cubes

a 3  b 3 = ( a  b ) ( a 2  ab  b 2 ) a 3  b 3 = a  b a 2  ab  b 2 6) Factoring by Grouping When the given expression is a multinomial, it may be factored by grouping. Examples:

a) 6 x 3  9 x 2  4 x  6

2 2 2 b) x  2 xy  y  b

Procedure:

a) 6 x 3  9 x 2  4 x  6 There is no factor common to all the terms other than 1. We can, however, group the terms as (6 x 3  9 x 2 ) and (4 x  6) and factor these separately. 3 2 = (6 x  9 x )  (4 x  6) 2 = 3x (2 x  3)  2(2 x  3)



2 = 2 x  3 3x  2

b) x  2 xy  y  b 2

2



Grouping the terms Factoring each binomial Factoring out the common factor (2x –3)

2

The terms do not have any common factor other than 1 but the terms can be grouped as

( x 2  2 xy  y 2 ) and b 2  . Both groups can be expressed as squares. Thus, x  y    b 2

= x  y

 b 2 b  x  y  b =  x  y   =  x  y b x  y  b 2


2

Rewriting the expression as DTS. Factoring the DTS Simplifying each factor 5


7) Factoring by Completing the Square There are polynomials that cannot be factored by using any of the preceding methods. Completing the square can factor some of these polynomials.

Examples: Factor a) x 4  64

b) 4 x  37 x y  49 y 4

2

2

4

Procedure: a) x 4  64 If the given expression is a binomial whose terms are perfect squares and positive, these two terms can be considered as the first and last terms of the perfect square trinomial. It means that in the binomial x 4  64 , the middle term is missing. The middle term (mt) can be found by using the formula: mt = 2

ft lt

where ft is the first term and lt is the last term. Using the formula,

 

4 2 2 mt = 2 x 64 = 2 x 8  16 x

So, = x 4  64  0 = x 4  16 x 2  64  16 x 2 4 2 2 = ( x  16 x  64)  (16 x )

 = x

  4x   8   4 x  x

= x2  8 2

2

Additive Identity Property 16 x 2  16 x 2  0 Grouping the terms

2

Rewriting the expression as DTS 2

 8  4 x

Factoring the DTS

 8 x 2 4 x  8 x 4  64  x 2  4 x 

Simplifying and arranging the terms

4 2 2 4 b) 4 x  37 x y  49 y 4 2 2 4 In the expression 4 x  37 x y  49 y , the first and last terms are squares . For the given to be a PST,

the middle term must be

  

2 2 2 2 mt = 2 4 x 4 49 y 4 = 2 2 x 7 y  28x y 2 2 2 2 2 2 Thus, we rename  37 x y as  28x y  9 x y , and we have

4 x 4  37 x 2 y 2  49 y 4 = 4 x 4  28x 2 y 2  49 y 4  9 x 2 y 2 4 2 2 4 2 2 = (4 x  28x y  49 y )  (9 x y )

 

= 2x  7 y 2

  3xy    

2 2

3

Grouping the terms

Rewriting the expression as DTS



2 2 2 2 = 2 x  7 y  3xy 2 x  7 y  3xy Factoring the DTS

4 2 2 4 2 2 2  xy  7 y 2 Simplifying & arranging the terms 4 x  37 x y  49 y = 2 x  3xy 7 y 2 x  3

IV. Rational Expressions Definition: A rational expression, or a fraction, is a quotient of algebraic expressions (remember that division by zero is not defined). A. Signs of Rational Expressions 1)

2)

a b



a b

 a  ( a ) a ( a ) a     b b  (b) b  (b) 

a

b



 ( a ) a a ( a )    b  (b) b (b)


6


Examples: x  (  x) x x a.    y  x  ( y  x)  ( y  x ) x  y

b.

y 3  x3 x y



 ( y 3  x3 )  ( x3  y 3 )  x y x y

B. Relations of Rational Expressions 1) Equivalent rational expressions

These are rational expressions with equal values. Note that

a



b

Examples: 1 4  a) 2 8

b)

1

a



2 a 3b 2 4

2a b

1

c)

2

x y



c

if and only if ad  bc .

d

x 2  xy  y 2 x3  y 3

2) Similar rational expressions These are rational expressions with the same denominator. Examples: 3 7 9 2 x 1 y  2 2x x 1 x  y , , a) , , b) c) , , 4 4 4 ab ab ab 2x  y 2x  y 2x  y 3) Dissimilar rational expressions These are rational expressions having different denominators. Examples: 5 3 1 3  x 3 x 2  1 2x  3y a) 9 , 8 , 6 b) a 2b , ab , ab2 c) x  y , x  y , x 2  y 2

C. Reduction of Rational Expression to Simplest Form A rational expression is said to be in lowest or simplest form if the numerator and denominator are relatively prime. The process of reducing fractions to their simplest form is dividing both numerator and denominator by their greatest common factor (GCF). Examples: 1) Reduce

35 a 4b 2

 42 a 3b 3

to lowest terms.

Solution: 4

35a b

2

 42a b

3 3

Or



57aa a abb 5a   23 7  a  a  a b b b 6b

4 2 3 2 35a b  7a b

 42a 3b 3  7a 3b 2



5a

 6b 3

2

where the greatest common factor (GCF)between the numerator and the denominator is 7 a b .

2) Reduce

x3  x2  6 x x3  3x 2  2 x

to lowest term.

Solution: We will first factor the members of the given fraction and then proceed as above.

x3  x 2  6 x x  3x  2 x 3

2




x( x 2  x  6) x( x 2  3x  2)

Common factoring by x

7


x( x  3)( x  2)



=

3) Reduce

3x  2 3x( x  1)  2( x  1)

Factoring quadratic trinomials

x( x  1)( x  2)

x3 x 1

Canceling x and (x – 2)

to simplest form.

Solution: =

3x  2

Given

3x( x  1)  2( x  1)

=

3x  2 ( x  1)(3 x  2)

=

1 x 1

Common factoring or by grouping

Canceling (3x

– 2)

D. Operations on Rational Expressions 1) Multiplication and Division of Rational Expressions a c ac a) Product of fractions:   b d bd The product of two or more given rational expres sions is a fraction whose numerator is the product of the numerators of the given fractions and whose denominator is the product of the given denominators. The product should be reduced to lowest terms. Examples:  9a 2b3 20c 6d 4 4b3c5  2    1) 4b 6c 4 18a 3d 2 5c 7 d 3 ad

2)

a 2  4b 2 2a  7 ab  3b 2

2



8a  4b 2a  4b



a 2  4ab  3b 2

=

=

a 2  ab  2b 2 (a  2b)(a  2b) (2a  b)(a  3b)

2( a  b )



4( 2 a  b ) 2(a  2b)



(a  b)(a  3b) (a  2b)(a  b)

Factoring

Canceling common factors and get the product

ab

a c a d ad     b d b c bc To obtain the quotient of two rational expressions, we multiply the dividend by the reciprocal of the divisor. Examples: 2 4 3 3 100a b 60a b  1) 3 30bc 42ac 4 Solution: 100a 2b 4 42ac 4  = Getting the reciprocal of the divisor 30bc 3 60a 3b3 then proceed to multiplication b) Quotient of fractions:

=

(5b)(20a 2b3 ) 3

(5b)(6c )



7c(a)(6c 3 ) 3(a)(20a 2b3 )

Factoring the numerator and the denominator


8


7c

=

Canceling the common factors of the numerators and the

3

denominators and simplify the result.

2) Divide

x 2  3x  2 2 x  3x  1 2

by

x2  x  2 2 x 2  3x  2

Solution: ( x  2)( x  1)

=

(2 x  1)( x  1)



( x  2)(2 x  1) ( x  1)( x  2)

Factoring and multiplying the dividend by the reciprocal of the divisor

x2 x 1

=

Canceling common factors

c) Addition and Subtraction of Rational expressions

Sum or difference of rational expressions: a c ac   Similar: d d d a c ad  bc   Dissimilar: b d bd Examples: 2x  1 7 x  5 9x  3   a) Find . 6x  1 6x  1 6x  1 Solution: =

2x  1  7 x  5  9x  3

Writing the rational expressions as a single

6x  1

fraction with a common denominator =

=

(2 x  7 x  9 x)  (1  5  3)

Collecting like terms in the numerator

6x  1

1

Performing the indicated operations in the

6x  1

numerator b) Express

2 3x  2 y



3x 3x 2  2 xy



8y 9x2  4 y2

in simplest form.

Solution: =

2 (3 x  2 y )



3x

x(3x  2 y )



8y (3x  2 y )(3x  2 y)

Factoring the

denominators Make the rational expressions similar by getting the LCD, divide the LCD by the given denominator and multiply the quotient by the given numerator =

=

2 x(3x  2 y )

x(3x  2 y )(3x  2 y)



3x(3x  2 y)

x(3x  2 y )(3x  2 y )

6 x 2  4 xy  9 x 2  6 xy  8 xy

x(3x  2 y )(3x  2 y )



8 y ( x)

x(3x  2 y )(3x  2 y )

Getting the products in the numerators


9


=

 3x 2  2 xy x(3x  2 y )(3x  2 y )

Combining like terms in the numerator

=

=

 ( x)(3x  2 y )

Factoring the numerator

( x)(3x  2 y )(3x  2 y)

1

Reducing the result to

(3 x  2 y )

simplest form E. Simplification of Complex Rational Expressions Complex rational expression is a rational expression in which the numerator or denominator is a rational expression. Examples:

y 1) Simplify the complex rational expression

x2 y

y2 x x

.

Solution:

y2  x2 =

y y x 2

2



y2  x2



y2  x2

y

x

x =

y 2  x2 y



x y 2  x2



x y

Canceling common factors and then multiplying the dividend by the reciprocal of the divisor.

V. Radical Expressions and Negative Exponents th

Any expression involving an n root can be written in radical form. The symbol

is called the radical

symbol. Radicals 1

If n is a positive integer and a is a real number for which a n is defined, then the expression 1

n

a is called a radical, and

n

a = an .

The number a is called the radicand. The number n is called the index of the radical. Remember that 1

n

a = a n . So

n

a is the positive nth root of a when n is even and a is positive, whereas

of a when n is odd and a is any real number. The expressions  4 , 4  16 , and because there are no even roots of negative numbers in the real number system. Whenever the exponent of a base is in rational form

6

n

a is the real nth root  2 are not real numbers

m where n  0, the expression can always be n

expressed in radical form.


10


Example 1: Write each exponential expression using radical notation. 5

3

b) a

a) a 4 Solution:

2

2

c) 8 3

5

3

a) a 4 =

4

a3

b) a

2

= (2 a )5

2

3

c) 8 3 =

82

Example 2: Write each radical expression using exponential notation. a) 36 Solution:

b)

3

8

b)

3

 8 = (8) 3

3

a6

c)

3

a6 = a 3

1

1

36 = 36 2

a)

c)

6

A. Simplifying Radical Expressions Product Rule for Radicals th

The n root of a product is equal to the product of the n n

ab =

th

roots of the f actors. In symbols,

a  b ,

n

n

provided that all of the expressions represent real numbers.

Examples: Simplify each of the following using the Product Rule for Radicals. a)

4

b.

32

12 x

6

c)

3

54 x5 y 9



4

2 = 24 2

Solution: a)

4

4



16

12 x 6 = 4 x

b) c)

32 =

5

3

4



6

4

2 =

9

24

2 3 2 3 = 2 (x )

3

54 x y = 3 27 x y



9

3

2

2x =

 3

3 = 2 x3 3 3

3

3 x y

9



3

3

2 x 2 = 3xy3 2 x 2

Quotient Rule for Radicals th th The n root of a quotient is equal to the quotient of the n roots of the numerator and denominator . In symbols

a

n



b

n

a

n

b

provided that all of the expressions are real numbers and b  0 .

Examples: Simplify each of the following using the Quotient Rule for Radicals. a)

3

 8x 3 y 5

b)

3

24 y 5 12 5

27 Solution: a)

b)

3

 8 x3 y 5



3

 8x3 y 3 y 2

27

3

24 y 5 125

3



3

5

(2) x y y 3

3

3

2



3

3

3

3

3

3

8  3y y 3



2



3

3

3

2 y y 5

2



 2 xy3 y 2 3

2 y3 y

2

5

Rationalizing the Denominator Multiplying both the numerator and denominator by another radical that makes the denominator


11


Examples: Rationalize the denominator of each of the following: 2

3

10

a)

24 x y

b)

6

3

4

c)

4 xy

2 1 3

Solution: 10

a)



6

3

6



6

24 x 2 y 4

3

b)

10

4 xy

60



6

6

24 x 2 y 4

3

=



4 xy

3

2 2  15



2 15



6

2

3

2x2 y2

3

2x y

2



15



6 48 x 4 y 6

3

2

3

3

3

8x y

3



3

23  6 x3 xy 6 3

3

3

2 x y



2 xy 2 3 6 x

3

2 xy

 y3 6 x

c) To rationalize letter c, use the conjugate of the two-term denominator as the multiplier of both numerator and denominator of the given expression.

2



1 3



1 3 1 3

2(1  3 ) 1 3



2(1  3 ) 1 3

2



2(1  3 )

2

 (1  3)  1  3

B. Operations on Radical Expressions 1) Addition and Subtraction Like radicals are radicals that have the same index and the same radicand. To simplify the sum of 3 2  5 2 , we can use the fact that 3x + 5x = 8x is true for any value of x. So, 3 2  5 2  8 2 . The

expression 3 2  2 3 cannot be simplified because they are unlike radicals. There are radicals that need to be simplified before adding or subtracting them. Examples: Find: 1) 2 x 3 -

4 x 2 + 5 18 x

3

2)

3

3

4

16 x y -

3

54 x3 y 4

Procedure: 3

1)

2x -

2

4 x + 5 18 x

2x  x -

Given

2 x + 5 3  2x  x

2

=

3

2

2

2

2

Product rule for radicals

= x 2 x - 2x + 15x 2 x

Simplifying each radical

= 15x- 2x 2x

3

2) =

Adding like radicals

16 x 3 y 4 -

3

3

54 x y

4

Given

2 2 x  y  y -

3

3

3

3

3

= 2 xy 3 2 y - 3xy 3 2 y

3 2 x  y  y Product rule for radicals Simplifying each radical 3

3

3

=  xy3 2 y


2) Multiplication of Radicals The product rule for radicals, n ab = n a  n b , allows multiplication of radicals with the same index. Examples: Find the product of the radicals in simplest form. 1)  2 6  4 3

2)

3

a (3 a  23 a 2 )

3) (2 3 + 5 ) (2 3 -

5)

Procedure: 1)  2 6  4 3 =  8 18  8 2  32  (8)(3) 2  24 2 2)

3

a (3 a  23 a 2 ) =

3

3

a2  2 a3 =

3

a 2  2a


12


5 2 = 4(3) – 5 = 12 – 5 = 7

2 5) = 4 3 -

3) (2 3 + 5 ) (2 3 -

C. Simplifying Expressions with Negative Exponents For any rational number a except 0, and for all whole numbers m, 1 m



a

am

To simplify algebraic expressions with negative exponents, mean to express the given expression into an equivalent quantity where the exponents become positive. Examples: Simplify the following expressions: a)

9 2 a 4bc 3 4

b)

3  2 3

3 a b c

9 x 2  4 y 4 3x 1  2 y  2

Procedure:

a)

b)

92 a 4bc 3 4

3  2 3



3 a b c

2 4 9x  4 y

1

3x  2 y

2



(32 ) 2 a 4bc 3 4

3  2 3

3 a b c

x2 y 4 2

x y

4



34 a 4bc 3



4

3 a b c

4 2 9 y  4x

3xy  2 x y 4



3  2 3

2

2



34 a 4b 2c 3 4

4

2 3



b3

3 a b c

2 2 (3 y  2 x)(3 y  2 x)

xy (3 y  2 x) 2

2

a



2 3y  2x

xy 2

VI. Relations, Functions, and their Zeros A. Definition of Relation, Function, Domain and Range A relation is a set of ordered pairs. The domain of a relation is the set of first coordinates. The range is the set of second coordinates. Relations are often defined by equations with no domain stated. If the domain is not stated, we agree that the domain consists of all real numbers that, when substituted for the independent variable, produce real n umbers for the dependent variable .

Example 1: The table of values below shows the relation between the distance of the movie projector from the screen and the size of a motion picture on the screen. Distance 1 2 3 4 Screen Size 1 4 9 16 The numbers in the table above could be written as ordered pairs (x, y) where x is the First member or first coordinate and y is the second member or second coordinate. We can express these numbers as the set of ordered pairs.

The set

1,1, 2.4 ,  3, 9 , 4,16  1,1, 2.4 ,  3, 9 , 4,16  is a relation. The domain of the relation is 1,

2, 3, 4 , and the range

is 1, 4, 9, 16 . Example 2: The set of ordered pairs below shows a similar relation. Each person is paired with a number representing his or her height. Person Height (cm)

Carl 202

Dan 142

Em 138

Frank 142

  Carl, 202 , Dan, 142 , Em  ,138 , Frank,142  . The domain is The relation is Carl , Dan, Em, Frank  and the range is 138, 142, 202 . Notice that for each person there is exactly one height. This is a special kind of relation called a function. St. Louis Review Center, Inc – San Pablo City Branch http://www.slreviewcenter.com| https://www.facebook.com/SLRCspc (049)562-2239 | 0917-584-8106

13


A function is a relation that assigns to each member of the domain exactly one member of the range. It is a set of ordered pairs of real numbers  x , y  in which no two distinct ordered pairs have the same first coordinate. The set of all permissible values of x is called the domain of the function, and the set of all resulting values of y is called the range of the function.

Example 3: 2 Let f be the function defined by f   x, y  y  x  3 . Find the domain and range of the function.

Solution: The domain is the set of all real numbers. Since the expression x 2 is nonnegative, the smallest value that y can take is 3 (when x  0 ). Hence the range is is the set of all real numbers greater than or equal to 3 or y y3 .





Example 4:



Find the domain and range of the function defined by f   x, y  y 



x  . x  1

Solution: The domain is the set of all real numbers except  1 , which is x  R x  1. To determine the range, express x in terms of y, that is x 

y y 1

. From this result, y cannot be equal to  1 . Therefore the range of

function is the set of all real numbers except  1 , which is y  R y  1 B. Values of Functions

The symbol f  x  (read " f of x" or " f at x" ) denotes the particular value of the function that corresponds to the given value of x . The variable x is called the independent variable while the variable y is called the dependent variable because y is usually expressed in terms of x when their relationship is given in the form of an equation. If there is an equation that is a function, we may replace y in the equation with f x  , since f x  represents y . For example, if the given function is y  3 x  1 , we may write it as f  x   3 x  1 . To evaluate a function for a specific value of x , replace each x in the function with the given value, and then perform the indicated operation/s. 2 Find the value of f  x   3x  5 x  2 when x  2 , denoted by f 2  .

Example 1: Solution:

f  x   3x 2  5 x  2

f 2 3 2  5 2  2 f 2  34  10  2 2

f 2   4

f 2   12  8

 In the example, the ordered pair 2, 4  belongs to the function f , in symbols 2, 4  f . We also say that 4 is the image of 2 under f , and 2 is the pre-image of 4 under f . 2 Example 2: Given that f is the function defined by f  x   x  4 x  3 , find the following:

a) f 0 

b) f  2 c) f 2 h 



2 d) f  3h



e) f  x  h 

VII. Equations Definition An equation is a mathematical sentence that uses an equal sign to state that two expressions represent the same number or are equivalent.

Examples: a) 3  2  5

b) 3x  4  2 x  5


c) x 2  9  0

14


An equation that contains at least one variable is called an open sentence. Equations b & c above are examples of open sentences. In equation b, only -1 makes the sentence true or satisfies the equation. However, more than one number might satisfy an equation. For example, +3 and -3 satisfy the equation x 2  9  0 . Any number that satisfies an equation is called a solution or root to the equation. The set of numbers from which you can select replacements for the variable is called the replacement set.

Solution set The set of all solutions to an equation is called the solution set to the equation.

1) Solving Equations

To solve an equation means to find all of its solutions. The most basic method for solving equations involves the properties of equality.

Properties of Equality For any rational numbers a, b and c, a) a  a is always true Reflexive Property b) If a  b , then b  a Symmetry Property c) If a  b , b  c , then a  c . Transitive Property d) Adding the same number to both sides of the equation does not change the solution set of the equation. In symbols, if a  b , then a  c  b  c . Addition Property e) Multiplying both sides of the equation by the same nonzero number does not change the solution set of the equation. In symbols, if a  b then ac  bc .

Multiplication Property

Equations that have the same solution set are called equivalent equations. Using the properties of equality, we can derive equations equivalent to the srcinal equation. The equations 2 x  4  10 and  x  21  4 x are eq uivalent equations because 7 satisfies both equations. 2) Types of Equations Identity is an equation that is satisfied by every number by which both sides of the equation are defined. The number of solutions is infinite.

Examples:

a) 23 x  1  6 x  2

b) x  1  x  1

c) x  1

x

Both equations 23 x  1  6 x  2 and x  1  x  1 are considered identities because both sides of each equation are identical. They can be satisfied by any real numbers. The equation real numbers except 0 because

0 0

x  1 can be satisfied by all x

is undefined.

Conditional equation is an equation that is satisfied by at least one number but is not an identity. The number of solutions is finite.

Examples:

a) 3x  4  2 x  5

2 b) x  9  0

2 The only value that makes equation 3x  4  2 x  5 true is – 1. While x  9  0 has the solution set  3, 3.


15


Inconsistent equation is an equation whose solution set is the empty set.

Examples:

a) x  x  2

b) 5  3 x 6  4 x  9  7 x

Simplifying either of the two equations using only the properties of equality will yield an equation that is false.

A. Linear Equations Definition An equation is linear if the variables occur as first powers only, there are no products of variables, and no variable is in a denominator. The graph of the linear equation is a straight line. A linear equation is also called a first-degree equation.

Examples: a) y  4 x  2

c) 3x  2  5 x  4

b) 2 x  5  9

d) 4 x  2 y  0

e.

2x 3

5 

x 2

1

Linear Equations in One Variable A linear equation in one variable x is one in th e form ax  b  0 where a and b are real numbers, with a  0 .

Techniques in Solving Equations

1) Simplifying the equation first Example: Solve the equation 2( x  4)  5 x  34. Solution: Before using the properties of equality, we simplify the expression on the left of the equation: 2( x  4)  5 x  34. Given

2 x  8  5x  34 Distributive Property 7 x  8  34 Combining like terms 7 x  8  8  34  8 Addition property by 8 7 x  42 Simplify 1 7 x  42 Multiplication property 7 x=6 Solution set is {6} 2) Using the properties of equality Example: Solve  x  21  4 x . Solution: We want to obtain an equivalent equation with only an x on the left side and a constant on the other.

 x  21  4 x Given  x  21  4 x  21  4 x  4 x  21 Addition Property 3x  21 Combining like terms 1 3

3x  21

Multiplication Property by

1 3

x=7 Checking: Replacing x by 7 in the srcinal equation gives us - 7 – 21 = - 4(7) - 28 = - 28 which is correct. So the solution set to the srcinal equation is {7}. St. Louis Review Center, Inc – San Pablo City Branch http://www.slreviewcenter.com| https://www.facebook.com/SLRCspc (049)562-2239 | 0917-584-8106

16


We may remove fractions by multiplying by the lowest common denominator (LCD). 3 5  3. Example: Solve the equation  4  x 2x Solution: Find the LCD first. The LCD is 2x. 3 5 4  3. Given x 2x 5 3  4  3 2x x 

2 x

Multiply the equation by 2x.

6  8x  5  6 x Distributive property Addition property 6  8x  (6)  (6 x)  5  6 x  (6)  (6 x) 2 x  1 Combining like terms 1 2 x  1 Multiplication property 2 1 1  }. x=  Solution set is { 2 2 Decimals may be removed from an equation before solving. Multiply by a power of 10 large enough to make all decimal numbers whole numbers. If you multiply by 10, you move the decimal point in all terms one place to the right. If you multiply by 100, you move the decimal point in all terms two places to the right. Example: Solve the equation x  0.1x  0.75x  4.5 Solution: Because the highest number of decimal places is 2 in the term 0.75x, we multiply the equation by 100, thus

100( x  0.1x  0.75x  4.5) Multiply the equation by 100 100x  10x  75x  450 Distributive Property 100x  10 x  (75x)  75x  450  (75x) Addition property 15x  450 Combining like terms 1 15 x  450 Multiplication property 15 x = 30 Solution set is {30} Applications of Linear Equations in One Variable 1) Number-Related problems Example: There are two numbers whose sum is 50. Three times the first is 5 more than twice the second. What

are the numbers? 2) Time, Rate, and Distance or Motion-Related Problems Example: A passenger bus starts from Tuguegarao City and heads for Santiago City at 40 kph. Two hours later, a car leaves the same station for Santiago City at 60 kph. How long will it be as the car overtakes the passenger bus? 3) Age- Related Problems Example: Mother is four times as old as Mary. Five years ago, she was seven times as old. How old will each be in 5 years? 4) Work-Related Problems Example:


17


Jerry can build a garage in 3 days, and Sam can build a similar garage in 5 days. How long would it take them to build a garage working together? 5) Investment-Related Problems Example: Gary invested P50,000, part of it at 6% and the other part at 8%. The annual interest on the 6% investment was P480 more than that from the 8% investment. How much was invested at each rate? 6) Digit-Related Problems Example: The tens digit of a certain number is 3 less than the units digit. The sum of the digits is 11. What is the number? 7) Geometry-Related Problems Example: The length of a rectangular piece of property is one meter longer than twice the width. If the perimeter of the property is 302 meters, find the length and width. 8) Mixture-Related Problems Example: What amounts (in ounce) of 50% and 75% pure silver must be mixed to produce a solution of 15 ounces with 70% pure silver?

B. Systems of Equations Definition: Any collection of two orthen morethe equations taken as onethat is called a system of equations equations.isIf the the system involves two variables, set of ordered pairs satisfy all of the solution set of the system.

Types of Systems of Linear Equations 1) A system of equations that has one or more solutions is called consistent. The graphs of the equations either intersect at a point or coincide. The set of coordinates of the intersection is the solution set of the system. a) Consistent Independent is a system of linear equations with only one point as its solution. The graphs of the equations in the system intersect at one point only.

Example: a) Find the solution set x  y  6 and x  y  2 by using the substitution method. Procedure: The second equation states that x and y  2 are equal, thus in the first equation, we can replace x with y  2 .

xy 6 y2 y  6

Equation 1 Replace x in equation 1 with y  2

Since this equation now has only one variable, we can solve for y . 2y  4 y2 Next, replace y with 2 in either equation to solve for x . Equation 1

xy 6 x26 x4


Equation 2

x  y2 x  22 x4 18


Checking: Replace x with 4 and y with 2 in either equation. Equation 1 xy 6 Equation 2 x  y2

42  6 66

4  2 24 = 2 + 2 44

The solution set of the system is {4, 2}. The system is consistent because there is only one solution. b) The sum of two numbers is 115. Their difference is 21. Find the numbers. Procedure: (By Elimination Method) st nd Let x = 1 number (greater number) & y = 2 number (smaller number) Formulate the equations:

x  y  115 Equation 1: x  y  21 Equation 2: Here, the elimination method can be done by adding the equations or by subtracting one equation from the other. Thus, +

x  y  115 Equation 1 x  y  21 Equation 2

2 x  136 From the sum, compute for the value of the retained variable x .

x  68

MPE

To compute for the value of y , replace x with 68 in either equation. x  y  115 Equation 1

68  y  115 y  115  68 y  47 The numbers are 68 and 47. The system is consistent because there is only one solution. b) A system of equations that has infinitely many solutions is called consistent dependent. The graphs of the equations coincide. The equations in the system are equivalent.

Examples: a) 2 y  2   x x  2y  4

Equation 1 Equation 2

3 y  1  2( x  3) 3 y  2 x  3

b)

Expressing y as a function of x in each of the equations in each system gives equal expressions. Graphing can also show dependence between the two equations in each system. The graphs of the equations will coincide. 2) A system of equations that has no solution is called inconsistent. The graphs of the equations do not intersect or are parallel. The solution set is an empty set.

Examples: a) 2 x  3 y  6

3y  2x  3

Equation 1

b)

Equation 2

 4 y  5x  7 4 y  5 x  12

Solving the system by elimination or by substitution will result into the inequality of two constants where both variables are dropped. The elimination of both variables implies that no solution can be obtained.


19


Philippine Normal University LICENSURE EXAMINATION FOR TEACHERS (LET)

Refresher Course

WHAT TO EXPECT MAJORSHIP Area:

MATHEMATICS

Focus: Advanced Algebra

LET Competencies         

Solve for the roots of a given quadratic equation Solve problems on quadratic equations Determine an equation given a set of roots which are imaginary/complex numbers Perform operations involving exponential and logarithmic functions Solve for the solution set of a given inequality n Determine the rth term of the expansion (a + b) Solve problems involving arithmetic and geometric progressions Solve problems involving variations Determine the number of positive and negative roots of a given polynomial

PART I - CONTENT UPDATE:

QUADRATIC EQUATION 2 An equation of the form ax + bx + c = 0 where a  0, a ,b, and c are constants, is a quadratic equation.

Examples: The following are quadratic equations: 2 1. 3x + 4x + 5 = 0 2. 2x2 - 21x = 0 2 3. 5x - 25 = 0 ROOTS OF QUADRATIC EQUATIONS To solve a quadratic equation in x means to find the value/s of x (unknown) that will satisfy the given equation. The values of x that will make the equation true ar e called the roots or solution s of the quadratic equation.

Methods of Finding the Roots of a Quadratic Equation 2 1. Factoring (Use this method if ax + bx + c is factorable) Example: Find the roots of x2 + 2x - 15 = 0 Factoring the left side of the equation, (x + 5) (x – 3) = 0 Equating each factor to zero, (x + 5) = 0 (x – 3) = 0 Solving the resulting linear equations, x=-5 x=3 2 2. Quadratic Formula (Use this method if ax + bx + c is not factorable) The quadratic formula is

x

 b  b 2  4a c 2a 1


Example: Solve 3x2 – 2x – 7 = 0 Substitute a = 3 b = -2 c = -7 in the quadratic formula,

x

x

2  ( 2) 2  4(3)(7) 2(3) 1  22 3

BINOMIAL FORMULA n To obtain the terms of the binomial expansion (a + b) , we use the binomial formula:

n ( n  1)a n  2 b 2

n 1

n

n

 na

(a + b) = a

b



2!

THE rth TERM OF THE EXPANSION (a + b)

rth term =

n ( n  1)(n  2)a n  3 b 3 3!

n 1

 ...  na b

n

b

n

n (n  1)(n  2)...(n  r  2)a n  r 1 b r 1 ( r  1)!

Example: 8

Find the fifth term of (2a – 3b) n=8 r=5 n–r+2=5 5th term =

(8)(7)(6)(5)( 2a ) 4 (5b) 4

th

4! 4 4

5 term = 90, 720 a b

EXPONENTIAL FUNCTIONS An exponential function with base b is defined by an equation of the form x

f (x) = b , where b and x are real numbers and b> 0,b ‡ 1. Note: An exponential function has a constant base and a variable exponent.

Examples: The following are examples of exponential functions 1. f(x) = 2x 2. f(x) = 32x - 2

1  2

4 x 3

f(x) = 

3.

PROPERTIES OF EXPONENTIAL FUNCTIONS The exponential functions f (x) = bx and g(x) = by where x and y are real numbers, b> 0, b  1 , satisfy the following properties x y x+y 1. b b = b x y 2. b = b , if and only if x = y -x

3. b =

1 bx

4. If b>1, and x< y, then bx < by x y 5. If 0 b

2


NATURAL EXPONENTIAL FUNCTIONS The natural exponential function is defined by the equation x f(x) = e where e = 2.71828… Notice that e is an irrational number. EXPONENTIAL EQUATIONS An equation where the unknown quantity appears in an exponent is called an exponential equation. Examples: The following are exponential equations: x-3 x-3 3x-2 1. 2. 60 5 ==22 2x – 6x+3 4x+5 3. 6 =2

SOLVING EXPONENTIAL EQUATIONS To solve an exponential equation is to find the value of the unknown quantity in the given equation. Example 1: Solve for x in 64 = 8x-2 Solution: Rewriting the given equation, we have 82 = 8x-2 Since the bases are equal, the exponents must be equal. Thus, 2=x–2 4=x Hence, x = 4 x2 4x

Example 2: Solve for x in the equation 2   2  Equating the exponents, we have 2 x + 4x= -3 x2 + 4x + 3 = 0 (x + 3) (x + 1) = 0 x = -3 x = -1. Therefore

x = -3

3

x = -1.

LOGARITHMIC FUNCTIONS The equation f(x) = logb x, where b  1 , x> 0 and b> 0 is called a logarithmic function. Examples: The following are logarithmic functions: 1. f(x) = log2 (3x -2) 2. f(x) = log2 x 3. f(x) = log8 (6x -3) Note: 1. If the base of the logarithm is not indicated it is understood that the base is 10. 2. If the base of the logarithm is the number e, then it is called a natural logarithm and it is written as f(x) = ln x.

LOGARITHMIC EQUATIONS A logarithmic equation is an equation that contains logarithm. Examples: The following are logarithmic equations 1. log3 81 = 4 2. log2 8 = 3 3. log2 16 = 4 4. log 100 = 2 3

| Mathematics Major [4] y

The logarithmic equation y = logb x can be written in exponential form as b = x and vice-versa.

Here are some examples: Logarithmic Form log2 8 = 3 log10 100 = 2 log4

1

Exponential Form 23 = 8 2 10 = 100

4 2 

=-2

16

1 16

34 = 81

Log3 81 = 4

PROPERTIES OF LOGARITHMS 1. logb x + log b y = log b (xy) Example: log2 (2x + 3) + log2 x = log2 [x(2x + 3)]

2. logb x – logb y = logb

x   y

Example:

 3x  5    5x 

log3 (3x -5) – log3 5x = log3  3. logb xn = nlogb x Example: log2 x3 = 3 log2 x

SOLVING LOGARITHMIC EQUATIONS To solve a logarithmic equation means to find the value of the unknown quantity in the given equation. Example 1: Solve for x in log (x + 2) – log x = log (x + 2) Rewriting the given equation, we have

log

x2 x

 log(x  2)

Since the logarithmic of both sides of the equation are equ al, then we have

x2 x

 x2

x(x + 2) = x + 2 2 x +x-2=0 (x + 2) (x-1) = 0 x = -2 x = 1 Thus, x = 1

Example 2 Solve for x in log( x – 4) + log(x – 3) = log 30 Solution Rewriting the given equation, we obtain log(x – 4)(x – 3) = log 30 (x – 4) (x – 3) = 30 2 x - 7x + 12 = 30 x2 - 7x -18 = 0 4


Thus,

(x – 9) (x + 2) = 0 x = 9 x= -2 x = 9.

INEQUALITIES Any relation expressed using the symbols <, >, > or < is called an inequality. An absolute inequality is an inequality which is always true. A conditional inequality is one which is true only for certain values of the variable involved. 1. 4 > 3 is an absolute inequality 2. x > 3 is a conditional inequality

PROPERTIES OF INEQUALITIES Let a, b, c, & d be real numbers. The following hold. 1. Trichotomy Property a > b or a b if a - b > 0 a 0 and b> 0, then a + b> 0 and ab>0. b. If a < 0 and b < 0, then a+b< 0 and ab> 0 4. Transitivity If a 0, then ac < bc If a bc

SOLVING INEQUALITIES To solve an inequality means to find the value of the unknown that will make th e inequality true. Examples: 1. Solve the inequality 4x + 3 < x + 8 Solution: 4x 8 4x + – x3 << 8x + –3 3x < 5 x<

5 3

2. Solve the inequality 2x - 3 < 5x + 7 Solution: 2x – 3 < 5x + 7 2x – 5x < 7 + 3 -3x < 10 x>

 10 3

3. Solve the inequality 3x2 – 5 x + 7 Solution: 2 3x –- 5x < 0 x(3x – 5) < 0 5


Case I x > 0 and 3x – 5 < 0 x > 0 and 3x < 5 x > 0 and x <

Thus, 0 < x <

5 3 5 3

Case II x < 0 and 3x – 5 . 0 x < 0 and 3x > 5 x < 0 and x >

5

Thus, the solution is a3null set. Therefore the general solution is 0 < x <

5 3

2

4. Solve the inequality 3x – 4 <– 3x Solution: 2 x – 4 < –3x 2 x + 3 x – 4< 0 (x + 4) ( x – 1) < 0 Case I x + 4 < 0 and x – 1 > 0 x < –4 and x > 1 Thus, there’s no solution.

Case II x + 4 > 0 and x – 1 < 0 x > -4 and x < 1 Thus, x = ( -4, 1) Therefore, the general solution set is (-4, 1) or -4 < x < 1.

SEQUENCES AND SERIES A sequence is an ordered list of numbers called terms. Example: For the sequence 1, 3, 5, 7, …, the first term is 1, the second term is 3, and so on and so forth.

The sum of the terms of a sequence is c alled a series. Example: The sum of the sequence 1, 3, 5, 7,… denoted by 1 + 3 + 5 + 7 +… is a series.

ARITHMETIC SEQUENCE

A sequence in which a constant d is added to the previous term to get the next is called an arithmetic sequence. The constant d is called the common difference. Examples:

1. The sequence 2, 4, 6, 8,… is an arithmetic sequence whose constant difference is 2. 2. The sequence 35, 30, 25, 20,… is an arithmetic sequence whose constant difference is -5 The nth Term of an Arithmetic Sequence (t n ) The nth term of an arithmetic sequence can be found by using the formula, 6


tn = t1 + (n – 1) d where th

tn = n term of the arithmetic sequence t1 = first term of the arithmetic sequence n = number of terms d = common difference Example 1 th Find the 20 term of the sequence 4, 14, 24, 34,… Solution: Here t1 = 4, n = 20, d = 10 Thus, – 1) (10) t20 ==4t1++(19) (20 (10)

= 4 + (190) t20 = 194 Example 2 th Find the 50 term of the sequence 13, 10, 7, 4,… Solution: Here t1 =13, n = 50, d = -3 Thus, t50 = t1 + (n – 1) (d) = 13 + (50 – 1) (-3) = 13 + (49) (-3) t50 = 134

ARITHMETIC SERIES The indicated sum of an arithmetic sequence is an Arithmetic Series. The sum of the first n terms of an arithmetic sequence, denoted by Sn is given by the formula

Sn=

n 2

Sn =

( t 1  t 2 ) or n 2

[(2t1 + (n – 1)d]

Example 1: Find the sum of the first 20 terms of the sequence 1, 8, 15 . . . Solution: S20=

20 2

[2(1) + (20 – 1) (7)]

=10 [2+ 133] =10 (135) S20=1350 Example 2: How many multiples of 4 are there between 15 and 94? Solution: Here= t16, t =n 92, d=4 1 Thus, tn = t1 + (n – 1)d 92 = 16 + (n – 1) 4 92 = 16 + 4n – 4 80 = 4n 20 = n So, there are 20 numbers which are multiples of 4 between 15 and 94.

7


GEOMETRIC SEQUENCE A sequence in which a constant r is multiplied by the previous term to get th e next term is called a g eometric sequence. The constant r is called the common ratio. Example: 1. The sequence 2, 4, 8, 16, 32,… is a geometric sequence whose constant ratio is 2. 2. The sequence 24, 8,

8 3

, … is a geometric sequence whose constant ratio is

1 3

.

The nth Term of a Geometric Sequence th The n term of a geometric sequence is given by the formula, tn = t, r n-1 th where tn= n term of the geometric sequence

ti = first term of the geometric sequence Example: th Find the 15 term of the sequence 1, 4, 16, 64,… Solution Here t1 = 1, n = 15, r=4 Thus, n-1 t15 = t1 r 15-1 = (1)(4 ) POLYNOMIAL FUNCTION The function defined by the equation n n-1 n-2 2 f(x) = a0x +a1x + a2x + . . .+ an-2x + an-1x + an where n is a nonnegative integer and a0, a1, . . .,an are constants, a0,  0 is a polynomial function in x of degree n. The zeros or roots of f(x) are the numbers x such that f(x) = 0. The Number of Positive and Negative Roots of a Polynomial Function If f(x) is a polynomial function with real coefficients, The number of positive real zeros of f(x) is either equal to the number of variations in sign in f(x), or to that number diminished by a positive even integer. The number of negative real zeros of f(x) is either equal to the number of variations in sign in f(-x), or to that number diminished by a positive even integer. Example: Determine the possible number of positive and negative zeros of f(x). 3 2 1. f(x) = 8x – 12x – 2x + 3 Because f(x) is of degree 3, there would be 3 zeros. f(x) has 2 variations in si gns, while f(-x) = -8x3 – 12x2 + 2x + 3 has 1 variation in sign. Thus, f(x) may have the following possible roots 1. 2 positive zeros and 1 negative zero 2. no positive zero, 1 negative zero, and 2 imaginary zeros 6 5 3 2. f(x) = 4x – 3x + 7x – 3 Because f(x) is of degree 6, the re would be 6 zeros. f(x) has 3 variations in signs, while f(-x) = -8x3 – 12x2 + 2x + 3 has 1 variation in sign. Thus, f(x) may have the following possible roots 1. 3 positive zeros, 1 negative zero, and 2 imaginary zeros 



2. 1 positive zero, 1 negative zero, and 4 imaginary zeros

VARIATION

The term use to denote the effect of changes among two or more related quantities. There are different types of variation: Direct Variation If y varies directly as x, then we write y  kx where k is the constant of variation. 



Inverse Variation

If y varies inversely as x, then we write y  



k x

where k is the constant of variation.

Joint Variation If y varies jointly as x and w, then we write y = kwx where k is the constant of v ariation. Combined Variation

If y varies jointly as w and x and inversely as z, then we write y  k

wx z

where k is the

constant of variation.

8


Philippine Normal University LICENSURE EXAMINATION FOR TEACHERS

Refresher Course for year 2015

WHAT TO EXPECT CONCENTRATION: MATHEMATICS

Focus: Trigonometry LET Competencies: 1. Show mastery of the basic terms, concepts and operations in Trigonometry involving 

Angles Trigonometric ratios  Conversion (degree to radian and vice-versa)  Pythagorean Theorem 2. Solve, evaluate and manipulate symbolic and numerical problems in Trigonometry by applying fundamental principles and processes. 

PART I. CONTENT UPDATE

1. Angles In General An angle is formed by two rays with the same endpoint. The common endpoint is called the vertex of the angle and the rays are called the sides of the angle.

B



O A

Figure 1

In Figure 1, the vertex of the angle  (read as theta) is labeled O. A and B are points on each side of . Angle  can also be denoted by AOB, where the letter associated with the vertex is written between the letters associated with the points on each side. We can think of  as having been formed by rotating side OA about the vertex to side OB. In this case, we shall call side OA theinitial side of  and side OB the terminal side of . When the rotation from the initial side takes place in a counterclockwise direction, the angle formed is considered a positive angle. If the rotation is in clockwise direction, the angle formed is a negative angle. 2. Degree Measure One way to measure the size of anangle is with degree measure. The angle formed by rotating a ray through one complete rotation has a measure of 360 degrees, written as 360.

1


One degree (1), then, is

1 36 0

of a full rotation. Likewise, 180 is one-half of a full rotation, and 90  is

half of that (or a quarter of a rotation). Angles that measure 90 are called right angles, while angles that measure 180 are called straight angles. Angles that measure between 0 and 90 are called acute angles, while angles that measure between 90  and 180 are called obtuse angles. A central angle in a circle is an angle with its vertex at the center of the circle. In a circle, a central angle that cuts off an arc equal in length to the radius of the circle has a measure of 1 radian (rad).



O

s

r

Figure 2

3. Conversions 



To convert degrees to radians, we consider the fact that 1 



180

rad

 180     



To convert radians to degrees, we consider the fact that 1 rad = 

We note the following: 1. If the unit of measure of an angle is not indicated, then it is understood to be in radians. 2. 1 rad is approximately 57, thus, 1 rad is much larger than 1 degree. Example 1. 1.1 Convert 60 to radians. Solution: Note that 1 



180

rad, so to convert from degrees to radians, we multiply by

 18 0

. Thus, we have

    rad  rad 180 3  

60  60

When our answer is in terms of , we are writing an exact value. If we wanted an approximation, we would replace  with 3.1416.  3.1416 Exact value   0.7854 Approximate value 4 4 1.2 Convert -270 to radians. Solution: We multiply by the given by



. Thus, we have

18 0

3     270  270 rad   rad 2  180  Example 2. 2.1 Convert

 4

to degrees.

Solution: To convert from radians to degrees, we multiply by  4

2.2 Convert 

7 6

rad =

180



.



 180     45 4  



to degrees. 18 0

Solution: We multiply the given by



. Hence, we have. 2






7

rad = 

6

7  180     210 6   

4. Pythagorean Theorem Right triangles are very important to the study of trigonometry. In every right triangle, the longest side is called the hypotenuse, and it is always opposite the right angle. The two other sides are called the legs of the right triangle. Y

z

x

X Z

y

Figure 3 Right triangle XZY (symbolized as right XZY) with right angle Z is drawn in Figure 3. Its hypotenuse is z while its legs are x and y. We now state the Pythagorean Theorem. Pythagorean Theorem: In any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides (called legs). B

If C = 90, then c2 = a2 + b2

c

a

A C

b

Figure 4 Example 3. Solve for x in the right triangle in Figure 4. B

x 9 C

A

12 Figure 4 Solution: Applying the Pythagorean Theorem, we have

x 2  9 2  12 2

x  9  12 = 15 Example 4. Solve for x in the right triangle in Figure 5. 2

2

B

13

x C

A

x+7 Solution: By applying the Pythagorean Theorem, we have

Figure 5

(  7)   13 x x 2

2

2

3


x 2  14 x  49  x 2  169 2 x 2  14 x  120  0 x 2  7 x  60  0 ( x  12)( x  5)  0 x+ 12 = 0

or

x-5=0

x = -12 or

x=5

Our only solution is x = 5. We do not accept x = -12 because x is the length of a side of triangle ACB and therefore, its value cannot be negative. 5. Trigonometric Functions If  is an angle in standard position, and the point ( x, y) is any point on the terminal side of  other than the srcin, then the six trigonometric functions of  are defined as follows:

2

The sine of  :

sin  

The cosine of  :

cos  

The tangent of  :

tan 

The cotangent of  :

cot  

The secant of  :

sec  

The cosecant of  :

csc  

2

2

where r = x + y or r 

y r x r y ; x0 x x ; y0 y r ; x0 x r ; y0 y

x 2  y 2 . That is, r is the distance of the point (x, y) from the srcin.

Example 5. Find the six trigonometric functions of  if  is in standard position and the point (2, -3) is on the terminal side of .

(-2,3) 

Figure 6 Solution: Using the values x = -2 and y = 3, we solve for r. 2

r

2

x  y  13

Applying the definition for the six trigonometric functions, we have sin 

3

y 



cos





3

y 



x

sec 

13

r

ta n



2

x 

cot 

13

r



csc 

2





x y r x x y

13 

3 13

 

2 2

 

3

6. Solving Right Triangles To solve a right triangle is to know the measurement or to approximate the sizes of its acute angles and sides when the measurements of at least one leg and an angle or of two of its sides are given. We now define the six trigonometric ratios as follows:

sin  

side opposite  hypotenuse

csc 

hypotenuse side opposite 

4

| Mathematics Major [8] cos 

side adjacent to  

tan  

sec 




cot  

side adjacent to 

hypotenuse side adjacent to  side adjacent to  side opposite 

Example 6. Find the value of sin B

A c = 6 cm

B Solution: The sine ratio is

b = 3 cm

a = 3 3 cm side opposite B hypotenuse

C

. The length of the side opposite B

is 3 cm., and the hypotenuse is 6 cm. Therefore, we have

sin B 

side opposite B hypotenuse



3 6

 0.5

Example 7. In right XYZ with right angle Z, mX = 25 and y = 10 units. Solve the triangle. Solutions: First, we draw the triangle.

X z=? y = 10 units Y

Z

x=?

We now solve for the unknown parts of the triangle. mY = 90 - 25 = 65. Now,

ta n X



x x 0 . By substitution, we have tan 25  y 10

x  10 tan 250 x  4.66 Similarly, sin Y 

y 10 0 . Thus, sin 65  z z z z

10 sin 65 0 10 0.906

z  11.04

5


WHAT TO EXPECT MAJORSHIP: MATHEMATICS Focus: Analytic Geometry LET Competencies: 1. Determine the equation of a line given: a. any two points on the line b. a point and the slope of the line c. a point and the slope of the line parallel to the desired line d. a point and the slope of the line perpendicular to the desired line e. the intercepts. 2. Solve involving a. problems the midpoint of a line segment, distance between two points, slopes of lines, distance between a point and a line, and segment division. b. a circle, parabola, ellipse, and hyperbola. 3. Determine the equations and graphs of a circle, parabola, ellipseand hyperbola.

PART 1: CONTENT UPDATE Introduction: This material includes a brief review of the basic terms concerning lines, circles, parabolas, ellipses, and hyperbolas. A straight line is represented by an equation of the first degree in one or two variables, while the circle, parabola, ellipse and hyperbola are represented by equations of the second degree in two variables. A. The Straight Line 1. The distance between two points A(x1,y1) and B(x2,y2) is

( x1  x2 ) 2  ( y1  y2 ) 2 .

2. Slope of a line a) The slope of the non-vertical line containing A(x 1,y1) and B(x2,y2) is m 

y1  y2 y  y1 or m  2 . x1  x2 x2  x1

b) The slope of a line parallel to the x-axis is 0. c) The slope of a line parallel to the y-axis is undefined. d) The slope of a line that leans to the right is positive. e) The slope of a line that leans to the left is negative. 3. The Equation of a line In general, a line has an equation of the formax + by + c = 0 where that a and b are not both zero.

a, b, c are real numbers and

4. Different forms of the equation of a line a) General form: ax + by + c = 0. b) Slope-intercept form: y = mx + b, where m is the slope and b the y-intercept. c) Point slope form: y  y1  m( x  x1 ) where (x1, y1) is any point on the line. d) Two point form: y  y1



y2  y1 ( x  x1 ) where (x1, y1) and (x2, y2) are any two points on the x2  x1

line. e) Intercept form:

x y  a b



1 where a is the x-intercept and b the y-intercept.

5. Parallel and Perpendicular lines Given two non-vertical lines p and q so that p has slope m 1 and q has slope m2. a) If p and q are parallel, then m1 = m2. b) If p and q are perpendicular to each other, then m1m2 = -1. 6. Segment division Given segment AB with A(x1,y1) and B(x2,y2). St. Louis Review Center, Inc – San Pablo City Branch http://www.slreviewcenter.com | https://www.facebook.com/SLRCspc (049)562-2239 | 0917-584-8106

1


x1  x2 y1  y2 , ). 2 2 r AP r1 b) If a point P divides AB in the ratio 1 so that  , then the coordinates of P( x,y) can be r2 PB r2 rx r x r y r y obtained using the formula x  1 2 2 1 and y  1 2 2 1 . r1  r2 r1  r2 a) The midpoint M of segment AB is M (

7. Distance of a point from a line The distance of a point A(x1,y1) from the line A x + By + C = 0 is given by

d



Ax1  By1  C A2  B 2

.

B. The Circle 1. Definition. A circle is the set of all points on a plane that are equidistant from a fixed point on the plane. The fixed point is called thecenter, and the distance from the center to any point of the circle is called the radius. 2. Equation of a circle a) general form: x2 + y2 + Dx + Ey + F = 0 b) center-radius form: (x – h)2 + (y – k)2 = r2 where the center is at (h,k) and the radius is equal tor. 3. Line tangent to a circle A line tangent to a circle touches the circle at exactly one point called the point of tangency. The tangent line is perpendicular to the radius of the circle, at the point of tangency. C. Conic Section A conic section or simply conic, is defined as the graph of a second-degree equation in x and y. In terms of locus of points, a conic is defined as the path of a point, which moves so that its distance from a fixed point is in constant ratio to its distance from a fixed line. The fixed point is called the focus of the conic, the fixed line is called thedirectrix of the conic, and the constant ratio is called theeccentricity, usually denoted by e. If e < 1, the conic is an ellipse. (Note that a circle has e=0.) If e = 1, the conic is a parabola. If e > 1, the conic is hyperbola.

D. The Parabola 1. Definition. A parabola is the set of all points on a plane that are equidistant from a fixed point and a fixed line of the plane. The fixed point is called thefocus and the fixed line is the directrix. 2. Equation and Graph of a) The equation ofaa Parabola parabola with vertex at the srcin and focus at (a,0) is

y2 = 4ax . The parabola

opens to the right if a > 0 and opens to the left ifa < 0. b) The equation of a parabola with vertex at the srcin and focus at (0,a) is

x2 = 4ay. The parabola

opens upward if a > 0 and opens downward ifa < 0. c) The equation of a parabola with vertex at (h , k) and focus at (h + a, k) is

(y – k)2 = 4a(x – h).

The parabola opens to the right ifa > 0 and opens to the left ifa < 0. d) The equation of a parabola with vertex at (h , k) and focus at (h, k + a) is

(x – h)2 = 4a(y – k).

e) The parabola opens upward if a > 0 and opens downward ifa < 0. f)

Standard form:

(y – k)2 = 4a(x – h) or (x – h)2 = 4a(y – k)

g) General form: y2 + Dx + Ey + F = 0, or x2 + Dx + Ey + F = 0 3. Parts of a Parabola a) The vertex is the point, midway between the focus and the directrix. b) The axis of the parabola is the line containing the focus and perpendicular to the directrix. The parabola is symmetric with respect to its axis. St. Louis Review Center, Inc – San Pablo City Branch http://www.slreviewcenter.com | https://www.facebook.com/SLRCspc (049)562-2239 | 0917-584-8106

2


c) The latus rectum is the chord drawn through the focus and parallel to the directrix (and therefore perpendicular to the axis) of the parabola. d) In the parabola y2=4ax, the length of latus rectum is4a, and the endpoints of the latus rectum are (a, -2a) and (a, 2a).

L

In the figure at the right, the vertex of the

Px

C

parabola is the srcin, the focus is F(a,o), the directrix is the line containing LL' , the axis is the x-axis, the latus rectum is

O

the line containing CC ' .

F(a, 0)

x

C’

L'

y

–

4/3 = 0

y

L

x1=0 O

y2=0

x P(x,y)

V(3,2)

F(0,-4/3)

F(5,2)

O

’

The graph of x

2

 

16 3

y. The graph of (y-2)2 = 8 (x-3).

E. Ellipse 1. Definition. An ellipse is the set ofall points P on a plane such that the sum of the distances ofP from two fixed points F’ and F on the plane is constant.Each fixed point is called focus (plural: foci). 2. Equation of an Ellipse a) If the center is at the srcin, the vertices are at (  a, 0), the foci are at ( c,0), the endpoints of the minor axis are at (0,  b) and b

2



a 2  c 2 , then the equation is

b) If the center is at the srcin, the vertices are at (0, minor axis are at ( b, 0) and b

2





x2 a2



y2 b2



1.

a), the foci are at (0,  c), the endpoints of the


x2 b2



y2 a2



1.

c) If the center is at (h, k), at (h a, k), the foci are at (h c,k), the endpoints of the minor axis are at

St. Louis Review Center, Inc – San Pablo City Branch http://www.slreviewcenter.com | https://www.facebook.com/SLRCspc (049)562-2239 | 0917-584-8106

3


(h, k b) the distance between the vertices is 2a, the principal axis is horizontal and b then the equation is

( x  h) 2

a

( y  k )2



2

b2



2



a2  c2 ,

1.

d) If the center is at (h, k), the distance between the vertices is2a, (h, k a), the foci are at (h, k c), 2 the endpoints of the minor axis are at (h  b, k),the principal axis is vertical and b

the equation is

( y  k)

2 

a2

( x  h)



a 2  c 2 , then

2 

b2

1.

3. Parts of an Ellipse. For the terms described below, refer to the ellipse shown with center at O, vertices at V’(-a,0) and V(a,0), foci atF’(-c,0) and F(c,0), endpoints of the minor axis at B’(0,-b) and B(0,b), endpoints of one latus rectum at G’ (-c, 

b

2

a

) and G(-c,

and the other at H’ ( c, 

b

b

2

a

)

2

a

) and G(c,

b

2

a

).

y

B(0,b)

2

2

(c, ba )

(c, ba )

V’(-a,0)

O

F’(-c,0)

V(a,0)

F(c,0)

x

2

2

(c, ba )

(c, ba ) B’(0,-b)

a) The center of an ellipse is the midpoint of the segment joining the two foci. It is the intersection of the axes of the ellipse. In the figure above, point O is the center. b) The principal axis of the ellipse is the line containing the fociand intersecting the ellipse at its vertices. The major axis is a segment of the principal axis whose endpoints are the vertices of the ellipse. In the figure, V 'V is the major axis and has length of 2a units. c) The minor axis s i the perpendicular bisector of themajor axis and whose endpoints are bothon the ellipse. In the figure, B' B is the minor axis and has length 2b units. d) The latus rectum is the chord through a focus and perpendicular to the major axis. G' G and H ' H 2b

2

are the latus rectum, each with a length of . a y

y (0, 3) (-4, 9 ) 5

(-5,0)

(2,6)

(4, 9 ) 5

(4,0)

(-4,0)

(8,5) (-6,4)

(5,0)

O

x

(2,1)

(-8 1)

(12 1)

,

(-4,- 9 ) 5

(0, -3)

(4,- 9 ) 5

,

x

O (8,3) (2,-4)


4


The graph of

x2 y 2  25 9



1.

( x  2)

The graph of

100

2 

( y  1)

2

25



1.

4. Kinds of Ellipses a) Horizontal ellipse. An ellipse is horizontal if its principal axis is horizontal. The graphs above are both horizontal ellipses. b) Vertical ellipse. An ellipse is vertical if its principal axis is vertical. F. The Hyperbola 1. Definition. A hyperbola isthe set of points on a plane such that the difference of the distances of each point 2.

on the set from two fixed points on the plane is constant. Each of the fixed points is called focus. Equation of a hyperbola a) If the center is at the srcin, the vertices are at (  a, 0), the foci are at ( c,0), the endpoints of the minor axis are at (0,



b) and b 2



c 2  a 2 , then the equation is

b) If the center is at the srcin, the vertices are at (0,



x2 a2



y2 b2



1.

a), the foci are at (0, c), the endpoints of the minor

x2 1. b2 c) If the center is at (h, k), the distance between the vertices is 2a, the vertices are at (h  a, k), the foci are at (h c,k), the endpoints of the minor axis are at (h,k b),the principal axis is horizontal and axis are at ( b, 0) and b

b2



2





( x  h) 2

a

2



( y  k )2

b2



y2 a2



1.

d) If the center is at (h, k), the distance between the vertices is 2a, the vertices are at (h, k  a), the foci are 

2



c), the endpoints of2 the minor2 axis are at (h b, k),the principal axis is vertical and b ( y  k) ( x  h) then the equation is  1 a2 b2 at (h,k



2

c



2

a ,

2. Parts of a hyperbola. For the terms described below, refer to the hyperbola shown which has its center at O, vertices at V’(-

a,0) and V(a,0), foci at F’(-c,0) and F(c,0) and endpoints of one latus rectum atG’ (-c,  b

2

a

H’ (c, 

) and the other at

b

2

a

) and

H(c,

b

b

2

a

) and

G(-c,

2

a

).

y

2

(c, ba )

B(0,b)

2

(c, ba )

V’(-a,0) F’(-c,0)

O V(a,0)

2

2

(c, ba )

x

F(c,0)

B’(0,-b)

(c, ba )

a) The hyperbola consists of two separate parts called branches. b) The two fixed points are called foci. In the figure, the foci are at  ( c,0). c) The line containing the two foci is called the principal axis. In the figure, the principal axis is the xaxis. St. Louis Review Center, Inc – San Pablo City Branch http://www.slreviewcenter.com | https://www.facebook.com/SLRCspc (049)562-2239 | 0917-584-8106

5


d) The vertices of a hyperbola are the points of intersection of the hyperbola and the principal axis. In the figure, the vertices are at ( a,0). e) The segment whose endpoints are the vertices is called thetransverse axis. In the figure V 'V is the transverse axis. 2

2

2

The line segment with endpoints (0,b) and (0,-b) where b  c  a is called the conjugate axis, and is a perpendicular bisector of the transverse axis. g) The intersection of the two axes is the center of the hyperbola . h) The chord through a focus and perpendicular to the transverse axis is called alatus rectum. In the f)

G’ (-c, 

figure, G' G is a latus rectum whose endpoints are length of

2b

b

2

a

) and

G(-c,

b

2

a

) and has a

2

.

a

3. The Asymptotes of a Hyperbola Shown in the figure belowis a hyperbola with two lines asextended diagonals of the rectangle shown.

y y

 

b a

y

x

b



a

x

P

(0,b)

(-a,0)

O

x

(a,0) (0,-b)

These two diagonal lines are said to be the asymptotes of the curve, and are helpful in sketching the graph of a hyperbola. The equations of the asymptotes associated with the equations of the asymptotes associated with

y2 a2



x2 b2



x2 a2



y2 b2

1 are y 



1 are y 

b b x and y   x . Similarly, a a

a a x and y   x . b b

y

y (6,9)

(-9,6)

F(0,6)

(9,6) 3y



x



0

(0,3) F (-6,0) (-3,0) ’

O

(3,0)

F(6,0)

x x

O (0,-3) 3y

(6,-9)




x



0

F (0,-6) ’

6


The graph of

x2 9



y2 27



1.


The graph of

y2 9



x2 27



1.

7


LICENSURE EXAM FOR TEACHERS (LET) Refresher Course for 2015

CALCULUS PART I

–

CONTENT UPDATE

Theorems Limits of Functions We use theon following theorems to evaluate limits of functions: 1. Uniqueness Theorem: If the limit of a function exists, then it is unique. That is, if lim f ( x )  L1 and lim f ( x )  L2 , then L1  L2 . xa

x a

2.

If m, b  R , then lim( mx  b)  ma  b

3.

If f ( x)  c , a constant, then lim c  c

4.

lim x  a

5.

If lim f ( x)  L1 and lim g ( x)  L2 then

xa

xa

xa

xa

x a

5.1 lim[ f ( x)  g ( x)]  L1  L2 xa

5.2 lim[ f ( x )  g ( x)]  L1  L2 xa

 f ( x)  L1 provided L2  0   g ( x)  L2  If lim f ( x )  L and n  Z then

5.3 lim x a 6.

xa

6.1 lim[ f ( x)]n  Ln x a

6.2 lim

n

x  a1

f ( x) 

n

L with the restriction that if n is even, L  0.

7.

lim f ( x )  L if and only if lim f ( x)  lim f ( x )  L

8.

If r is any positive integer, then 1 8.1 lim r  

x a

xa

x0

8.2 lim x 0

8.3 lim x 0

9.

x

1

xr 1 xr

  r , odd    r , even  

x  a f ( x )  0 and lim x  a g ( x )  c where c Let a  R, lim  R, c  0

9.1

9.2

9.3

9.4 10.

xa

Let

 g ( x)      f ( x)   g ( x)  If c  0 and if f ( x) approaches 0 through negative values of f ( x) , then lim    x  a f ( x)    g ( x)  If c  0 and if f ( x) approaches 0 through positive values of f ( x) , then lim    x  a f ( x)    g ( x)  If c  0 and if f ( x) approaches 0 through negative values of f ( x) , then lim    x  a f ( x)   lim f ( x)   , lim g ( x)   and lim h( x )  c where c is any If c  0 and if f ( x) approaches 0 through positive values of f ( x) , then lim x a

xa

xa

xa

constant, then 1


10.1 lim[ f ( x )  h( x)]   x a

10.2 lim[ g ( x)  h( x)]   xa

10.3 if c  0, lim[ f ( x)  h( x)]   x a

10.4 if c  0, lim[ f ( x)  h( x )]   xa

10.5 if c  0, lim[ g ( x)  h( x )]   xa

10.6 if c  0, lim[ g ( x)  h( x)]   xa

If p  Z , c  R then lim

11.

II.

c



x  

xp

 0 and lim

x  

c xp

0

Theorems on Differentiation In getting the derivative of a function, one or more of the following theorems may be applied: 1. If f ( x)  c, where c is a constant, then f ' ( x)  0 n 2. If f ( x)  x , where n  Z  0, then f ' ( x)  nx n

1

3. If f ( x)  g ( x)  h( x), then f ( x)  g ' ( x)  h' ( x) ,

4. If f ( x)  g ( x)  h( x), then f ' ( x)  g ( x)h' ( x)  h( x) g ' ( x) 5. If f ( x)  cg ( x), where c is a constant, then f ' ( x)  cg ' ( x) 6. If f ( x) 

g ( x) h( x) g ' ( x)  g ( x) h' ( x ) , where h( x)  0 , then f , ( x)  h( x ) h( x)2

 [ g ( x)]r , then f ' ( x)  r[ g ( x)]r 1 g ' ( x) 8. If f ( x)  sin[g ( x)], then f ' ( x)  g ' ( x) cos[g ( x)] 9. If f ( x)  cos[g ( x)], then f ' ( x)   g ' ( x) sin[g ( x)] 2 10. If f ( x)  tan[g ( x)], then f ' ( x)  g ' ( x) sec [ g ( x)] 7. If r is any rational number, f ( x)

2

11. If f ( x)  cot[g ( x)], then f ' ( x)   g ' ( x) csc [ g ( x)] 12. If f ( x)  sec[g ( x)], then f ' ( x)  g ' ( x) sec[g ( x)] tan[g ( x)] 13. If f ( x)  csc[g ( x)], then f ' ( x)   g ' ( x) csc[g ( x)] cot[g ( x)] 14. If f ( x)  e

g ( x)

15. If f ( x)  a

g ( x)

then f ' ( x)  e

g ( x)

where a is any constant not equal to 0, then f ' ( x)  a

16. If f ( x)  ln[g ( x)] then f ' ( x) 

III.

g ' ( x) 1

g ( x)

g ( x)

ln(a) g ' ( x)

g ' ( x)

Formulas for Antiderivatives The following are known formulas for antiderivatives 1. 2. 3.

 dx  x  c  af ( x)dx  a  f ( x)dx where a is a constant.  [ f ( x)  f ( x)]dx   f ( x)dx   f ( x)dx 1

2

1

2

 x n 1   c, n  1 4. If n is a rational number ,  x dx   n  1  ln x  c, n  1 5.  sin udu   cos u  c n

 cos udu  sin u  c 7.  sec udu  tan u  c 8.  sec u tan udu  sec u  c 9.  csc udu   cot u  c 10.  csc u cot udu   csc u  c 11.  tan udu  ln sec u  c 6.

2

2

cot udu  ln sin u  c

12.

 2


13. 14.

 sec udu  ln sec u  tan u  c  csc udu  ln csc u  cot u  c

 u n 1   c , n  1 15.  u du   n  1  ln u  c, n  1 16.  eu du  eu  c n

17.

IV.

1

 a du  ln a a u

u

c

Areas of Plane Regions Below are some formulas for finding the areas of plane regions 1. Area (A) bounded by the curve y  f ( x) , the x-axis and the lines x  a and x  b b

A

 f ( x)dx a

2. Area (A) bounded by the two curves y  f1( x)

and

y  f2 ( x) for all

x  [a , b] b



A  [f2 ( x )  f2 ( x )]dx a

V.

Volumes of Solids of revolution We use the following methods to find the volumes of solids of revolution 1. Disk Method The region bounded by the curve y  f ( x) , the x-axis and the lines x  a and x  b , is revolved about the x-axis. b



V   [f ( x )]2 dx a

2. Ring Method The region bounded by the two curves y  f1( x)

and

y  f2 ( x) ,

where f1(x) > f2(x) for all x  [a , b], is revolved about the x-axis b





V   [ f1( x )]2  [ f2 ( x )]2 dx a

3. Shell Method 3.1 The region bounded by the curve y  f ( x ) , the x-axis and the lines x  a and x  b , is revolved about the y-axis. b



V  2 xf ( x )dx a

3.2 The region bounded by the two curves y  f1( x)

and

y  f2 ( x) , where f1(x) > f2(x) for

all x  [a , b], is revolved about the y-axis b



V  2 x[f1( x )  f2 ( x )]dx a

3

Geometry and Measurement

Mathematics Major

Basic Algebra

Arithmetic and Business Mathematics

Advanced Algebra

Basic Calculus

Trigonometry

Licensure Examination for Teachers Prepared by: Prof. Henry P. Contemplacion

Analytic Geometry

COMPLEX NUMBERS

25

COMPLEXNUMBERS

RADICALS

RADICALS

ALGEBRA

ALGEBRA

ALGEBRA

What must be subtracted from

Find the sum of (a2 – b2 + 2ab)

3x3 –

x2

and (a2 –

x3

x2 –

+

+ 5x – 2 to get

2ab + b2).

2x + 1 ?

2x3 – 2x2 + 7x

–

3

ALGEBRA

If x = -2, y = -1 and z = 2, What is the value of 2x2y3z2?

2a2

1. What fraction of 3 1/4 is ELEM. MATHEMATICS 1 1/3? 2. Find the smallest positive integer that has factors of 9,12 and 18.

-32

3. Give the value: 33 1/3% of 12 ELEM. MATHEMATICS plus 66 2/3% of 18 minus 12 ½% of 16 4. Write 0.243243243243243.... in fraction. 5. How much interest would be paid on a bank loan of P50,000 for 9 months at 12% annual interest?

Number Theory The simplest expression for 340/920 is _____________.

1

Number Theory Find the difference between the largest and smallest prime no. from 1 to 100.

95

Number Theory How many factors of set X are factors of any members of set Y? X = (2,3,4,5,6) Y=(3,4,8,9,10)

4

If x³ is Number an odd number Theory and y³ is an even number, which of the ff. is/are TRUE ? I. II. III. IV.

x³ - y³ is an even number x³ + y³ is an odd number x³ y³ is an even number x³ - y³ + 1 is an odd number

II & III Kevin has collection of match boxes. If NUMBER he counts the match THEORY boxes by twos, threes, fours, fives, or sixes, there is always one left over. If he counts the match boxes by sevens, none remain. What is the fewest number of match boxes Kevin could have?

The green neon lights flash every 4 NUMBER THEORY seconds; the blue, every 5 seconds; the red, every 6 seconds. If all the lights flash together at 8:00 PM, at what time will they flash together again?

8:01 PM Arithmetic and Business Mathematics

A 25-hectare farm land produces 350 cavans of palay. How many cavans can 60-hectare land produce?

Answer: 840

301 Arithmetic and Business Mathematics


What is the smallest number that can be divided by all the numbers 1 to 10?

How much is 30% of 70% of 500?

Answer: 2520


What is the difference between 38 and 83 ?

Answer: 105


A water pipe 10 feet long was cut into 2 with the shorter part measuring 2.5 ft. Find the ratio of the smaller part to the larger part.

Answer: 6049

Answer: 1:3



A bond paper is folded in half, after which it is folded again in half, and again in half. If this folding is done 5 times, how many rectangles on the srcinal sheet of paper will be?

A merchant sold an item for P35,000, in the process earning for himself 15% profit. How much is the profit?

Answer: 32


A graduating class of 160 boys and 180 girls held a seniors’ ball. Approximately what percent attended if 40% of the girls and 30% of the boys were there?

Answer: 35.29%


A large container contains 12 liters when 30% full. How much does it contain when it is 90% full?

Answer: 36 liters


If you are currently earning P22,000 a month, how much will new salary be if you are your to receive a 4% increase next month?

Answer: P22,880

Answer: P5,250


How much would P180,000 amount to in 20 years if it earns 9% per annum?

Answer: P504,000


A book is 2 ½ cm thick. How many copies of this book can be placed in a shelf one meter long?

Answer: 40


Find the least common multiple of the numbers 18, 24 and 54?

Answer: 216


28 students passed while 20% failed in the midterm exam. How many students were tested?

Answer: 35


What is the number 2/3 of whose square root is 10?

Answer: 225


A farmer has enough feeds for 72 ducks for 14 days. If he sells 16 ducks, how long will the feeds last?

Answer: 18


A gas tank of Mr. Reyes car is 5/8 full. If he used 3/5 of this amount travelling from his house to his province, what part of the tank has gas?

Answer: 1/4


How many boys are there if 33 1/3% of 1500 first year high school students are girls?

Answer: 1000


A housewife deposited P150,000 at 6% compounded annually for 5 years. What is the amount in 5 years?

Answer: P200,733.84


In how many years will P50,000 grow to P60,000. If it is invested at 6% compounded monthly?

Answer: 3.05


Successive discounts of 20% and 30% is equivalent to what single discount?

Answer: 44%



A defective ruler was to be found 11.5 in. long. Using this ruler a boy was found to be 4 ft. tall. What is the actual height of the boy?

The average of 6 consecutive no. is 17.5. What is the average of 5 consecutive no.?

Answer: 4ft. 2 in.

FRACTION If 2/5 of x is 20, what is 1/2 of x?

Answer: 17

FRACTION A water tank is 5/6 full. If 1/3 of the water was used for cleaning the car, what part of the tank has water?

25

5/9

FRACTION I used 2/5 of the money in first store and 1/3 in a second store. If I had P56 left how much money did I have at first?

In one class, 1/4 of the students FRACTION

210

RATIO A 10-meter log was cut off at 75 dm. mark. What is the ratio of the smaller piece to the larger piece?

1:3

are honor students. Of these honor students, 2/7 are varsity athletes. If there are 6 athletes in the class how many students are there in the class?

84

RATIO Divide P4340 among A, B and C so that A’s share; B’s share; C’s 1 1 1 share is 2 : 3 : 5 , respectively. How much is the share of each?

2100, 1400, 840


Half of the people in the room left. One-fourth of the remaining started to dance, there were then 12 people who are not dancing. Find the srcinal number of the people in the room.

Answer: 32

PERCENT An employee spends about P3,332 a month. This sum is 70% of his monthly salary. How much does he receive a month?

P4,760 An agency with 3,200 PERCENT employees plans to reduce its staff by 35%. How many employees are to be retained?

2080

PERCENT There are 1400 pupils in Kaiba Central School. Of these students, 12.5% are in Grade VI and 20% themany rest are in are Grade V. of How pupils there in Grade V?

245


What number when increased by 90% of itself equals 133?

Answer: 70

The srcinal price of a pair PERCENT of jogging pants was P300. What was the rate of discount after paying P195?

35%

PERCENT A tank was 50% full of water. 600 liters of water are added making it ¾ full. How many liters does it hold when full?

2400 L

FRACTION/ PERCENT If the length of the rectangle is reduced by 1/5 and width is also reduced by 1/5, what percent of the srcinal area is the new area of the rectangle?

64%


In the Education Dept., 3/5 are BSED students and the rest are BEED students. Of these BSED students, 1/3 are Math majors. If there are 20 Math majors, how many BEED students are there?

Answer: 40

Plane and Solid Geometry Measurement

How many hectares are in a rectangular field that is 750 m wide and 800 m long?

Answer: 60 ha


The outer diameter of a metal pipe is 3 ¼ cm and the inner diameter is 2 ¾ cm. Find the thickness of the pipe.

Answer: ¼ cm


The supplement of an angle is three times its measure. What is the supplement?


A mile is about 1.609 km. How many meters is a mile longer than a kilometer?

Answer: 609


The length and width of a rectangle are 3x –2 and 2x + 1, respectively. What is the perimeter of the rectangle?

Answer: 10x – 2


If the perimeter of a rectangle is 72 cm and its length is 6 cm longer than its width. What is its length?

Answer: 21 cm


Rico cuts out the largest circle possible from a square whose side is 10 m. What is the area of the circle?

Answer: 25π or Answer: 135°

78.54 sq.m.


If the width of a rectangle is denoted by w and the length is 4 times the width, what is the perimeter of the rectangle?


Find the measure of an angle if the measure of its supplement is 50 more than twice the measure of its complement.

Answer: 10w

Answer: 50°



The length of an equilateral triangle is 16 cm. Find its area.

Answer: 64


What do you call a regular polygon with 170 diagonals?

Answer: Icosagon

Each interior angle of a regular polygon measures 162°. How many sides does it have?

Answer: 20


If arc AB is 72°, then the measure of angle α?

Answer: 36°



If arc AB has a length of 36° and arc CD 72°, then the measure of angle AOB is ____. (AB and CD intersect in the


interior of the circle)

Answer: 54°

exterior of the circle)

Answer: 50°


If arc AC has a length of 228° and arc CA 132°, then the measure of angle ABC is ____. (Angles formed by two tangent lines)

Answer: 48°


A rectangular lot has a perimeter of 120 m. If the length of the lot is 20 m. more than its width, what is its area?

Answer: 800 sq. m.


A picture 10 cm x 8 ½ cm is mounted on a piece of hard cardboard. If there is a margin of 2 ½ cm around the picture, what is the perimeter of the cardboard used?

Answer: 57 cm


How much larger is the supplement of 54° angle than the complement of a 65° angle?


A room of 10 m by 7m. There is 7.5 m by 5 m carpet in the middle. What percent of the room is uncovered? ?

Answer: 46.4%


If the length of the rectangle is reduced by 1/5 and width is also reduced by 1/5, what percent of the srcinal area is the new area of the rectangle?

Answer: 101°

Answer: 64%



What is the sum of 43 dm and 2.5 m in centimeters?

What is the edge of a cube that has the same volume as a rectangular box with dimensions 18 in by 8 in by 1.5 in?

Answer: 680 cm

Answer: 6 inches

Algebra

What is the value of (16 1/3)(16 1/16)?

Algebra

(16 ½)

Answer: 4

Algebra

Find the numerical value of 36 3/2?

Answer: 216

Algebra

The product of two numbers is 60 and their reciprocal is 4/5, what are the numbers? Answer: 10 and 6

Algebra If Maria can address a box envelopes in 5 hours and Jane can address the same box of envelopes in 10 hours, how many minutes will it take Maria and Jane working together to address all envelopes in the box?

Answer: 200

If (3 100) (4 100)(5 100) = 2 2a 15 50b then a + b = ____.

Answer: 12

Algebra

Given: f(x) = - 2x 2 - 3x, f(-5)= ______.

Answer: - 35

Algebra

If the sum of two numbers is 5 and the difference of the two number is 5, what is the product of the numbers? Answer: 0

Algebra Mario sold five pigs and four goats for P21,000. Selling the animals at the same , Ben sold three pigs and 2 goats for P12,000. What is the selling price of each animal?

Answer: 3000 and 1500

Algebra

Algebra

Find the sum of the first 50 multiples of 4, 4, 8, 12, 16,…,

a50

Answer: 5100

Algebra

In the expansion of (x + the fourth term is:

Simplify the expression

log4 3 * log3 32

Answer: 5/2

Algebra

3y)7

Answer: 945x4y3

Algebra

If y = 3x and 2x + 3y = 22, then y = ____.

Answer: 6

Algebra

If f(x) = - 2x 2 - 3x, then f(-5) = ____.

Answer: - 35

Algebra At an amusement park, Leo bought 3 hamburgers and 4 sodas for a total of P150. While paying the same prices Kaye bought 2 hamburgers and 3 sodas for P105. What is the total cost of 1 hamburger and 1 soda?

Answer: P45

Algebra

If x > 0 and (22x - 1 )(4x + 2) =

If x > 0 and (22x - 1 )(4x + 2) =

(8x + 3)

, then x = _____.

(8x + 3) , then x = _____.

Answer: 3

Answer: 3

Basic Trigonometry

What is the value of sin 75°?

Trigonometry

Determine the length of the arc cut by a central angle of 60° in a circle with radius of 2 cm.

Answer: 2π/3

Answer:

Trigonometry

Trigonometry

An angle of one revolution is equal to an angle whose measure in terms of radians as:

A ladder 18 feet long leans against a building forming angle 60° with the ground. How high up the side of the building?

Answer: 2π

Answer:

Trigonometry From a point level with 1000 ft. away from the base of a monument, the angle of elevation to the top of the monument is 35°. Determine the height in the nearest feet.

Trigonometry If in ΔABC, sin A = 3/5, then what is the value of sin B?

Answer: 700 ft.

Answer: 4/5

Trigonometry In ΔABC, C s a right triangle and tan A = 1. What is the value of sin A + cos A?

Trigonometry

If the ratio of sec x to csc x is 1:4 then the ratio of tan x to cot x is

Answer:

Answer: 1:16

Trigonometry

Find the exact value of tan - 5π/3.

Answer:

Trigonometry

A ladder 30 ft. long leans against a wall, with its foot 8 ft. from the wall. Find the nearest degree which the ladder makes with the ground? Answer: 75°

Basic Analytic Geometry

Find the midpoint of the

segment joining A(-4,5) and B(2,-3).

Answer: (-1,1)


Find the slope of the line joining the points (3,2) and (-1,-2).

Answer: 1

Trigonometry

Find the nearest feet the height of the flagpole which cast a shadow 44.8 ft. long when the angle of elevation of the sun is 36°? Answer: 32 ft.


Find the distance between

A(3,2) and B(-1,4).

Answer:


The midpoint of a line segment M(1,2). One of the endpoints is A(0,3). Find the other endpoint B. Answer: (2,-7)


Find the equation of the line through point (1,-2) of slope 2.

Answer: 2x - y = -4

Basic Calculus

Basic Calculus

Find the derivative of

Differentiate

f(x) = 4x3

f(x) = 6x4 – 5x3 + x2 – 6x +7

Answer: 12x2

Answer: 24x3 – 15x2 + 2x – 6

Basic Calculus

Find the derivative of f(x) = (6x2 + 2x)(3x – 1)

Answer: 54x2 – 2

Basic Calculus

Differentiate y = (2x2 – 4x)2

Answer: y’ = 16x3 – 48x2 +32x

Basic Calculus

Find the derivative of y=

Answer:

Basic Calculus

Evaluate

Answer:

Basic Calculus

Evaluate

Answer:

Basic Calculus

Evaluate

Answer:


Philippine Normal University LICENSURE EXAMINATION FOR TEACHERS (LET) Refresher Course for Year 2015

WHAT TO EXPECT MAJORSHIP Area: Mathematics Focus: ARITHMETIC AND BUSINESS MATH LET Competencies: 1. Simplifying expressions involving series of operations 2. Solve problems involving a. GFC and LCMF b. prime and composite c. divisibility d. inverse and partitive proportions e. compound interest

PART I

– CONTENT

UPDATE

INTEGERS The set of integers (also called the signed numbers) is the union of the set of counting numbers { 1, 2, 3, 4, 5…}, – – – – – the set of their opposites { 1, 2, 3, 4, 5…} and zero {0}. It is usually denoted by Z. Thus, Z = {… –5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5…} The set of positive integers is {1, 2, 3, 4, 5…}. – – – – – The set of negative integers is {… 5, 4, 3, 2, 1}. The integer zero (0) is neither positive nor negative. OPERATIONS ON SIGNED INTEGERS Addition. 1. To add two integers with like signs, add their absolute values and affix their common sign. – – – – + + + + ( 5) + ( 9) = │ 5│ + │ 9│ ( 3) + ( 7) = │ 3│ + │ 7│ =5+9 =3+7 – = +14 = 10 2. To add two integers with unlike signs, find the difference of their absolute values and use the sign of the number with the larger absolute value. – – – – + + + + ( 15) + ( 9) = │ 15│ – │ 9│ ( 21) + ( 9) = │ 21│ – │ 9│ 9 = 15 – = 21 –9 – = +6 = 12 Subtraction. To subtract two integers, change the sign of the subtrahend and then proceed to the rules for addition. – – – – (+15) – ( 9) = │+15│ + │ +9│ ( 28) – (+23) = │ 28│ + │ 23│ = 15 + 9 = 28 + 23 – = +34 = 51

1

| Mathematics Major [1] Multiplication and Division 1. To multiply/divide two integers with like signs, multiply/divide the absolute values of the given integers and affix positive sign. + + + + (–28) x (–21) = │ –28│ x │–21│ ( 5) x ( 29) = │ 5│ x │ 29│ = 5 x 29 = 28 x 21 + + = 145 = 588 – – – – (+125) ÷ (+5) = │+125│ ÷ │+5│ ( 861) ÷ ( 123) = │ 861│ ÷ │ 123│ = 125 ÷ 5 = 861 ÷ 123 + + = 25 = 7 2. To multiply/divide two integers with unlike signs, multiply/divide the absolute values of the given integers and affix the negative sign. – – – – + + + + ( 32) x ( 14) = │ 32│ x │ 14│ ( 15) x ( 31) = │ 15│ x │ 31│ = 32 x 14 = 15 x 31 –

–

= 448 – = –465 – – ( 2706)  (+11) = │ 2706│  │+11│ (+2184)  ( 12) = /+2184/  / 12/ = 2706 ÷ 11 = 2184 ÷ 12 – – = 246 = 182 Odd and Even Numbers Any integer that ends with 0, 2, 4, 6 or 8 is called an even number, and any integer that ends with 1, 3, 5, 7 or 9 is called an odd number. Example: 4, 20, 36, 772, 1958 are even numbers. 11, 223, 805, 907, 8009 are odd numbers. Three important facts about even and odd numbers a) If two integers are both even or both odd, their sum or difference is even. b) If one integer is even and the other is odd, their sum or difference is odd. c) The product of two integers is even unless both of them are odd.

+/Even – Even Odd

Odd Even Odd

x

Even Even Odd

Odd Even

Odd Even Even

Even Odd

PRIME AND COMPOSITE NUMBERS Any positive integer with exactly two distinct positive factors in the set of integers is called a prime number. Any positive integer that has more than two distinct positive factors in the set of integers is called composite number.

Example: Which of the following numbers are prime, composite, or neither? a) 53 prime b) 421 prime c) 24,638 composite d) 43,101 composite e) 1 neither

Fundamental Theorem of Arithmetic Every composite whole numbers can be expressed as the product of primes in exactly one way (the order of the factors is disregarded). Example: Find the prime factorization of 300. 300 5

60 10 5

6 2

2

3 2

2

the prime factors of 300: 5 x 5 x 2 x 2 x 3 = 5 x 2 x 3

2

| Mathematics Major [1] Factors and Multiples If a and b are whole numbers and a  0, then a is a factor of b if and only if there is a whole number c such that ac = b. If a is a factor of b, we can also say that a divides b (a b), or b is a multiple of a. Every number has a finite set of factors (or divisors) and an infinite set of multiples.

Examples: Classify each of the following as true or false. a) –3 is a factor of 12. true b) 03 c) 30 d) 2 is a multiple of 8. e) for all integers a, 1a.

false true false true

GREATEST COMMON FACTOR Let a and b be natural numbers. The greatest natural number d that divides both a and b is called their greatest common factor (divisor), and we write d = (a, b). Example: Find the greatest common factor. (180, 220) 180 = 2 x 2 x 3 x 3 x 5 220 = 2 x 2 x 5 x 11

2x2

x5

So, (180, 220) = 2 x 2 x 5 = 20

LEAST COMMON MULTIPLE Let a and b be natural numbers. The least natural number m that is a multiple of both a and b is called their least common multiple and we write m = [a, b]. Example: Find the least common multiple. [28, 44] 28 = 2 x 2 x 7 44 = 2 x 2 x 11

2 x 2 x 7 x 11 So, [28, 44] = 2 x 2 x 7 x 11 = 308 Remember: For positive integers a and b,

[a, b] =

ab (a, b)

and when (a, b) = 1, then [a, b] = a x b

DIVISIBILITY RULES Divisibility Properties a) If a number divides each of two other numbers, then it divides their sum. If ab and ac, then a(b + c). b) If a number divides one of two numbers but not the other, then it will not divide their sum. If ab and a ł c, then a ł (b + c) . c) If one number divides another number, then it will divide the product of that number with any other whole number. If ab, then abk. Divisibility A number is divisible by 2 if the number ends with an even number (0,2,4,6,8). Example: 158 and 5792 are divisible by 2 since the numbers end with 8 and 2 respectively which are both even numbers. A number is divisible by 3 if the sum of its digits is divisible by 3. Example: 2301 is divisible by 3 since the sum of its digits 2+3+0+1 = 6 is divisible by 3. A number is divisible by 4 if the last two digits of the number form a number which is divisible by 4. Example: 2900, 3136, and 745084 are divisible by 4. A number is divisible by 5 if the number ends with 0 or 5. Example: 1015, 2890, and 802525 are divisible by 5.

3


A number is divisible by 6 if the number is divisible by both 2 and 3. Example: 1350 and 201564 are divisible by 6 because they are both divisible by 2 and by 3. A number is divisible by 7 if the difference between twice the unit digit and the number formed by the remaining digits is divisible by 7. Example: 2191 is divisible by 7, since 2191  219 – 2 (twice 1) 217  21 –14 (twice 7) 7



divisible by 7

A number is divisible by 8 if the last three digits of the number form a number which is divisible by 8. Example: 413000, 6739048, and 9013816 are divisible by 8. A number is divisible by 9 if the sum of its digits is divisible by 9. Example: 19008 and 324594 are divisible by 9, since 1+9+0+0+8 = 18 and 3+2+4+5+9+4 = 27 are divisible by 9 A number is divisible by 10 if the number ends with 0. Example: 47020 and 390580 are both divisible by 10. A number is divisible by 11 if the difference between the sum of the digits in the even position and the sum of the digits in the odd position is divisible by 11. Example: 7654235050 is divisible by 11 since, (7+5+2+5+5) – (6+4+3+0+0) = 24 – 13 = 11 is divisible by 11 even position odd position A number is divisible by 12 if the number is divisible by both 3 and 4. Example: 16178904 is divisible by 12 because it is divisible by 3 (since 1+6+1+7+8+9+0+4 = 36 is divisible by 3) and by 4 (since the last 2 digits 04 form a number which is divisible by 3). A number is divisible by 13 if the sum of four times the unit digits and the number formed by the remaining digits is divisible be 13. Example: 195 is divisible by 13, since 195  19 + 20 (four times 5) 39



is divisible by 13.

RATIO AND PROPORTION A ratio is a comparison of 2 numbers a and b, with b ≠ 0, and may be expressed in the form “a to b”, “a : b” or “a/b”.

Ratio compares two quantities with same units.

“12 Math books to 15 English books” is a ratio “10 kilometers per liter” is a rate. Example: In 1993, for every woman arrested in the United States, 4 men were arrested. 1. What is the ratio of the number of men arrested to the number of women arrested? Ans.

4 1

or 4:1

2. What is the ratio of the number of women arrested to the number of men arrested? Ans.

1 4

or 1:4

For any two equal ratios a/b and c/d, a/b = c/d is called a proportion. This is also written as a:b = c:d. In this form, a and d are called extremes and b and c are called the means, and the rule states that “the product of the m eans equals the product of the extremes.” This can also be written as; if a : b = c : d, then ad = bc.

4


Example: If the ratio of teachers to students in a school is 1 to 18 and there are 360 students, how many teachers are there? Let x be the number of teachers,

x

1 18



360

or 1 : 18 = x : 360

18x = 360 x = 20 teachers PARTITIVE PROPORTION If a quantity q is to be partitioned into p1, p2, p3, . . . p n, so that the partitions are in the ratio a1 : a2 : a3 : . . . : an, then the size of the kth partition may be computed as follows:

q Pk = a1



a2



a3

 ... 

an



ak

Example: Joshua divides his day into leisure, sleep, and work. In the ratio 1 : 2 : 3. How many hours does he spend working? 24

P3 =

1 2  3

3

= 12 hours.

DIRECT PROPORTION

If the ratio of two quantities being compared is constant, then they are directly proportional. y1 = kx1 and y2 = kx2, then

y1 x1



k and

y2 x2



k , therefore

y1 x1

y2 

x2

Example: Junior paid 125php for 14 chocolate candies. How much would 25 of such chocolate candies cost? 125 

14

x 25

→ 14x = 125(25) = 223.21php

INVERSE PROPORTION If the product of two quantities being compared is constant, then they are inversely proportional

x1y1 = k and x2y2 = k, then x1y1 = x2y2 or

x1 y2



x2 y1

Example: It takes 20 men to build a house for 60 days. How many men will be needed to build it in 15 days?

x

20 15



60

→ 15x = 20(60) = 1 200 → x = 80 men

COMPUTING INTEREST Simple Interest ( Simple Interest = Prt ) Simple interest is an interest computed on the srcinal principal. The srcinal amount deposited or borrowed is called the principal. The percent used to determine the interest is called the interest rate. Interest rates are given for specific periods of time such as years, months or days. Example: Christian opens a savings account that pays simple interest at the rate of 5¼% per year. If he deposits 2 000php and makes no other deposits, find the interest and the final amount for 90 days.

I = (2 000php) ● (5¼%) ● (90/365) = (2 000php) ● 0.0525) ● (90/365) = 25.89php Final Amount = Principal + I = P(1 + rt) = 2 000php + 25.89php = 2 025.89php n Compound Interest ( Final Amount = P[ 1 + r ] ) Compound interest is different from simple interest because after the first interest calculation, the interest is added to the principal, so interest is earned on previous interest in addition to the principal. Compound Interest rates are usually given as annual (1 time a year), semiannual (2 times a year), quarterly (4 times a year), monthly (12 times a year), and daily (365 times a year). Example: If 500php is invested at 8% compounded semiannually, what will the final amount be after three years? Final Amount = P[ 1 + r ]n = 500[ 1 + (8% / 2)] 3 * 2 = 500[ 1 + 0.04 ] 6 = 500[1.27] = 635

5



PLANE AND SOLID GEOMETRY –

PART I

I.

CONTENT UPDATE

Basic Ideas The undefined terms, point, line, and plane are geometric ideas and they are visually represented by a tiny dot, a thin wire, and a smooth flat surface, respectively. Points are labeled by means of capital letters, lines by naming any two of its points, and planes by naming at least three of its points. The subsets of a line are ray, segment, and the line itself.

A . .

B. Line

AB

A.

B.

A.

AB

B. Line

Ray

AB

Space is the set of all points.

Some postulates on points, lines and planes:  An infinite number of lines may pass through a given point.  To every pair of different points there corresponds a unique positive number.  This number is called the distance between the two points.     

For every two different points there is exactly one line that contains both points. If two points of a line lie in a plane, then the line lies in the same plane. Any three points lie in at least one plane, and any three non-collinear points lie in exactly one plane. If two planes intersect, then their intersection is a line. An infinite number of planes may pass through a given line.

Further:  Every segment has exactly one midpoint.  If a line intersects a plane not containing it, then the intersection is a point.  Given a line and a point not on the line, there is exactly one plane containing both.  Given two intersecting lines, there is exactly one plane containing both. II.

ANGLES If two rays have a common endpoint, but do not lie on the same line, then their union is an angle. Their common end point is called its vertex and the two rays are called its sides. The following are angles:

A point may be on the angle, in the interior or neither on the angle nor in its interior called the exterior. P. P. P.

The unit of measure for an angle is called a degree. An angle is measured with a protractor. Angle Measurement Postulate. To every angle there corresponds a real number between 0 and 180.

1

| Mathematics Major [2] Kinds of Angles 1. Acute Angle. An acute is an angle whose measure is less than 90. 2. Right Angle. A right angle is one that measures 90. It is usually represented by a small the vertex. Two lines that intersect and form right angles are called perpendicular (  ) 3. Obtuse Angle. This is an angle whose measure is more than 90 but less than 180.

square at lines.

Remarks:In plane Geometry, an angle is simply a set of points. This should be distinguished from the angles in trigonometry where we speak of directed angles. When we use directed angles, we a llow “zero angles” and “straight angles”. In the study of directed angles, we seldom use the degree as a unit measure. Instead, we use the radian. FACTS ABOUT PAIRS OF ANGLES 1. Vertical Angles. Two angles are vertical angles if and only if their sides form two pairs of opposite rays. When two lines intersect, they form two pairs of vertical angles. 2. Complementary Angles . Two angles whose measures total 90 . 3. Supplementary Angles. Two angles whose measures total 180 . C 4. Linear Pair.Two angles that are formed by two opposite rays and a third common ray. Two angles,  BAC and  CAD, form a linear pair if only if B, A, and D

.

.

are collinear and C is not on BA . A linear pair is a supplementary pair.

A

B

.

D

An angle bisector is a ray in the interior of the angle dividing it into two congruent parts. III. EQUIVALENCE RELATIONS Equality between two numbers has the following properties  Reflexive Property: a  a , for every a 

Symmetric Property: If



Transitive property: If

a  b , then b  a . a  b and b  c , then a  c .

IV. CONGRUENCE RELATIONS Congruence Properties between Angles (or between segments): 

Reflexive Property: A  A for every A .



Symmetric property: If A  B , then B  A . Transitive Property: If A  B and B  C , then A  C .



AB  CD implies that AB  CD and vice versa. Similarly, A  B implies that

mA  mB

and vice versa.

Equality is used for measures while congruence is used for figures. Some Theorems on Angles  If the angles in a linear pair are congruent, then each of them is a right angle.  If two angles are complementary, then they are both acute.  Any two right angles are congruent.  If two angles are both congruent and supplementary, then each is a right angle.  Supplements of congruent angles are congruent.  Complements of congruent angles are congruent.  Vertical angles are congruent.  Two perpendicular lines form four right angles.  The angles in a linear pair are supplementary.  The shortest segment from a point not on a line to the line is the perpendicular () segment. Examples: 1. Estimate the measure of an angle if it is three times the measure of its supplement. Solution: x = 3(180 – x) 4x = 3(180) x = 135

2. On the edge of a half-plane, take points, M, K, A such that A is between M and K. Take ray

AT

so that m TAK = 35. In the

same half-plane take ray AV such that m MAV = 85. What is the measure of TAV? ( Ans. 60) 3. If mA = 64, find the measures of the angles that are complementary and supplementary to A. Solutions: The measures of two complementary angles must add to 90, so the measure of the complement of A is given by 90 - 64 = 26. The measures of two supplementary angles must add to 180, so the measure of the supplement of A is given by 180 - 64 = 116. 4. In the figure, 1 is complementary to2, and m3 = 141. Find the measure of each numbered angle in the figure.

2


Solutions:  Since 3 and m6 are vertical angles, then these angles are congruent. Thus, if m3 = 141, therefore, m6 = 141 .     

5.

1

and 3 form linear pair and therefore they are supplementary. Thus, m1 = 39. 1 and 5 are vertical angles and vertical angles are congruent. Thus, m5 = 39. Since 1 is complementary to 2, then m2 = 51. Since 2 and 8 are vertical angles, then m 8 = 51. Now, m4 = m7 = 129. Why? ____________________________________

In the figure below, the measures of angles 1, 2, and 3 are in the ratio 1:2:3, respectively. Find the measure of each angle. Solution: Adding the terms of the ratio 1,2, and 3 gives 6.

12

1/6 of 90 = 15; 2/6 of 90 = 30; and 3/6 of 90 = 45 Thus, the measures of the angles are 15, 30 and 45.

6.

3

Find the measure of an angle whose measure is 40 more than the measure of its supplement.

Let x = the measure of the supplement of the angle x + 40 = the measure of the angle x + (x + 40) = 140 2x = 140 x = 70 and x + 40 = 100 Thus, the measure of the angle is 110.

H

C

G

F

IV. PARALLEL LINES Facts about parallel lines: 1. Parallel lines are coplanar lines that do not intersect 2. Skew lines are noncoplanar and nonintersecting lines. Examples:

a.)

B

A

Solution:

D

E

In the parallelepiped above, name all the lines that contain the edges parallel to AB .

b.) Name all the lines containing the edges that are skew to EF . Solutions: a.) The lines parallel to AB are FG , HC and ED b.) The lines that are skew to EF are AB , HC , BG and CD . 3. A transversal is a line that intersects two coplanar lines at two different points. Angles formed by Transversals a.) Alternate Interior Angles

4 8

b.) Alternate Exterior Angles

1 2 3

3 5

5 6 7

4

and 6; 3 and 5 are alternate interior angles

c.) Same-side Interior angles

1 and 7; 2 and 8 are alternate exterior angles

d.) Corresponding angles 3

1 2

1 2


4 and 5; 3

and 6 are angles on the same side of the transversal

1 and 5; 2 and 6; 4 and 8; 3 and 7 are corresponding angles

PRINCIPLES ON PARALLEL LINES Principle 1: Parallel – lines postulates: Through a given point P, not on a line l, exactly one

P

.

line maybe drawn parallel to line l.

l

Angles Relationships for Parallel Lines that are C ut by a Transversal

Principle 2: If parallel lines are cut by a transversal, then the alternate interior angles are congruent. Principle 3: If parallel lines are cut buy a transversal, then the corresponding angles are congruent. Principle 4: If parallel lines are cut by a transversal, then the alternate exterior angles are congruent. Principle 5: If parallel lines are cut by a transversal, then the same-side interior angles are supplementary. Principles on Proving Lines to be Parallel

Principle 6: If two lines are cut by a transversal so that a pair of alternate interior angles are congruent, then the lines are parallel. Principle 7: If two lines are cut by a transversal so that a pair of corresponding angles are congruent, then the lines are parallel. Principle 8: If two lines are cut by a transversal so that a pair of alternate exterior angles are congruent, then two lines are parallel. Principle 9: If two lines are cut by a transversal so that a pair of same-side interior angles are supplementary then two lines are parallel. Principle 10: Lines are parallel if they are parallel to the same line. Examples: A. Use the given information to decide which lines are parallel. Justify your answers with a principle for parallel lines.

1. 2. 3. 4. 5. 6.

6  9

1 2 4 3

4  8

m3 + m8 = 180 m5 + m7 = 180  10  11  2   10

a

12 5

10

Solutions:

9 8 d

6 7 11

b c

e

1. d ║e by Congruent Corresponding Angles (Principle 7) 2. a ║ c by Alternate Interior Angles Postulate (Principle 6) 3. a║c by Same-Side Interior Angles are Supplementary (Principle 9) 4. b ║c by Same-Side Interior Angles are Supplementary (Principle 9) 5. d ║e by Congruent Corresponding Angles (Principle 7) 6. a ║c by Congruent Alternate Exterior Angle (Principle 8)

B. In the figure at the right, find the value of x given s║t cut by a transversal l 4

| Mathematics Major [2] 1. m 2 = 2x, m3 = 4x

l

2. m1 = 2x, m6 = 136

1 2

3. m1 = 3x, m5 = 60 4.

s

3 5 4 6

m1 = 6x, m3 = 120

t

Solutions:

1. Since s║t, then 2 is supplementary to 3, (by Principle 5) Thus, 2x + 4x = 180 6x = 180 x = 30 2. Since s║t, then 1  6 (Principle 4) Thus, 2x = 136 x = 68 3. Since s║t, then 2  5 (Principle 2) Thus, 2 = 60 But 1and 2 are supplementary (Def. of linear pair) Therefore, from the given 1 = 3x and 2 = 60 Then 3x + 60 = 180 3x = 120 x = 40 10. Since s║t, then 1  3 (Principle 3) Thus, 6x = 120 x = 20 V.

TRIANGLES

A triangle is the union of the segments determined by three non-collinear points. Triangles may be classified according to the congruence or noncongruence of their sides. 1. 2. 3.

Scalene triangle. A scalene triangle is a triangle having no congruent sides. Isosceles triangle. An isosceles triangle is a triangle having at least two congruent sides Equilateral triangle. An equilateral triangle is a triangle having three congruent sides.

Scalene

Isosceles

Equilateral

Triangles may also be classified according to the kinds of angles they have. 1. Right triangle. A triangle having one right angle. 2. 3.

Obtuse triangle. A triangle having one obtuse angle. Acute triangle. A triangle having three acute angles.

Right

Acute

Obtuse

Some Theorems about Triangles

1. 2. 3.

Isosceles Triangle Theorem. If two sides of a triangle are congruent, then the angles opposite these sides are congruent. Conversely, if two angles of a triangle are congruent, then the sides opposite them are congruent Every equilateral triangle is equiangular and conversely. The sum of the measures of the angles of a triangle is 180.

Special Lines in a Triangle 5


Angle bisector of a triangle . An angle bisector of a triangle is a segment or ray that bisects an angle and extends to the opposite side.

2. Median of a triangle . A median of a triangle is a segment from a vertex to the midpoint of the opposite side. 3. Perpendicular bisector of a side . A perpendicular bisector of a side of a triangle is a line and is perpendicular to a side.

that bisects

4. Altitude to a side of a triangle . An altitude of a triangle is segment from a vertex perpendicular to the opposite side. 5. Altitude of obtuse triangle . In an obtuse triangle, the two altitudes fall outside the 6. The altitudes to the congruent sides of an isosceles triangle are congruent. 7. The altitudes of an equilateral triangle are congruent.

triangle.

Theorems on Right Triangles 1. In a 30-60-90 triangle, a. the hypotenuse is twice as long as the shorter leg (the leg opposite the 30 angle), and

b.

the longer leg is

60

2a

a

3 times as long as the shorter leg.

30 a 3

2.

In a 45-45-90 triangle, the hypotenuse is leg.

2 times as long as either

45

a 2

a

45

a 3. Pythagorean Theorem In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

c

a b

a2 + b2 = c2 4. Angles Outside the Triangle The measure of an exterior angle of a triangle is equal to the sum of the remote interior angles.

2

1

4

3

m4 = m1 + m2 5. Exterior Angle Inequality Theorem The measure of the exterior angle of a triangle is greater than the measure of either remote interior angle.

2

1

4

3 m4 > m1 m4 > m2

6. Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

c

a

b a+b>c

Examples:

6



b ____

c ____

b. ____ ____

6 2

a.

3

c

a

b a=b

Solutions:

By Theorem 1, a) b = 3 and c = 3 2 2.

b) a = 6 and b = 6


b

a. ____

4 3

____

b. ____

____

10

60

c

a

c

30 b

Solutions:

By Theorem 2, a) a = 4 and c = 8 3.

b) a = 5 and b = 5 3 .

A pole is braced up by wires tied to its top portion from pegs on the ground each 6 ft from the foot of the pole. If a wire is 16 ft long, how tall is the pole? Solution:

c2

a2  b2 2 2 16  6  b 2 2 2 b  16  6 b 2  256  36 b 2  220  14.83 

2

4.

Find the measures of the labeled angles in the marked figure at the right. Solutions:

mx  60; my  120; mz  30; mw  30

TRIANGLE CONGRUENCE POSTULATES

Given two triangles. There are four ways to show that they are congruent using only three pairs of corresponding congruent parts: SAS Congruence Postulate . If two sides and the included angle of one triangle are congruent respectively to the

corresponding two sides and the included angle of another triangle, then the two triangles are congruent. ASA Congruence Postulate . If two angles and the included side of one triangle are congruent respectively to the corresponding two angles and the included side of another triangle, then the two triangles are congruent. SSS Congruence Postulate . If the three sides of one triangle are congruent respectively to the corresponding three sides of another triangle, then the two triangles are congruent. SAA Congruence Theorem. . If a side and two angles adjacent angles of one triangle are congruent respectively to the corresponding side and two adjacent angles of another triangle, then the two triangles are congruent. Corresponding Parts Principle. If two triangles are congruent by SAS, ASA, SSS, or SAA, then their remaining corresponding parts are also congruent

Examples:

7

| Mathematics Major [2] Each pair of marked triangles are congruent by the indicated congruence postulate.

60

60 32

32 SAS

ASA

65 95

95 65 SAA

SSS

Triangle Congruence for Right Triangles

From the triangle congruence postulates, any two right triangles may be congruent by any of the following principles: LL Congruence.Two right triangles are congruent if the two legs of one are congruent, respectively, to the corresponding two legs of the other. (By SAS) LA Congruence.Two right triangles are congruent if a leg and an adjacent acute angle of one are congruent, respectively, to the corresponding leg and an adjacent acute angle of the other. (By ASA; by SAA if the acute angles are not adjacent) HL Congruence.Two right triangles are congruent if the hypotenuse and a leg of one are congruent, respectively, to the corresponding hypotenuse and a leg of the other. (By Transitivity)

40 40

First Minimum Theorem

The shortest segment joining a point to a line is the perpendicular segment. Thus, the distance between a line and an external point is the length of the perpendicular segment from the point to the line. Similar Triangles

Two triangles are said to be similar if a. their corresponding angles are congruent, and b. their corresponding sides are proportional. Examples of similar triangles:

Similarity Postulates: 1. AA Similarity.If two angles of one triangle are congruent to two corresponding angles of another triangle, then the triangles are similar. 2. SAS Similarity.If an angle of one triangle is congruent to a corresponding angle of another triangle and the sides that include these angles are proportional, then the triangles are similar. 3. SSS Similarity.If all the three sides of one triangle are proportional to the lengths of the corresponding sides of another triangle, then the triangles are similar.

8


Midsegment Theorem for Triangles.

A

A segment whose endpoints are the midpoints of two sides of a triangle is

D

parallel to the third side, and half the length of the third side.

a. b.

E

B

C and DE

DE // BC

5.

Side-Splitting Theorem If a line parallel to a side of a triangle intersects the other two sides in distinct points, then it cuts off segments which are proportional to these sides.

1 

A

D

E

B

C AB AD

6. 7.

BC

2

AC 

AE

Similarity in a right triangle. The altitude to the hypotenuse of a right triangle forms two triangles that are each similar to the srcinal triangle and to each other. Given a right triangle and the altitude to the hypotenuse. (a) The altitude to the hypotenuse is the geometric mean of the segments into which it separates the hypotenuse. (b) Each leg is the geometric mean of the hypotenuse and the segment adjacent to the leg.

C

Thus, in the marked adjoining figure, ABC  ACD  CBD. Thus,

AD

CD



CD Hence,

2

CD

Hence,

B

D 

AD

Moreover,

A

BD

AD  BD



AC

AC AB

Finally,

BD

.

BC

AC2  AD DC.

And



BC BA

BC2  BD BA

Examples:

1. Two angles of ABC have measures, 45 and 15, while two angles of DEF have measures 120 and 45. Are the triangles similar? By what Similarity theorem or definition? Solution: YES, by the AAA Similarity Theorem_ 2. One angle of PQR measures 40 and the sides that include the angle measures 5 each. Another triangle has an angle that measures 70 and the sides that include these angle measures 8 each. Are the triangles similar?

Solution: YES, by the SAS Similarity Theorem 3. Given the figure as marked. Find PS. Solution:

S x

Q x

15 P

9 R

15 PS

12 T

9 

12

PS  (15)(12) 9  20

C

4. In the marked figure at the right, AD = 8 and DB = 18. Find CD. Solution:

(CD)2



(CD)2

 8  18

CD = 12

AD  BD

A

B

D 9

| Mathematics Major [2] QUADRILATERALS

A quadrilateral is a four-sided figure.

   

If all four angles of a quadrilateral are right angles, then the quadrilateral is a rectangle. If all four angles of a quadrilateral are right angles, and all four sides are congruent, then the quadrilateral is a square. If both pairs of opposite sides are parallel, then the quadrilateral is a parallelogram. If one and only one pair of opposite sides are parallel, then the quadrilateral is a trapezoid.

THEOREMS ON QUADRILATERALS

1. 2.

Each diagonal separates a parallelogram into two congruent triangles. In a parallelogram, any two opposite sides are congruent. Corollary: If two lines are parallel, then all points of each line are equidistant from the other line. Recall: The distance between a line and an external point is the length of the perpendicular segment from the point to the line.

3.

The distance between any two parallel lines is the distance from any point of one to the other. In a parallelogram, any two opposite angles are congruent.

4. 5.

In a parallelogram, any two consecutive angles are supplementary. The diagonals of a parallelogram bisect each other.

Application: 6. The segment joining the midpoints of two sides of a triangle is parallel to the third side and half as long. 7. A rhombus is a parallelogram all of whose sides are congruent. 8. A rectangle is a parallelogram all of whose angles are congruent. 9. A square is a rectangle all of whose sides are congruent. 10. If a parallelogram has one right angle, then it is a rectangle. 11. In a rhombus, the diagonals are perpendicular to each other. 12. If the diagonals of a quadrilateral bisect each other and are perpendicular, then quadrilateral is a rhombus.

the

CIRCLES Definition Let P be a point in a given plane, and r be a positive number. The circle with center P and radius is the set of all points of the plane whose distance from P is r. Definition Let P be a point, and let r be a positive number. The sphere with center P and radius r is the set of all points of space whose distance from P is r.

Basic Terms on Circles and Spheres   

  

Two or more spheres or two or more circles with t he same center are called concentric. A chord of a circle is a segment whose endpoints lie on the circle. A line which intersects a circle in two points is called a secant of the

circle.

A chord of a sphere is a segment whose endpoints lie on the A diameter of a circle or sphere is a chord containing the center.

sphere.

A radius of a circle or a sphere is a segment from the center to a the sphere.

point of

10


The interior of a circle is the set of all points of the plane whose distance from the center is less than the radius. The exterior of a circle is the set of all points of the plane whose distance from the center is greater than the radius.

 

Definition

A tangent to a circle is a line (in the same plane) which intersects the circle in one and only one point. This point is called the point of tangency. Theorems Circles and Spheres 1. The intersection of a sphere with a plane through its center is a circle with the same center and the same radius. 2. The intersection of a sphere with a plane through its center is called a great circle of the sphere. 3. A line perpendicular to a radius at its outer end is tangent to the circle.

4. Every tangent to a circle is perpendicular to the radius drawn to the point of tangency. 5. The perpendicular from the center of a circle to a chord bisects the chord. 6. The segment from the center of a circle to the midpoint of a chord which is not a diameter is perpendicular to the chord. 7. In the plane of a circle, the perpendicular bisector of a chord passes through the center. 8. In the same circle or in congruent circles, chords equidistant from the center are congruent. 9. In the same circle or in congruent circles, any two congruent chords are equidistant from the center. 10. If the line and the circle are coplanar, and line intersects the interior of the circle, then it intersects the circle in two and only two points. Definition

Two circles are tangent if they are tangent to the same line at the same point. If two tangent circles are coplanar, and their centers are on the same side of their common tangent, then their internally tangent. If two tangent circles if two tangents are coplanar, and their canters are on opposite sides of their common tangent, then their externally tangent

ARCS OF CIRCLES

In the adjoining circle at the right, P is the center. The set of points (darkened) on the circle in the interior of APB is the minor arc AB . The remaining set of points on the circle is the major arc AB . A and B are the endpoints

of the arcs. Definition

A central angle of a circle is an angle whose vertex is the center of the circle.

In the figure,

APB is a central angle.

Definition

Let C be a is the AB

circle, and let A and B be the endpoints of a diameter. A semicircle union of A, B, and the points of C that lie in a given half-plane with

as edge. The points A and B are the end points of the semicircle.

11

| Mathematics Major [2] Definition

1. 2.

The degree measure of an arc is the measure of the corresponding central angle. The degree measure of a semicircle is 180.

INSCRIBED ANGLES AND INTERCEPTED ARCS Definition

An angle is inscribed in an arc if 1. The sides of the angle contain the end points of the arc and 2. The vertex of the angle is a point, but not an end point, of the arc.

Definition

An angle intercepts an arc if 1. The end points of the arc lie on the angle, 2. All other points of the arc are in the interior of the angle, and 3. Each side of the angle contains an end point of the arc.

Theorems on Inscribed Angles and Intercepted Arcs

1. The measure of an inscribed angle is half the measure of the intercepted arc. 2. An angle inscribed in a semicircle is a right angle. 3. Every two angles inscribed in the same arc are congruent. Definitions A triangle is inscribed in a circle if the vertices of the triangle lie on the circle. If each side of the triangle is tangent to the circle, then the quadrilateral is circumscribed about the circle.

SOLIDS AND THEIR VOLUMES

12


Theorems

1. 2. 3. 4.

The volume of the prism is the product of the altitude and the area of the base. The volume of a triangular pyramid is one-third the product of its altitude and its base area. The volume of a pyramid is one-third the product of its altitude and its base area. The volume of a circular cylinder is the product of its and the area of its base. The volume of a circular cone is one-third the of its altitude and the area of its base.

5.

altitude product

TRANSFORMATIONS Starting point

These triangles are congruent.

If you cut out ΔDEF and place it onto ΔABC, they should have the same size and shape. The movement may be a slide, a flip, or a turn.

Look at the kite at the right. If we fold the kite over BD , there is to-one correspondence between the points of the kite. A  C, points along

a oneAB

and CB correspond, and points along AD and CD correspond.

Transformation

A transformation is a one-to-one correspondence between points in the plane such that each point P is associated with a '

unique point P , called the imageof P . Transformations that preserve the size and shape of geometric figures are called isometries(iso means “same” and metry means “measure”) orrigid motions . Types of isometries: translation, rotation and reflection. Translation Translation is a transformation that acts like a “slide”. 

Example:

Describe a transformation that will move ABC to

' ' ' ABC



coincide with Solution:

Slide the triangle so that A moves to A’. Since B’ and C’

are

the same distance and direction from B and C, respectively, as A’ is from point A, point B’ is the image of B and point C’ is the image

' ' '

of C. Thus, ABCmoves to A B C . Trace ABC and slide it using the arrow from A to A’. To define translation , we need the concept of directed line segment. Informally, a line segment AB can be directed in two ways: (1) pointing from A to B denoted as AB or (2) pointing from B to A denoted as BA . Further, two line segments are said to be equivalentif they are parallel, have the same length and point in the same direction.

13

| Mathematics Major [2] Definition Translation

Suppose that A and B are points in the plane. The translationassociated with directed line segment AB , denoted TAB , is '


the transformation that maps each point P to the point P such that

Directed segment


AB so that

AB .

'

PP ' ║ AB and P P = AB. Thus quadrilateral

'

P P BA is a parallelogram, since it has a pair of opposite sides that are parallel and congruent. We can imagine that P is “slid” by the translation TAB in the direction from A to B for a distance equal to AB. 

Rotation

This is an isometry that corresponds to turning the plane around a fixed point. Example '

'

'

Describe a transformation that will move ABC to coincide with A B C .

Solution: '

'

'

'

We can turn ABC 180º around point P, the midpoint of segment BB to coincide with A B C .

To define rotation, we need the concept of a directed angle. An angle ABC is said to be a directed angle if it satisfies the following properties:

mABC  0 , then the measure of the directed angle is 0º.

1.

If

2. 3.

If ABC is a straight angle, then the measure of the directed angle is 180º. In the adjoining figure, a)

Let BA be turned about B through the smallest possible angle so that the ray BA coincides with

b)

image of

BC .

If the direction of the turn is counterclockwise, the measure of the directed angle is the positive number mABC. If the direction is clockwise, the measure is the negative number  mABC. The directed angle ABC is denoted by ABC.

For the directed angle by ABC, ray BA is called the initial side and ray BC is called the terminal side. Definition Rotation

The rotationwith center O and angle with measure a, denoted RO,a , is the '

transformation that maps each point P other than O to the point P such that '

1.

The measure of directed angle POP is a, and

2.

OP  OP'

Point O is mapped to itself by RO,a . 14

| Mathematics Major [2] 

Reflection This isometry corresponds to flipping the plane over a fixed line.

Example: '

'

'

Describe a transformation that will move ABC to coincide with A B C

Solution: '

Flip ABCover the perpendicular bisector of AA .

'

'

'

'

'

'

Then point A moves to point A , point B to B , and C to C . Hence, ABC moves to coincide with A B C 

Reflection

A transformation that “flips” the plane over a fixed line is called a reflection. Definition

Suppose that a line l is a line in the plane. The reflection in line l, denoted by

Mt ,

is the

transformation that maps points as follows: '

1.

Each point P not on line l is mapped to the point P such that l is the

2.

perpendicular bisector of segment PP . Each point Q on line l is mapped to itself.

'

Examples of transformations

15



WHAT TO EXPECT MAJORSHIP: MATHEMATICS FOCUS: Basic Algebra LET COMPETENCIES: 1. Perform operations on Algebraic Expressions 2. Simplify a given algebraic expression with series of operations 3. 4. 5. 6. 7. 8. 9. 10.

Apply the Laws of Exponents in Multiplying and Dividing Algebraic Expressions Factor polynomials Use factoring in simplifying rational expressions Perform operations on Rational Expressions Perform operations on Radical Numbers Identify the domain and/or the range of a given function Identify/ describe the graph of a function Solve problems on a) Linear equations b) Systems of linear equations 11. Compute the value of a function f(n), where n is a counting number

PART I – CONTENT UPDATE

I. Algebraic Expressions A mathematical phrase that contains a variable is an open phrase. A number phrase is an expression that does not contain a variable. It is also referred to as a numerical expression. The English phrase ‘a certain number added to 5’ may be translated to the open phrase ‘n + 5’ where n stands for a certain number. The English phrase ‘seven added to 5’ may be translated to the number phrase ‘7 + 5’. Expressions like 8 + 2, 12 – 2, 5 x 2, and 20 ÷ 2 are some number phrases for the number 10. Expressions like

x

5 , - 2a, 2n + 8, 2(l + w) are examples of open phrases. Another name for open phrase is algebraic expression. An expression composed of constants, variables, grouping symbols, and operation symbols, is called an algebraic expression. It is the result of adding, subtracting, multiplying, dividing (except by 0), or taking roots on any combination of constants and variables. 2 x3 1 1 1  4x2  2x  5 Example a) 2 x 2  3x b. 2 x  x  c) d)  2 3 x x A. Polynomials A term of a polynomial is a constant, a variable or the product of a constant and one or more variables raised to whole number exponents. The constant preceding the variable in each term is called coefficient of 3 2 3 2 the variable. In 3x  15x  2 , the coefficient of x is 3, and the coefficient of x is –15. In algebra, a number is 3 2 constant 3 x  15 x  2 frequently referred to as a , and so the last term –2 in is called the constant term. A polynomial is a term or a finite sum of terms, with only non-negative integer exponents permitted on the variables. If the terms of a polynomial contain only the variable x, then the polynomial is called polynomial in x.


1


Examples: 2

a)  8r 2 s 3 d) 8 x  3 y

2

b) x – 2xy – 3y

3

e)  4r s  2 2

3

c)

1

2 x3 3

 4x2  2x  5

f) x  8 x  11x  5 4

2

B. Degree of a Polynomial The degree of a polynomial in one variable is the highest exponent of the variable in the polynomial. A term containing more than one variable has degree equal to the sum of all the exponents appearing on the variables in the term. The degree of a polynomial in more than one variable is the highest degree of all the terms appearing in the polynomial. If a monomial consists of a constant term then the expression is called constant polynomial . The degree of a constant polynomial is zero. Examples: 1) The degree of the polynomial 6 x 3 y 2  3 xy is 5. 2) The degree of the polynomial  4 x 2  5 x  3 is 2. 2 2 3) The degree of the polynomial 2 x  5 x y  7 y is 3. C. Like Terms If two terms contain the same powers of the same variables, they are called like terms or similar 2 2 2 3 terms. For example, 3x and  5 x are like terms, whereas 3x and  2 x are not like terms. Other examples 2 2  7 x x  y . are 4 xy and  5 yx ,  3x y and 5 yx , and  4 x x  y and D. Kinds of Polynomials Certain polynomials have special names depending on the number of terms they have. 1. Monomial is a polynomial that has only one term. a)  8r 2 s 3

Examples:

b) 

5x 2 y 3 4

2. Binomial is a polynomial that has two terms.

6x 2 y

 3xy 5 3. Trinomial is a polynomial that has three terms. 3 a) 8 x  3 y

Examples:

b)

2x

2

3  5x 2 y  3 5 4. Multinomial is a polynomial that has four or more terms

Examples:

a)  4 x 2  5 x  3

b)

Examples:

a) 4 x 3  5 x 2  x  7

b)

2a 4 b 3



a 3b 2

5

4

 2a 2 b  5a

E. Evaluating Algebraic Expressions In Algebra, we replace a variable with a number. This is called substituting the variable. To evaluate an algebraic expression, substitute the variable by a number and then calculate. Evaluating an algebraic expression means obtaining or computing the value of the expression where value/s of the variable/s is/are assigned. Examples: Evaluate. x y x y  1) for x = 12 & y = 8 4 4

=

=

12  8



4

20 4

2)

12  8

4 4

=



3y

x

2

=

4

= 5 + 1

yx

42



2

6



for x = 2 & y = 4

3( 4) 2

12

2

2

=3+6

18 =6 = =9 4 2 F. Operations on Algebraic Expressions H. Simplifying Algebraic Expressions Involving Grouping Symbols =

24


2


II. Laws of Exponents A. Product Law If m and n are integers and a

0, then

a m  a n  a m n Examples: 1) x 2  x 3  x 5

2) 32  34  36

B. Power of a Power Law If m and n are integers and a

a 

m n

0, then

 a mn

Examples:

  

2 4

1) x 3

 x 6   x 24 4

 

3

2) 32

C. Power of a Product Law If m is an integer and a 0 and b

 36

0, then

abm  a mb m Examples:

 

2

1) 2 x 3

 22 x 6  4x 6





2) 32 x 4 y 2

   3

D. Quotient Law If m and n are integers and m > n, and a

3 3

2

x12 y 6   36 x 36 y 18 3

0, then

am  a mn an If m and n are integers and m < n, and a am 1

0, then



an anm If m and n are integers and m = n, and a am  amn  a0  1 an Examples: 1)

29 2

4

 29  4  25  32

2)

E. Power of a Quotient Law If n, a, and b are integers, and b n

x 4 x

2

0, then

1

 x

 2  ( 4 )

1

 x

2 4



1

x2

0, then

n

a a    n b b Examples: 3

x x3   3 y  y

1) 

4

24 16 2 2)    4  3 81 3

III. Special Products and Factoring A. Special Products A. Product of the Sum and Difference of Two Terms The product of the sum and difference of t wo terms is obtained by subtracting the product of the last terms from the product of th e first terms, and is called the difference of two squares (DTS).

Examples: b a  b  a 2  b 2 a) a  

5b  3a  5b  9a 2  25b 2 b)  3a  


3


Square of a Binomial The square of a binomial is the sum of the square of the first term, twice the product of the two terms, and the square of the last term.

a  b 2  a 2 2 a b  b 2  a 2  2ab  b 2 a  b 2  a 2 2 a  b  b 2  a 2  2ab  b 2 Product of Two Binomials of the Form ax by  and cx  dy The product of the binomials ax by  and cx  dy where a, b, c, & d are real numbers, is

 cx dy by cx  dy . equal to ax



Product of a Binomial and Trinomial of the Forma  b  and a  ab  b



2

2





2 2 The product of a binomial a  b  and a trinomial a  ab  b where the first term of the trinomial is

the square of the first term of the binomial, the middle term of the trinomial is the negative of the product of the two terms of the binomial, and the third term of the trinomial is the square of the second term of the binomial, is equal to the product of the first terms plus or minus the product of the last terms of the binomial and trinomial factors. 1)

2)

a  ba 2  ab  b 2   a 3  b 3 a  b a 2  ab  b 2   a 3  b 3

(Sum of Two Cubes) (Difference of Two Cubes)

Cube of a Binomial The cube of a binomial a  b  is equal to a  b  a  b  a  b  or a  b  . 3

a  b3

= a  b2 a  b  = a 2  ab  b 2 a  b  = a 3  3a 2 b  3ab 2  b 3





B. Factoring

Factoring is the reverse of multiplying. To factor an expression means to write an equivalent expression that is a product of two or more expressions. 1) Common Monomial Factoring 1 Get the GCF of the terms in the polynomial. 2 Divide the polynomial by the GCF of the terms in the polynomial. 3 Write as factors the GCF and the quotient. Examples: Factor

a)

5 x 4  20 x 3 = 5 x  x  4 3

2 2 4 2 2 2 2 b) 16 x y  24x y  48x y = 8 x y 2 x  3  6 y 4

2

2





2) Factoring the Difference of Two Squares (DTS) For a binomial to be a difference of two squares, two conditions must hold. 1. The given binomial is a difference of two terms. 2. The two terms must be perfect squares. Examples: Factor 8 6 b) 36 x  25 y

2 a) 4 x  25,

Procedure: 2

a   

b2



a



b



a



b



2 a) 4 x 2  25 = (2 x)2 - (5) = (2 x + 5) (2 x - 5) 2

a  

2

b  

a



b



a



b



b) 36 x  25 y = (6 x ) - (5 y 3 ) 2 = (6 x 4 + 5 y 3 ) (6 x 4 - 5 y 3 ) 8

6


4 2

4


3) Factoring the Perfect Square Trinomial (PST) The square of a binomial is often called the perfect square trinomial. Use the following to help recognize a perfect square trinomial: st

rd

a) Two of the terms (1 & 3 ) must be squares, a 2 and b 2 , and are both positive. b) If we multiply a and b and double the result, we get the middle term, 2ab, or its additive inverse, - 2ab. To factor perfect square trinomial, use the following relationships: a) a 2  2ab  b 2  a b

   a  b 2 a b a b   a  b  a  b 2 2 a 2  2ab  b 2  a b    a  b  a b a b   a  b  a  b 2

b) Remember to factor out a common factor first, if there is any. 4) Factoring the Quadratic Trinomial 2

bx  c , recall that c is the constant term. If that c is not a perfect square, the In the polynomial x trinomial cannot be factoredusing perfect square trinomial type. It may, however, be possible to factor it as the product of two different binomials. Some points to consider in factoring x 2  bx  c , where b and c are constants. a) If the sign of the constant term is positive, look for its factors whose sum is the numerical coefficient of the middle. The signs of these factors must be the same. The sign of the middle term becomes the signs of the factors of the constant term. Some points to consider in factoring ax 2  bx  c , where a, b and c are constants.

To factor ax 2  bx  c , we look for binomials in the form __ x  __ __ x  __ where the products of constants in the blanks are as follows. a) The constants in the first blanks of the binomials have product a. b) The constants in the last blanks of the binomials have product c. c) The product of the constants in the extremes and the product of the constants in the means have a sum of b. 5) Factoring the Sum or Difference of Two Cubes If we divide a 3  b 3 by a  b , we get the quotient a 2  ab  b 2 and no remainder. So a  b 2 2 and a  ab  b are factors of a 3  b 3 . Similarly, if we divide a 3  b 3 by a  b , we get the quotient a 2  ab  b 2 and no remainder. So a  b and a 2  ab  b 2 are factors of a 3  b 3 . Factoring a Difference or a Sum of Two Cubes

a 3  b 3 = ( a  b ) ( a 2  ab  b 2 ) a 3  b 3 = a  b a 2  ab  b 2 6) Factoring by Grouping When the given expression is a multinomial, it may be factored by grouping. Examples:

a) 6 x 3  9 x 2  4 x  6

2 2 2 b) x  2 xy  y  b

Procedure:

a) 6 x 3  9 x 2  4 x  6 There is no factor common to all the terms other than 1. We can, however, group the terms as (6 x 3  9 x 2 ) and (4 x  6) and factor these separately. 3 2 = (6 x  9 x )  (4 x  6) 2 = 3x (2 x  3)  2(2 x  3)



2 = 2 x  3 3x  2

b) x  2 xy  y  b 2

2



Grouping the terms Factoring each binomial Factoring out the common factor (2x –3)

2

The terms do not have any common factor other than 1 but the terms can be grouped as

( x 2  2 xy  y 2 ) and b 2  . Both groups can be expressed as squares. Thus, x  y    b 2

= x  y

 b 2 b  x  y  b =  x  y   =  x  y b x  y  b 2


2

Rewriting the expression as DTS. Factoring the DTS Simplifying each factor 5


7) Factoring by Completing the Square There are polynomials that cannot be factored by using any of the preceding methods. Completing the square can factor some of these polynomials.

Examples: Factor a) x 4  64

b) 4 x  37 x y  49 y 4

2

2

4

Procedure: a) x 4  64 If the given expression is a binomial whose terms are perfect squares and positive, these two terms can be considered as the first and last terms of the perfect square trinomial. It means that in the binomial x 4  64 , the middle term is missing. The middle term (mt) can be found by using the formula: mt = 2

ft lt

where ft is the first term and lt is the last term. Using the formula,

 

4 2 2 mt = 2 x 64 = 2 x 8  16 x

So, = x 4  64  0 = x 4  16 x 2  64  16 x 2 4 2 2 = ( x  16 x  64)  (16 x )

 = x

  4x   8   4 x  x

= x2  8 2

2

Additive Identity Property 16 x 2  16 x 2  0 Grouping the terms

2

Rewriting the expression as DTS 2

 8  4 x

Factoring the DTS

 8 x 2 4 x  8 x 4  64  x 2  4 x 

Simplifying and arranging the terms

4 2 2 4 b) 4 x  37 x y  49 y 4 2 2 4 In the expression 4 x  37 x y  49 y , the first and last terms are squares . For the given to be a PST,

the middle term must be

  

2 2 2 2 mt = 2 4 x 4 49 y 4 = 2 2 x 7 y  28x y 2 2 2 2 2 2 Thus, we rename  37 x y as  28x y  9 x y , and we have

4 x 4  37 x 2 y 2  49 y 4 = 4 x 4  28x 2 y 2  49 y 4  9 x 2 y 2 4 2 2 4 2 2 = (4 x  28x y  49 y )  (9 x y )

 

= 2x  7 y 2

  3xy    

2 2

3

Grouping the terms

Rewriting the expression as DTS



2 2 2 2 = 2 x  7 y  3xy 2 x  7 y  3xy Factoring the DTS

4 2 2 4 2 2 2  xy  7 y 2 Simplifying & arranging the terms 4 x  37 x y  49 y = 2 x  3xy 7 y 2 x  3

IV. Rational Expressions Definition: A rational expression, or a fraction, is a quotient of algebraic expressions (remember that division by zero is not defined). A. Signs of Rational Expressions 1)

2)

a b



a b

 a  ( a ) a ( a ) a     b b  (b) b  (b) 

a

b



 ( a ) a a ( a )    b  (b) b (b)


6


Examples: x  (  x) x x a.    y  x  ( y  x)  ( y  x ) x  y

b.

y 3  x3 x y



 ( y 3  x3 )  ( x3  y 3 )  x y x y

B. Relations of Rational Expressions 1) Equivalent rational expressions

These are rational expressions with equal values. Note that

a



b

Examples: 1 4  a) 2 8

b)

1

a



2 a 3b 2 4

2a b

1

c)

2

x y



c

if and only if ad  bc .

d

x 2  xy  y 2 x3  y 3

2) Similar rational expressions These are rational expressions with the same denominator. Examples: 3 7 9 2 x 1 y  2 2x x 1 x  y , , a) , , b) c) , , 4 4 4 ab ab ab 2x  y 2x  y 2x  y 3) Dissimilar rational expressions These are rational expressions having different denominators. Examples: 5 3 1 3  x 3 x 2  1 2x  3y a) 9 , 8 , 6 b) a 2b , ab , ab2 c) x  y , x  y , x 2  y 2

C. Reduction of Rational Expression to Simplest Form A rational expression is said to be in lowest or simplest form if the numerator and denominator are relatively prime. The process of reducing fractions to their simplest form is dividing both numerator and denominator by their greatest common factor (GCF). Examples: 1) Reduce

35 a 4b 2

 42 a 3b 3

to lowest terms.

Solution: 4

35a b

2

 42a b

3 3

Or



57aa a abb 5a   23 7  a  a  a b b b 6b

4 2 3 2 35a b  7a b

 42a 3b 3  7a 3b 2



5a

 6b 3

2

where the greatest common factor (GCF)between the numerator and the denominator is 7 a b .

2) Reduce

x3  x2  6 x x3  3x 2  2 x

to lowest term.

Solution: We will first factor the members of the given fraction and then proceed as above.

x3  x 2  6 x x  3x  2 x 3

2




x( x 2  x  6) x( x 2  3x  2)

Common factoring by x

7


x( x  3)( x  2)



=

3) Reduce

3x  2 3x( x  1)  2( x  1)

Factoring quadratic trinomials

x( x  1)( x  2)

x3 x 1

Canceling x and (x – 2)

to simplest form.

Solution: =

3x  2

Given

3x( x  1)  2( x  1)

=

3x  2 ( x  1)(3 x  2)

=

1 x 1

Common factoring or by grouping

Canceling (3x

– 2)

D. Operations on Rational Expressions 1) Multiplication and Division of Rational Expressions a c ac a) Product of fractions:   b d bd The product of two or more given rational expres sions is a fraction whose numerator is the product of the numerators of the given fractions and whose denominator is the product of the given denominators. The product should be reduced to lowest terms. Examples:  9a 2b3 20c 6d 4 4b3c5  2    1) 4b 6c 4 18a 3d 2 5c 7 d 3 ad

2)

a 2  4b 2 2a  7 ab  3b 2

2



8a  4b 2a  4b



a 2  4ab  3b 2

=

=

a 2  ab  2b 2 (a  2b)(a  2b) (2a  b)(a  3b)

2( a  b )



4( 2 a  b ) 2(a  2b)



(a  b)(a  3b) (a  2b)(a  b)

Factoring

Canceling common factors and get the product

ab

a c a d ad     b d b c bc To obtain the quotient of two rational expressions, we multiply the dividend by the reciprocal of the divisor. Examples: 2 4 3 3 100a b 60a b  1) 3 30bc 42ac 4 Solution: 100a 2b 4 42ac 4  = Getting the reciprocal of the divisor 30bc 3 60a 3b3 then proceed to multiplication b) Quotient of fractions:

=

(5b)(20a 2b3 ) 3

(5b)(6c )



7c(a)(6c 3 ) 3(a)(20a 2b3 )

Factoring the numerator and the denominator


8


7c

=

Canceling the common factors of the numerators and the

3

denominators and simplify the result.

2) Divide

x 2  3x  2 2 x  3x  1 2

by

x2  x  2 2 x 2  3x  2

Solution: ( x  2)( x  1)

=

(2 x  1)( x  1)



( x  2)(2 x  1) ( x  1)( x  2)

Factoring and multiplying the dividend by the reciprocal of the divisor

x2 x 1

=

Canceling common factors

c) Addition and Subtraction of Rational expressions

Sum or difference of rational expressions: a c ac   Similar: d d d a c ad  bc   Dissimilar: b d bd Examples: 2x  1 7 x  5 9x  3   a) Find . 6x  1 6x  1 6x  1 Solution: =

2x  1  7 x  5  9x  3

Writing the rational expressions as a single

6x  1

fraction with a common denominator =

=

(2 x  7 x  9 x)  (1  5  3)

Collecting like terms in the numerator

6x  1

1

Performing the indicated operations in the

6x  1

numerator b) Express

2 3x  2 y



3x 3x 2  2 xy



8y 9x2  4 y2

in simplest form.

Solution: =

2 (3 x  2 y )



3x

x(3x  2 y )



8y (3x  2 y )(3x  2 y)

Factoring the

denominators Make the rational expressions similar by getting the LCD, divide the LCD by the given denominator and multiply the quotient by the given numerator =

=

2 x(3x  2 y )

x(3x  2 y )(3x  2 y)



3x(3x  2 y)

x(3x  2 y )(3x  2 y )

6 x 2  4 xy  9 x 2  6 xy  8 xy

x(3x  2 y )(3x  2 y )



8 y ( x)

x(3x  2 y )(3x  2 y )

Getting the products in the numerators


9


=

 3x 2  2 xy x(3x  2 y )(3x  2 y )

Combining like terms in the numerator

=

=

 ( x)(3x  2 y )

Factoring the numerator

( x)(3x  2 y )(3x  2 y)

1

Reducing the result to

(3 x  2 y )

simplest form E. Simplification of Complex Rational Expressions Complex rational expression is a rational expression in which the numerator or denominator is a rational expression. Examples:

y 1) Simplify the complex rational expression

x2 y

y2 x x

.

Solution:

y2  x2 =

y y x 2

2



y2  x2



y2  x2

y

x

x =

y 2  x2 y



x y 2  x2



x y

Canceling common factors and then multiplying the dividend by the reciprocal of the divisor.

V. Radical Expressions and Negative Exponents th

Any expression involving an n root can be written in radical form. The symbol

is called the radical

symbol. Radicals 1

If n is a positive integer and a is a real number for which a n is defined, then the expression 1

n

a is called a radical, and

n

a = an .

The number a is called the radicand. The number n is called the index of the radical. Remember that 1

n

a = a n . So

n

a is the positive nth root of a when n is even and a is positive, whereas

of a when n is odd and a is any real number. The expressions  4 , 4  16 , and because there are no even roots of negative numbers in the real number system. Whenever the exponent of a base is in rational form

6

n

a is the real nth root  2 are not real numbers

m where n  0, the expression can always be n

expressed in radical form.


10


Example 1: Write each exponential expression using radical notation. 5

3

b) a

a) a 4 Solution:

2

2

c) 8 3

5

3

a) a 4 =

4

a3

b) a

2

= (2 a )5

2

3

c) 8 3 =

82

Example 2: Write each radical expression using exponential notation. a) 36 Solution:

b)

3

8

b)

3

 8 = (8) 3

3

a6

c)

3

a6 = a 3

1

1

36 = 36 2

a)

c)

6

A. Simplifying Radical Expressions Product Rule for Radicals th

The n root of a product is equal to the product of the n n

ab =

th

roots of the f actors. In symbols,

a  b ,

n

n

provided that all of the expressions represent real numbers.

Examples: Simplify each of the following using the Product Rule for Radicals. a)

4

b.

32

12 x

6

c)

3

54 x5 y 9



4

2 = 24 2

Solution: a)

4

4



16

12 x 6 = 4 x

b) c)

32 =

5

3

4



6

4

2 =

9

24

2 3 2 3 = 2 (x )

3

54 x y = 3 27 x y



9

3

2

2x =

 3

3 = 2 x3 3 3

3

3 x y

9



3

3

2 x 2 = 3xy3 2 x 2

Quotient Rule for Radicals th th The n root of a quotient is equal to the quotient of the n roots of the numerator and denominator . In symbols

a

n



b

n

a

n

b

provided that all of the expressions are real numbers and b  0 .

Examples: Simplify each of the following using the Quotient Rule for Radicals. a)

3

 8x 3 y 5

b)

3

24 y 5 12 5

27 Solution: a)

b)

3

 8 x3 y 5



3

 8x3 y 3 y 2

27

3

24 y 5 125

3



3

5

(2) x y y 3

3

3

2



3

3

3

3

3

3

8  3y y 3



2



3

3

3

2 y y 5

2



 2 xy3 y 2 3

2 y3 y

2

5

Rationalizing the Denominator Multiplying both the numerator and denominator by another radical that makes the denominator


11


Examples: Rationalize the denominator of each of the following: 2

3

10

a)

24 x y

b)

6

3

4

c)

4 xy

2 1 3

Solution: 10

a)



6

3

6



6

24 x 2 y 4

3

b)

10

4 xy

60



6

6

24 x 2 y 4

3

=



4 xy

3

2 2  15



2 15



6

2

3

2x2 y2

3

2x y

2



15



6 48 x 4 y 6

3

2

3

3

3

8x y

3



3

23  6 x3 xy 6 3

3

3

2 x y



2 xy 2 3 6 x

3

2 xy

 y3 6 x

c) To rationalize letter c, use the conjugate of the two-term denominator as the multiplier of both numerator and denominator of the given expression.

2



1 3



1 3 1 3

2(1  3 ) 1 3



2(1  3 ) 1 3

2



2(1  3 )

2

 (1  3)  1  3

B. Operations on Radical Expressions 1) Addition and Subtraction Like radicals are radicals that have the same index and the same radicand. To simplify the sum of 3 2  5 2 , we can use the fact that 3x + 5x = 8x is true for any value of x. So, 3 2  5 2  8 2 . The

expression 3 2  2 3 cannot be simplified because they are unlike radicals. There are radicals that need to be simplified before adding or subtracting them. Examples: Find: 1) 2 x 3 -

4 x 2 + 5 18 x

3

2)

3

3

4

16 x y -

3

54 x3 y 4

Procedure: 3

1)

2x -

2

4 x + 5 18 x

2x  x -

Given

2 x + 5 3  2x  x

2

=

3

2

2

2

2

Product rule for radicals

= x 2 x - 2x + 15x 2 x

Simplifying each radical

= 15x- 2x 2x

3

2) =


16 x 3 y 4 -

3

3

54 x y

4

Given

2 2 x  y  y -

3

3

3

3

3

= 2 xy 3 2 y - 3xy 3 2 y

3 2 x  y  y Product rule for radicals Simplifying each radical 3

3

3

=  xy3 2 y


2) Multiplication of Radicals The product rule for radicals, n ab = n a  n b , allows multiplication of radicals with the same index. Examples: Find the product of the radicals in simplest form. 1)  2 6  4 3

2)

3

a (3 a  23 a 2 )

3) (2 3 + 5 ) (2 3 -

5)

Procedure: 1)  2 6  4 3 =  8 18  8 2  32  (8)(3) 2  24 2 2)

3

a (3 a  23 a 2 ) =

3

3

a2  2 a3 =

3

a 2  2a


12


5 2 = 4(3) – 5 = 12 – 5 = 7

2 5) = 4 3 -

3) (2 3 + 5 ) (2 3 -

C. Simplifying Expressions with Negative Exponents For any rational number a except 0, and for all whole numbers m, 1 m



a

am

To simplify algebraic expressions with negative exponents, mean to express the given expression into an equivalent quantity where the exponents become positive. Examples: Simplify the following expressions: a)

9 2 a 4bc 3 4

b)

3  2 3

3 a b c

9 x 2  4 y 4 3x 1  2 y  2

Procedure:

a)

b)

92 a 4bc 3 4

3  2 3



3 a b c

2 4 9x  4 y

1

3x  2 y

2



(32 ) 2 a 4bc 3 4

3  2 3

3 a b c

x2 y 4 2

x y

4



34 a 4bc 3



4

3 a b c

4 2 9 y  4x

3xy  2 x y 4



3  2 3

2

2



34 a 4b 2c 3 4

4

2 3



b3

3 a b c

2 2 (3 y  2 x)(3 y  2 x)

xy (3 y  2 x) 2

2

a



2 3y  2x

xy 2

VI. Relations, Functions, and their Zeros A. Definition of Relation, Function, Domain and Range A relation is a set of ordered pairs. The domain of a relation is the set of first coordinates. The range is the set of second coordinates. Relations are often defined by equations with no domain stated. If the domain is not stated, we agree that the domain consists of all real numbers that, when substituted for the independent variable, produce real n umbers for the dependent variable .

Example 1: The table of values below shows the relation between the distance of the movie projector from the screen and the size of a motion picture on the screen. Distance 1 2 3 4 Screen Size 1 4 9 16 The numbers in the table above could be written as ordered pairs (x, y) where x is the First member or first coordinate and y is the second member or second coordinate. We can express these numbers as the set of ordered pairs.

The set

1,1, 2.4 ,  3, 9 , 4,16  1,1, 2.4 ,  3, 9 , 4,16  is a relation. The domain of the relation is 1,

2, 3, 4 , and the range

is 1, 4, 9, 16 . Example 2: The set of ordered pairs below shows a similar relation. Each person is paired with a number representing his or her height. Person Height (cm)

Carl 202

Dan 142

Em 138

Frank 142

  Carl, 202 , Dan, 142 , Em  ,138 , Frank,142  . The domain is The relation is Carl , Dan, Em, Frank  and the range is 138, 142, 202 . Notice that for each person there is exactly one height. This is a special kind of relation called a function. St. Louis Review Center, Inc – San Pablo City Branch http://www.slreviewcenter.com| https://www.facebook.com/SLRCspc (049)562-2239 | 0917-584-8106

13


A function is a relation that assigns to each member of the domain exactly one member of the range. It is a set of ordered pairs of real numbers  x , y  in which no two distinct ordered pairs have the same first coordinate. The set of all permissible values of x is called the domain of the function, and the set of all resulting values of y is called the range of the function.

Example 3: 2 Let f be the function defined by f   x, y  y  x  3 . Find the domain and range of the function.

Solution: The domain is the set of all real numbers. Since the expression x 2 is nonnegative, the smallest value that y can take is 3 (when x  0 ). Hence the range is is the set of all real numbers greater than or equal to 3 or y y3 .





Example 4:



Find the domain and range of the function defined by f   x, y  y 



x  . x  1

Solution: The domain is the set of all real numbers except  1 , which is x  R x  1. To determine the range, express x in terms of y, that is x 

y y 1

. From this result, y cannot be equal to  1 . Therefore the range of

function is the set of all real numbers except  1 , which is y  R y  1 B. Values of Functions

The symbol f  x  (read " f of x" or " f at x" ) denotes the particular value of the function that corresponds to the given value of x . The variable x is called the independent variable while the variable y is called the dependent variable because y is usually expressed in terms of x when their relationship is given in the form of an equation. If there is an equation that is a function, we may replace y in the equation with f x  , since f x  represents y . For example, if the given function is y  3 x  1 , we may write it as f  x   3 x  1 . To evaluate a function for a specific value of x , replace each x in the function with the given value, and then perform the indicated operation/s. 2 Find the value of f  x   3x  5 x  2 when x  2 , denoted by f 2  .

Example 1: Solution:

f  x   3x 2  5 x  2

f 2 3 2  5 2  2 f 2  34  10  2 2

f 2   4

f 2   12  8

 In the example, the ordered pair 2, 4  belongs to the function f , in symbols 2, 4  f . We also say that 4 is the image of 2 under f , and 2 is the pre-image of 4 under f . 2 Example 2: Given that f is the function defined by f  x   x  4 x  3 , find the following:

a) f 0 

b) f  2 c) f 2 h 



2 d) f  3h



e) f  x  h 

VII. Equations Definition An equation is a mathematical sentence that uses an equal sign to state that two expressions represent the same number or are equivalent.

Examples: a) 3  2  5

b) 3x  4  2 x  5


c) x 2  9  0

14


An equation that contains at least one variable is called an open sentence. Equations b & c above are examples of open sentences. In equation b, only -1 makes the sentence true or satisfies the equation. However, more than one number might satisfy an equation. For example, +3 and -3 satisfy the equation x 2  9  0 . Any number that satisfies an equation is called a solution or root to the equation. The set of numbers from which you can select replacements for the variable is called the replacement set.

Solution set The set of all solutions to an equation is called the solution set to the equation.

1) Solving Equations

To solve an equation means to find all of its solutions. The most basic method for solving equations involves the properties of equality.

Properties of Equality For any rational numbers a, b and c, a) a  a is always true Reflexive Property b) If a  b , then b  a Symmetry Property c) If a  b , b  c , then a  c . Transitive Property d) Adding the same number to both sides of the equation does not change the solution set of the equation. In symbols, if a  b , then a  c  b  c . Addition Property e) Multiplying both sides of the equation by the same nonzero number does not change the solution set of the equation. In symbols, if a  b then ac  bc .

Multiplication Property

Equations that have the same solution set are called equivalent equations. Using the properties of equality, we can derive equations equivalent to the srcinal equation. The equations 2 x  4  10 and  x  21  4 x are eq uivalent equations because 7 satisfies both equations. 2) Types of Equations Identity is an equation that is satisfied by every number by which both sides of the equation are defined. The number of solutions is infinite.

Examples:

a) 23 x  1  6 x  2

b) x  1  x  1

c) x  1

x

Both equations 23 x  1  6 x  2 and x  1  x  1 are considered identities because both sides of each equation are identical. They can be satisfied by any real numbers. The equation real numbers except 0 because

0 0

x  1 can be satisfied by all x

is undefined.

Conditional equation is an equation that is satisfied by at least one number but is not an identity. The number of solutions is finite.

Examples:

a) 3x  4  2 x  5

2 b) x  9  0

2 The only value that makes equation 3x  4  2 x  5 true is – 1. While x  9  0 has the solution set  3, 3.


15


Inconsistent equation is an equation whose solution set is the empty set.

Examples:

a) x  x  2

b) 5  3 x 6  4 x  9  7 x

Simplifying either of the two equations using only the properties of equality will yield an equation that is false.

A. Linear Equations Definition An equation is linear if the variables occur as first powers only, there are no products of variables, and no variable is in a denominator. The graph of the linear equation is a straight line. A linear equation is also called a first-degree equation.

Examples: a) y  4 x  2

c) 3x  2  5 x  4

b) 2 x  5  9

d) 4 x  2 y  0

e.

2x 3

5 

x 2

1

Linear Equations in One Variable A linear equation in one variable x is one in th e form ax  b  0 where a and b are real numbers, with a  0 .

Techniques in Solving Equations

1) Simplifying the equation first Example: Solve the equation 2( x  4)  5 x  34. Solution: Before using the properties of equality, we simplify the expression on the left of the equation: 2( x  4)  5 x  34. Given

2 x  8  5x  34 Distributive Property 7 x  8  34 Combining like terms 7 x  8  8  34  8 Addition property by 8 7 x  42 Simplify 1 7 x  42 Multiplication property 7 x=6 Solution set is {6} 2) Using the properties of equality Example: Solve  x  21  4 x . Solution: We want to obtain an equivalent equation with only an x on the left side and a constant on the other.

 x  21  4 x Given  x  21  4 x  21  4 x  4 x  21 Addition Property 3x  21 Combining like terms 1 3

3x  21

Multiplication Property by

1 3

x=7 Checking: Replacing x by 7 in the srcinal equation gives us - 7 – 21 = - 4(7) - 28 = - 28 which is correct. So the solution set to the srcinal equation is {7}. St. Louis Review Center, Inc – San Pablo City Branch http://www.slreviewcenter.com| https://www.facebook.com/SLRCspc (049)562-2239 | 0917-584-8106

16


We may remove fractions by multiplying by the lowest common denominator (LCD). 3 5  3. Example: Solve the equation  4  x 2x Solution: Find the LCD first. The LCD is 2x. 3 5 4  3. Given x 2x 5 3  4  3 2x x 

2 x

Multiply the equation by 2x.

6  8x  5  6 x Distributive property Addition property 6  8x  (6)  (6 x)  5  6 x  (6)  (6 x) 2 x  1 Combining like terms 1 2 x  1 Multiplication property 2 1 1  }. x=  Solution set is { 2 2 Decimals may be removed from an equation before solving. Multiply by a power of 10 large enough to make all decimal numbers whole numbers. If you multiply by 10, you move the decimal point in all terms one place to the right. If you multiply by 100, you move the decimal point in all terms two places to the right. Example: Solve the equation x  0.1x  0.75x  4.5 Solution: Because the highest number of decimal places is 2 in the term 0.75x, we multiply the equation by 100, thus

100( x  0.1x  0.75x  4.5) Multiply the equation by 100 100x  10x  75x  450 Distributive Property 100x  10 x  (75x)  75x  450  (75x) Addition property 15x  450 Combining like terms 1 15 x  450 Multiplication property 15 x = 30 Solution set is {30} Applications of Linear Equations in One Variable 1) Number-Related problems Example: There are two numbers whose sum is 50. Three times the first is 5 more than twice the second. What

are the numbers? 2) Time, Rate, and Distance or Motion-Related Problems Example: A passenger bus starts from Tuguegarao City and heads for Santiago City at 40 kph. Two hours later, a car leaves the same station for Santiago City at 60 kph. How long will it be as the car overtakes the passenger bus? 3) Age- Related Problems Example: Mother is four times as old as Mary. Five years ago, she was seven times as old. How old will each be in 5 years? 4) Work-Related Problems Example:


17


Jerry can build a garage in 3 days, and Sam can build a similar garage in 5 days. How long would it take them to build a garage working together? 5) Investment-Related Problems Example: Gary invested P50,000, part of it at 6% and the other part at 8%. The annual interest on the 6% investment was P480 more than that from the 8% investment. How much was invested at each rate? 6) Digit-Related Problems Example: The tens digit of a certain number is 3 less than the units digit. The sum of the digits is 11. What is the number? 7) Geometry-Related Problems Example: The length of a rectangular piece of property is one meter longer than twice the width. If the perimeter of the property is 302 meters, find the length and width. 8) Mixture-Related Problems Example: What amounts (in ounce) of 50% and 75% pure silver must be mixed to produce a solution of 15 ounces with 70% pure silver?

B. Systems of Equations Definition: Any collection of two orthen morethe equations taken as onethat is called a system of equations equations.isIf the the system involves two variables, set of ordered pairs satisfy all of the solution set of the system.

Types of Systems of Linear Equations 1) A system of equations that has one or more solutions is called consistent. The graphs of the equations either intersect at a point or coincide. The set of coordinates of the intersection is the solution set of the system. a) Consistent Independent is a system of linear equations with only one point as its solution. The graphs of the equations in the system intersect at one point only.

Example: a) Find the solution set x  y  6 and x  y  2 by using the substitution method. Procedure: The second equation states that x and y  2 are equal, thus in the first equation, we can replace x with y  2 .

xy 6 y2 y  6

Equation 1 Replace x in equation 1 with y  2

Since this equation now has only one variable, we can solve for y . 2y  4 y2 Next, replace y with 2 in either equation to solve for x . Equation 1

xy 6 x26 x4


Equation 2

x  y2 x  22 x4 18


Checking: Replace x with 4 and y with 2 in either equation. Equation 1 xy 6 Equation 2 x  y2

42  6 66

4  2 24 = 2 + 2 44

The solution set of the system is {4, 2}. The system is consistent because there is only one solution. b) The sum of two numbers is 115. Their difference is 21. Find the numbers. Procedure: (By Elimination Method) st nd Let x = 1 number (greater number) & y = 2 number (smaller number) Formulate the equations:

x  y  115 Equation 1: x  y  21 Equation 2: Here, the elimination method can be done by adding the equations or by subtracting one equation from the other. Thus, +

x  y  115 Equation 1 x  y  21 Equation 2

2 x  136 From the sum, compute for the value of the retained variable x .

x  68

MPE

To compute for the value of y , replace x with 68 in either equation. x  y  115 Equation 1

68  y  115 y  115  68 y  47 The numbers are 68 and 47. The system is consistent because there is only one solution. b) A system of equations that has infinitely many solutions is called consistent dependent. The graphs of the equations coincide. The equations in the system are equivalent.

Examples: a) 2 y  2   x x  2y  4

Equation 1 Equation 2

3 y  1  2( x  3) 3 y  2 x  3

b)

Expressing y as a function of x in each of the equations in each system gives equal expressions. Graphing can also show dependence between the two equations in each system. The graphs of the equations will coincide. 2) A system of equations that has no solution is called inconsistent. The graphs of the equations do not intersect or are parallel. The solution set is an empty set.

Examples: a) 2 x  3 y  6

3y  2x  3

Equation 1

b)

Equation 2

 4 y  5x  7 4 y  5 x  12

Solving the system by elimination or by substitution will result into the inequality of two constants where both variables are dropped. The elimination of both variables implies that no solution can be obtained.


19


Philippine Normal University LICENSURE EXAMINATION FOR TEACHERS (LET)

Refresher Course

WHAT TO EXPECT MAJORSHIP Area:

MATHEMATICS

Focus: Advanced Algebra

LET Competencies         

Solve for the roots of a given quadratic equation Solve problems on quadratic equations Determine an equation given a set of roots which are imaginary/complex numbers Perform operations involving exponential and logarithmic functions Solve for the solution set of a given inequality n Determine the rth term of the expansion (a + b) Solve problems involving arithmetic and geometric progressions Solve problems involving variations Determine the number of positive and negative roots of a given polynomial

PART I - CONTENT UPDATE:

QUADRATIC EQUATION 2 An equation of the form ax + bx + c = 0 where a  0, a ,b, and c are constants, is a quadratic equation.

Examples: The following are quadratic equations: 2 1. 3x + 4x + 5 = 0 2. 2x2 - 21x = 0 2 3. 5x - 25 = 0 ROOTS OF QUADRATIC EQUATIONS To solve a quadratic equation in x means to find the value/s of x (unknown) that will satisfy the given equation. The values of x that will make the equation true ar e called the roots or solution s of the quadratic equation.

Methods of Finding the Roots of a Quadratic Equation 2 1. Factoring (Use this method if ax + bx + c is factorable) Example: Find the roots of x2 + 2x - 15 = 0 Factoring the left side of the equation, (x + 5) (x – 3) = 0 Equating each factor to zero, (x + 5) = 0 (x – 3) = 0 Solving the resulting linear equations, x=-5 x=3 2 2. Quadratic Formula (Use this method if ax + bx + c is not factorable) The quadratic formula is

x

 b  b 2  4a c 2a 1


Example: Solve 3x2 – 2x – 7 = 0 Substitute a = 3 b = -2 c = -7 in the quadratic formula,

x

x

2  ( 2) 2  4(3)(7) 2(3) 1  22 3

BINOMIAL FORMULA n To obtain the terms of the binomial expansion (a + b) , we use the binomial formula:

n ( n  1)a n  2 b 2

n 1

n

n

 na

(a + b) = a

b



2!

THE rth TERM OF THE EXPANSION (a + b)

rth term =

n ( n  1)(n  2)a n  3 b 3 3!

n 1

 ...  na b

n

b

n

n (n  1)(n  2)...(n  r  2)a n  r 1 b r 1 ( r  1)!

Example: 8

Find the fifth term of (2a – 3b) n=8 r=5 n–r+2=5 5th term =

(8)(7)(6)(5)( 2a ) 4 (5b) 4

th

4! 4 4

5 term = 90, 720 a b

EXPONENTIAL FUNCTIONS An exponential function with base b is defined by an equation of the form x

f (x) = b , where b and x are real numbers and b> 0,b ‡ 1. Note: An exponential function has a constant base and a variable exponent.

Examples: The following are examples of exponential functions 1. f(x) = 2x 2. f(x) = 32x - 2

1  2

4 x 3

f(x) = 

3.

PROPERTIES OF EXPONENTIAL FUNCTIONS The exponential functions f (x) = bx and g(x) = by where x and y are real numbers, b> 0, b  1 , satisfy the following properties x y x+y 1. b b = b x y 2. b = b , if and only if x = y -x

3. b =

1 bx

4. If b>1, and x< y, then bx < by x y 5. If 0 b

2


NATURAL EXPONENTIAL FUNCTIONS The natural exponential function is defined by the equation x f(x) = e where e = 2.71828… Notice that e is an irrational number. EXPONENTIAL EQUATIONS An equation where the unknown quantity appears in an exponent is called an exponential equation. Examples: The following are exponential equations: x-3 x-3 3x-2 1. 2. 60 5 ==22 2x – 6x+3 4x+5 3. 6 =2

SOLVING EXPONENTIAL EQUATIONS To solve an exponential equation is to find the value of the unknown quantity in the given equation. Example 1: Solve for x in 64 = 8x-2 Solution: Rewriting the given equation, we have 82 = 8x-2 Since the bases are equal, the exponents must be equal. Thus, 2=x–2 4=x Hence, x = 4 x2 4x

Example 2: Solve for x in the equation 2   2  Equating the exponents, we have 2 x + 4x= -3 x2 + 4x + 3 = 0 (x + 3) (x + 1) = 0 x = -3 x = -1. Therefore

x = -3

3

x = -1.

LOGARITHMIC FUNCTIONS The equation f(x) = logb x, where b  1 , x> 0 and b> 0 is called a logarithmic function. Examples: The following are logarithmic functions: 1. f(x) = log2 (3x -2) 2. f(x) = log2 x 3. f(x) = log8 (6x -3) Note: 1. If the base of the logarithm is not indicated it is understood that the base is 10. 2. If the base of the logarithm is the number e, then it is called a natural logarithm and it is written as f(x) = ln x.

LOGARITHMIC EQUATIONS A logarithmic equation is an equation that contains logarithm. Examples: The following are logarithmic equations 1. log3 81 = 4 2. log2 8 = 3 3. log2 16 = 4 4. log 100 = 2 3

| Mathematics Major [4] y

The logarithmic equation y = logb x can be written in exponential form as b = x and vice-versa.

Here are some examples: Logarithmic Form log2 8 = 3 log10 100 = 2 log4

1

Exponential Form 23 = 8 2 10 = 100

4 2 

=-2

16

1 16

34 = 81

Log3 81 = 4

PROPERTIES OF LOGARITHMS 1. logb x + log b y = log b (xy) Example: log2 (2x + 3) + log2 x = log2 [x(2x + 3)]

2. logb x – logb y = logb

x   y

Example:

 3x  5    5x 

log3 (3x -5) – log3 5x = log3  3. logb xn = nlogb x Example: log2 x3 = 3 log2 x

SOLVING LOGARITHMIC EQUATIONS To solve a logarithmic equation means to find the value of the unknown quantity in the given equation. Example 1: Solve for x in log (x + 2) – log x = log (x + 2) Rewriting the given equation, we have

log

x2 x

 log(x  2)

Since the logarithmic of both sides of the equation are equ al, then we have

x2 x

 x2

x(x + 2) = x + 2 2 x +x-2=0 (x + 2) (x-1) = 0 x = -2 x = 1 Thus, x = 1

Example 2 Solve for x in log( x – 4) + log(x – 3) = log 30 Solution Rewriting the given equation, we obtain log(x – 4)(x – 3) = log 30 (x – 4) (x – 3) = 30 2 x - 7x + 12 = 30 x2 - 7x -18 = 0 4


Thus,

(x – 9) (x + 2) = 0 x = 9 x= -2 x = 9.

INEQUALITIES Any relation expressed using the symbols <, >, > or < is called an inequality. An absolute inequality is an inequality which is always true. A conditional inequality is one which is true only for certain values of the variable involved. 1. 4 > 3 is an absolute inequality 2. x > 3 is a conditional inequality

PROPERTIES OF INEQUALITIES Let a, b, c, & d be real numbers. The following hold. 1. Trichotomy Property a > b or a b if a - b > 0 a 0 and b> 0, then a + b> 0 and ab>0. b. If a < 0 and b < 0, then a+b< 0 and ab> 0 4. Transitivity If a 0, then ac < bc If a bc

SOLVING INEQUALITIES To solve an inequality means to find the value of the unknown that will make th e inequality true. Examples: 1. Solve the inequality 4x + 3 < x + 8 Solution: 4x 8 4x + – x3 << 8x + –3 3x < 5 x<

5 3

2. Solve the inequality 2x - 3 < 5x + 7 Solution: 2x – 3 < 5x + 7 2x – 5x < 7 + 3 -3x < 10 x>

 10 3

3. Solve the inequality 3x2 – 5 x + 7 Solution: 2 3x –- 5x < 0 x(3x – 5) < 0 5


Case I x > 0 and 3x – 5 < 0 x > 0 and 3x < 5 x > 0 and x <

Thus, 0 < x <

5 3 5 3

Case II x < 0 and 3x – 5 . 0 x < 0 and 3x > 5 x < 0 and x >

5

Thus, the solution is a3null set. Therefore the general solution is 0 < x <

5 3

2

4. Solve the inequality 3x – 4 <– 3x Solution: 2 x – 4 < –3x 2 x + 3 x – 4< 0 (x + 4) ( x – 1) < 0 Case I x + 4 < 0 and x – 1 > 0 x < –4 and x > 1 Thus, there’s no solution.

Case II x + 4 > 0 and x – 1 < 0 x > -4 and x < 1 Thus, x = ( -4, 1) Therefore, the general solution set is (-4, 1) or -4 < x < 1.

SEQUENCES AND SERIES A sequence is an ordered list of numbers called terms. Example: For the sequence 1, 3, 5, 7, …, the first term is 1, the second term is 3, and so on and so forth.

The sum of the terms of a sequence is c alled a series. Example: The sum of the sequence 1, 3, 5, 7,… denoted by 1 + 3 + 5 + 7 +… is a series.

ARITHMETIC SEQUENCE

A sequence in which a constant d is added to the previous term to get the next is called an arithmetic sequence. The constant d is called the common difference. Examples:

1. The sequence 2, 4, 6, 8,… is an arithmetic sequence whose constant difference is 2. 2. The sequence 35, 30, 25, 20,… is an arithmetic sequence whose constant difference is -5 The nth Term of an Arithmetic Sequence (t n ) The nth term of an arithmetic sequence can be found by using the formula, 6


tn = t1 + (n – 1) d where th

tn = n term of the arithmetic sequence t1 = first term of the arithmetic sequence n = number of terms d = common difference Example 1 th Find the 20 term of the sequence 4, 14, 24, 34,… Solution: Here t1 = 4, n = 20, d = 10 Thus, – 1) (10) t20 ==4t1++(19) (20 (10)

= 4 + (190) t20 = 194 Example 2 th Find the 50 term of the sequence 13, 10, 7, 4,… Solution: Here t1 =13, n = 50, d = -3 Thus, t50 = t1 + (n – 1) (d) = 13 + (50 – 1) (-3) = 13 + (49) (-3) t50 = 134

ARITHMETIC SERIES The indicated sum of an arithmetic sequence is an Arithmetic Series. The sum of the first n terms of an arithmetic sequence, denoted by Sn is given by the formula

Sn=

n 2

Sn =

( t 1  t 2 ) or n 2

[(2t1 + (n – 1)d]

Example 1: Find the sum of the first 20 terms of the sequence 1, 8, 15 . . . Solution: S20=

20 2

[2(1) + (20 – 1) (7)]

=10 [2+ 133] =10 (135) S20=1350 Example 2: How many multiples of 4 are there between 15 and 94? Solution: Here= t16, t =n 92, d=4 1 Thus, tn = t1 + (n – 1)d 92 = 16 + (n – 1) 4 92 = 16 + 4n – 4 80 = 4n 20 = n So, there are 20 numbers which are multiples of 4 between 15 and 94.

7


GEOMETRIC SEQUENCE A sequence in which a constant r is multiplied by the previous term to get th e next term is called a g eometric sequence. The constant r is called the common ratio. Example: 1. The sequence 2, 4, 8, 16, 32,… is a geometric sequence whose constant ratio is 2. 2. The sequence 24, 8,

8 3

, … is a geometric sequence whose constant ratio is

1 3

.

The nth Term of a Geometric Sequence th The n term of a geometric sequence is given by the formula, tn = t, r n-1 th where tn= n term of the geometric sequence

ti = first term of the geometric sequence Example: th Find the 15 term of the sequence 1, 4, 16, 64,… Solution Here t1 = 1, n = 15, r=4 Thus, n-1 t15 = t1 r 15-1 = (1)(4 ) POLYNOMIAL FUNCTION The function defined by the equation n n-1 n-2 2 f(x) = a0x +a1x + a2x + . . .+ an-2x + an-1x + an where n is a nonnegative integer and a0, a1, . . .,an are constants, a0,  0 is a polynomial function in x of degree n. The zeros or roots of f(x) are the numbers x such that f(x) = 0. The Number of Positive and Negative Roots of a Polynomial Function If f(x) is a polynomial function with real coefficients, The number of positive real zeros of f(x) is either equal to the number of variations in sign in f(x), or to that number diminished by a positive even integer. The number of negative real zeros of f(x) is either equal to the number of variations in sign in f(-x), or to that number diminished by a positive even integer. Example: Determine the possible number of positive and negative zeros of f(x). 3 2 1. f(x) = 8x – 12x – 2x + 3 Because f(x) is of degree 3, there would be 3 zeros. f(x) has 2 variations in si gns, while f(-x) = -8x3 – 12x2 + 2x + 3 has 1 variation in sign. Thus, f(x) may have the following possible roots 1. 2 positive zeros and 1 negative zero 2. no positive zero, 1 negative zero, and 2 imaginary zeros 6 5 3 2. f(x) = 4x – 3x + 7x – 3 Because f(x) is of degree 6, the re would be 6 zeros. f(x) has 3 variations in signs, while f(-x) = -8x3 – 12x2 + 2x + 3 has 1 variation in sign. Thus, f(x) may have the following possible roots 1. 3 positive zeros, 1 negative zero, and 2 imaginary zeros 



2. 1 positive zero, 1 negative zero, and 4 imaginary zeros

VARIATION

The term use to denote the effect of changes among two or more related quantities. There are different types of variation: Direct Variation If y varies directly as x, then we write y  kx where k is the constant of variation. 



Inverse Variation

If y varies inversely as x, then we write y  



k x

where k is the constant of variation.

Joint Variation If y varies jointly as x and w, then we write y = kwx where k is the constant of v ariation. Combined Variation

If y varies jointly as w and x and inversely as z, then we write y  k

wx z

where k is the

constant of variation.

8


Philippine Normal University LICENSURE EXAMINATION FOR TEACHERS

Refresher Course for year 2015

WHAT TO EXPECT CONCENTRATION: MATHEMATICS

Focus: Trigonometry LET Competencies: 1. Show mastery of the basic terms, concepts and operations in Trigonometry involving 

Angles Trigonometric ratios  Conversion (degree to radian and vice-versa)  Pythagorean Theorem 2. Solve, evaluate and manipulate symbolic and numerical problems in Trigonometry by applying fundamental principles and processes. 

PART I. CONTENT UPDATE

1. Angles In General An angle is formed by two rays with the same endpoint. The common endpoint is called the vertex of the angle and the rays are called the sides of the angle.

B



O A

Figure 1

In Figure 1, the vertex of the angle  (read as theta) is labeled O. A and B are points on each side of . Angle  can also be denoted by AOB, where the letter associated with the vertex is written between the letters associated with the points on each side. We can think of  as having been formed by rotating side OA about the vertex to side OB. In this case, we shall call side OA theinitial side of  and side OB the terminal side of . When the rotation from the initial side takes place in a counterclockwise direction, the angle formed is considered a positive angle. If the rotation is in clockwise direction, the angle formed is a negative angle. 2. Degree Measure One way to measure the size of anangle is with degree measure. The angle formed by rotating a ray through one complete rotation has a measure of 360 degrees, written as 360.

1


One degree (1), then, is

1 36 0

of a full rotation. Likewise, 180 is one-half of a full rotation, and 90  is

half of that (or a quarter of a rotation). Angles that measure 90 are called right angles, while angles that measure 180 are called straight angles. Angles that measure between 0 and 90 are called acute angles, while angles that measure between 90  and 180 are called obtuse angles. A central angle in a circle is an angle with its vertex at the center of the circle. In a circle, a central angle that cuts off an arc equal in length to the radius of the circle has a measure of 1 radian (rad).



O

s

r

Figure 2

3. Conversions 



To convert degrees to radians, we consider the fact that 1 



180

rad

 180     



To convert radians to degrees, we consider the fact that 1 rad = 

We note the following: 1. If the unit of measure of an angle is not indicated, then it is understood to be in radians. 2. 1 rad is approximately 57, thus, 1 rad is much larger than 1 degree. Example 1. 1.1 Convert 60 to radians. Solution: Note that 1 



180

rad, so to convert from degrees to radians, we multiply by

 18 0

. Thus, we have

    rad  rad 180 3  

60  60

When our answer is in terms of , we are writing an exact value. If we wanted an approximation, we would replace  with 3.1416.  3.1416 Exact value   0.7854 Approximate value 4 4 1.2 Convert -270 to radians. Solution: We multiply by the given by



. Thus, we have

18 0

3     270  270 rad   rad 2  180  Example 2. 2.1 Convert

 4

to degrees.

Solution: To convert from radians to degrees, we multiply by  4

2.2 Convert 

7 6

rad =

180



.



 180     45 4  



to degrees. 18 0

Solution: We multiply the given by



. Hence, we have. 2






7

rad = 

6

7  180     210 6   

4. Pythagorean Theorem Right triangles are very important to the study of trigonometry. In every right triangle, the longest side is called the hypotenuse, and it is always opposite the right angle. The two other sides are called the legs of the right triangle. Y

z

x

X Z

y

Figure 3 Right triangle XZY (symbolized as right XZY) with right angle Z is drawn in Figure 3. Its hypotenuse is z while its legs are x and y. We now state the Pythagorean Theorem. Pythagorean Theorem: In any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides (called legs). B

If C = 90, then c2 = a2 + b2

c

a

A C

b

Figure 4 Example 3. Solve for x in the right triangle in Figure 4. B

x 9 C

A

12 Figure 4 Solution: Applying the Pythagorean Theorem, we have

x 2  9 2  12 2

x  9  12 = 15 Example 4. Solve for x in the right triangle in Figure 5. 2

2

B

13

x C

A

x+7 Solution: By applying the Pythagorean Theorem, we have

Figure 5

(  7)   13 x x 2

2

2

3


x 2  14 x  49  x 2  169 2 x 2  14 x  120  0 x 2  7 x  60  0 ( x  12)( x  5)  0 x+ 12 = 0

or

x-5=0

x = -12 or

x=5

Our only solution is x = 5. We do not accept x = -12 because x is the length of a side of triangle ACB and therefore, its value cannot be negative. 5. Trigonometric Functions If  is an angle in standard position, and the point ( x, y) is any point on the terminal side of  other than the srcin, then the six trigonometric functions of  are defined as follows:

2

The sine of  :

sin  

The cosine of  :

cos  

The tangent of  :

tan 

The cotangent of  :

cot  

The secant of  :

sec  

The cosecant of  :

csc  

2

2

where r = x + y or r 

y r x r y ; x0 x x ; y0 y r ; x0 x r ; y0 y

x 2  y 2 . That is, r is the distance of the point (x, y) from the srcin.

Example 5. Find the six trigonometric functions of  if  is in standard position and the point (2, -3) is on the terminal side of .

(-2,3) 

Figure 6 Solution: Using the values x = -2 and y = 3, we solve for r. 2

r

2

x  y  13

Applying the definition for the six trigonometric functions, we have sin 

3

y 



cos





3

y 



x

sec 

13

r

ta n



2

x 

cot 

13

r



csc 

2





x y r x x y

13 

3 13

 

2 2

 

3

6. Solving Right Triangles To solve a right triangle is to know the measurement or to approximate the sizes of its acute angles and sides when the measurements of at least one leg and an angle or of two of its sides are given. We now define the six trigonometric ratios as follows:

sin  

side opposite  hypotenuse

csc 


4

| Mathematics Major [8] cos 

side adjacent to  

tan  

sec 




cot  

side adjacent to 

hypotenuse side adjacent to  side adjacent to  side opposite 

Example 6. Find the value of sin B

A c = 6 cm

B Solution: The sine ratio is

b = 3 cm

a = 3 3 cm side opposite B hypotenuse

C

. The length of the side opposite B

is 3 cm., and the hypotenuse is 6 cm. Therefore, we have

sin B 

side opposite B hypotenuse



3 6

 0.5

Example 7. In right XYZ with right angle Z, mX = 25 and y = 10 units. Solve the triangle. Solutions: First, we draw the triangle.

X z=? y = 10 units Y

Z

x=?

We now solve for the unknown parts of the triangle. mY = 90 - 25 = 65. Now,

ta n X



x x 0 . By substitution, we have tan 25  y 10

x  10 tan 250 x  4.66 Similarly, sin Y 

y 10 0 . Thus, sin 65  z z z z

10 sin 65 0 10 0.906

z  11.04

5


WHAT TO EXPECT MAJORSHIP: MATHEMATICS Focus: Analytic Geometry LET Competencies: 1. Determine the equation of a line given: a. any two points on the line b. a point and the slope of the line c. a point and the slope of the line parallel to the desired line d. a point and the slope of the line perpendicular to the desired line e. the intercepts. 2. Solve involving a. problems the midpoint of a line segment, distance between two points, slopes of lines, distance between a point and a line, and segment division. b. a circle, parabola, ellipse, and hyperbola. 3. Determine the equations and graphs of a circle, parabola, ellipseand hyperbola.

PART 1: CONTENT UPDATE Introduction: This material includes a brief review of the basic terms concerning lines, circles, parabolas, ellipses, and hyperbolas. A straight line is represented by an equation of the first degree in one or two variables, while the circle, parabola, ellipse and hyperbola are represented by equations of the second degree in two variables. A. The Straight Line 1. The distance between two points A(x1,y1) and B(x2,y2) is

( x1  x2 ) 2  ( y1  y2 ) 2 .

2. Slope of a line a) The slope of the non-vertical line containing A(x 1,y1) and B(x2,y2) is m 

y1  y2 y  y1 or m  2 . x1  x2 x2  x1

b) The slope of a line parallel to the x-axis is 0. c) The slope of a line parallel to the y-axis is undefined. d) The slope of a line that leans to the right is positive. e) The slope of a line that leans to the left is negative. 3. The Equation of a line In general, a line has an equation of the formax + by + c = 0 where that a and b are not both zero.

a, b, c are real numbers and

4. Different forms of the equation of a line a) General form: ax + by + c = 0. b) Slope-intercept form: y = mx + b, where m is the slope and b the y-intercept. c) Point slope form: y  y1  m( x  x1 ) where (x1, y1) is any point on the line. d) Two point form: y  y1



y2  y1 ( x  x1 ) where (x1, y1) and (x2, y2) are any two points on the x2  x1

line. e) Intercept form:

x y  a b



1 where a is the x-intercept and b the y-intercept.

5. Parallel and Perpendicular lines Given two non-vertical lines p and q so that p has slope m 1 and q has slope m2. a) If p and q are parallel, then m1 = m2. b) If p and q are perpendicular to each other, then m1m2 = -1. 6. Segment division Given segment AB with A(x1,y1) and B(x2,y2). St. Louis Review Center, Inc – San Pablo City Branch http://www.slreviewcenter.com | https://www.facebook.com/SLRCspc (049)562-2239 | 0917-584-8106

1


x1  x2 y1  y2 , ). 2 2 r AP r1 b) If a point P divides AB in the ratio 1 so that  , then the coordinates of P( x,y) can be r2 PB r2 rx r x r y r y obtained using the formula x  1 2 2 1 and y  1 2 2 1 . r1  r2 r1  r2 a) The midpoint M of segment AB is M (

7. Distance of a point from a line The distance of a point A(x1,y1) from the line A x + By + C = 0 is given by

d



Ax1  By1  C A2  B 2

.

B. The Circle 1. Definition. A circle is the set of all points on a plane that are equidistant from a fixed point on the plane. The fixed point is called thecenter, and the distance from the center to any point of the circle is called the radius. 2. Equation of a circle a) general form: x2 + y2 + Dx + Ey + F = 0 b) center-radius form: (x – h)2 + (y – k)2 = r2 where the center is at (h,k) and the radius is equal tor. 3. Line tangent to a circle A line tangent to a circle touches the circle at exactly one point called the point of tangency. The tangent line is perpendicular to the radius of the circle, at the point of tangency. C. Conic Section A conic section or simply conic, is defined as the graph of a second-degree equation in x and y. In terms of locus of points, a conic is defined as the path of a point, which moves so that its distance from a fixed point is in constant ratio to its distance from a fixed line. The fixed point is called the focus of the conic, the fixed line is called thedirectrix of the conic, and the constant ratio is called theeccentricity, usually denoted by e. If e < 1, the conic is an ellipse. (Note that a circle has e=0.) If e = 1, the conic is a parabola. If e > 1, the conic is hyperbola.

D. The Parabola 1. Definition. A parabola is the set of all points on a plane that are equidistant from a fixed point and a fixed line of the plane. The fixed point is called thefocus and the fixed line is the directrix. 2. Equation and Graph of a) The equation ofaa Parabola parabola with vertex at the srcin and focus at (a,0) is

y2 = 4ax . The parabola

opens to the right if a > 0 and opens to the left ifa < 0. b) The equation of a parabola with vertex at the srcin and focus at (0,a) is

x2 = 4ay. The parabola

opens upward if a > 0 and opens downward ifa < 0. c) The equation of a parabola with vertex at (h , k) and focus at (h + a, k) is

(y – k)2 = 4a(x – h).

The parabola opens to the right ifa > 0 and opens to the left ifa < 0. d) The equation of a parabola with vertex at (h , k) and focus at (h, k + a) is

(x – h)2 = 4a(y – k).

e) The parabola opens upward if a > 0 and opens downward ifa < 0. f)

Standard form:

(y – k)2 = 4a(x – h) or (x – h)2 = 4a(y – k)

g) General form: y2 + Dx + Ey + F = 0, or x2 + Dx + Ey + F = 0 3. Parts of a Parabola a) The vertex is the point, midway between the focus and the directrix. b) The axis of the parabola is the line containing the focus and perpendicular to the directrix. The parabola is symmetric with respect to its axis. St. Louis Review Center, Inc – San Pablo City Branch http://www.slreviewcenter.com | https://www.facebook.com/SLRCspc (049)562-2239 | 0917-584-8106

2


c) The latus rectum is the chord drawn through the focus and parallel to the directrix (and therefore perpendicular to the axis) of the parabola. d) In the parabola y2=4ax, the length of latus rectum is4a, and the endpoints of the latus rectum are (a, -2a) and (a, 2a).

L

In the figure at the right, the vertex of the

Px

C

parabola is the srcin, the focus is F(a,o), the directrix is the line containing LL' , the axis is the x-axis, the latus rectum is

O

the line containing CC ' .

F(a, 0)

x

C’

L'

y

–

4/3 = 0

y

L

x1=0 O

y2=0

x P(x,y)

V(3,2)

F(0,-4/3)

F(5,2)

O

’

The graph of x

2

 

16 3

y. The graph of (y-2)2 = 8 (x-3).

E. Ellipse 1. Definition. An ellipse is the set ofall points P on a plane such that the sum of the distances ofP from two fixed points F’ and F on the plane is constant.Each fixed point is called focus (plural: foci). 2. Equation of an Ellipse a) If the center is at the srcin, the vertices are at (  a, 0), the foci are at ( c,0), the endpoints of the minor axis are at (0,  b) and b

2




b) If the center is at the srcin, the vertices are at (0, minor axis are at ( b, 0) and b

2





x2 a2



y2 b2



1.

a), the foci are at (0,  c), the endpoints of the


x2 b2



y2 a2



1.

c) If the center is at (h, k), at (h a, k), the foci are at (h c,k), the endpoints of the minor axis are at


3


(h, k b) the distance between the vertices is 2a, the principal axis is horizontal and b then the equation is

( x  h) 2

a

( y  k )2



2

b2



2



a2  c2 ,

1.

d) If the center is at (h, k), the distance between the vertices is2a, (h, k a), the foci are at (h, k c), 2 the endpoints of the minor axis are at (h  b, k),the principal axis is vertical and b

the equation is

( y  k)

2 

a2

( x  h)



a 2  c 2 , then

2 

b2

1.

3. Parts of an Ellipse. For the terms described below, refer to the ellipse shown with center at O, vertices at V’(-a,0) and V(a,0), foci atF’(-c,0) and F(c,0), endpoints of the minor axis at B’(0,-b) and B(0,b), endpoints of one latus rectum at G’ (-c, 

b

2

a

) and G(-c,

and the other at H’ ( c, 

b

b

2

a

)

2

a

) and G(c,

b

2

a

).

y

B(0,b)

2

2

(c, ba )

(c, ba )

V’(-a,0)

O

F’(-c,0)

V(a,0)

F(c,0)

x

2

2

(c, ba )

(c, ba ) B’(0,-b)

a) The center of an ellipse is the midpoint of the segment joining the two foci. It is the intersection of the axes of the ellipse. In the figure above, point O is the center. b) The principal axis of the ellipse is the line containing the fociand intersecting the ellipse at its vertices. The major axis is a segment of the principal axis whose endpoints are the vertices of the ellipse. In the figure, V 'V is the major axis and has length of 2a units. c) The minor axis s i the perpendicular bisector of themajor axis and whose endpoints are bothon the ellipse. In the figure, B' B is the minor axis and has length 2b units. d) The latus rectum is the chord through a focus and perpendicular to the major axis. G' G and H ' H 2b

2

are the latus rectum, each with a length of . a y

y (0, 3) (-4, 9 ) 5

(-5,0)

(2,6)

(4, 9 ) 5

(4,0)

(-4,0)

(8,5) (-6,4)

(5,0)

O

x

(2,1)

(-8 1)

(12 1)

,

(-4,- 9 ) 5

(0, -3)

(4,- 9 ) 5

,

x

O (8,3) (2,-4)


4


The graph of

x2 y 2  25 9



1.

( x  2)

The graph of

100

2 

( y  1)

2

25



1.

4. Kinds of Ellipses a) Horizontal ellipse. An ellipse is horizontal if its principal axis is horizontal. The graphs above are both horizontal ellipses. b) Vertical ellipse. An ellipse is vertical if its principal axis is vertical. F. The Hyperbola 1. Definition. A hyperbola isthe set of points on a plane such that the difference of the distances of each point 2.

on the set from two fixed points on the plane is constant. Each of the fixed points is called focus. Equation of a hyperbola a) If the center is at the srcin, the vertices are at (  a, 0), the foci are at ( c,0), the endpoints of the minor axis are at (0,



b) and b 2




b) If the center is at the srcin, the vertices are at (0,



x2 a2



y2 b2



1.

a), the foci are at (0, c), the endpoints of the minor

x2 1. b2 c) If the center is at (h, k), the distance between the vertices is 2a, the vertices are at (h  a, k), the foci are at (h c,k), the endpoints of the minor axis are at (h,k b),the principal axis is horizontal and axis are at ( b, 0) and b

b2



2





( x  h) 2

a

2



( y  k )2

b2



y2 a2



1.

d) If the center is at (h, k), the distance between the vertices is 2a, the vertices are at (h, k  a), the foci are 

2



c), the endpoints of2 the minor2 axis are at (h b, k),the principal axis is vertical and b ( y  k) ( x  h) then the equation is  1 a2 b2 at (h,k



2

c



2

a ,

2. Parts of a hyperbola. For the terms described below, refer to the hyperbola shown which has its center at O, vertices at V’(-

a,0) and V(a,0), foci at F’(-c,0) and F(c,0) and endpoints of one latus rectum atG’ (-c,  b

2

a

H’ (c, 

) and the other at

b

2

a

) and

H(c,

b

b

2

a

) and

G(-c,

2

a

).

y

2

(c, ba )

B(0,b)

2

(c, ba )

V’(-a,0) F’(-c,0)

O V(a,0)

2

2

(c, ba )

x

F(c,0)

B’(0,-b)

(c, ba )

a) The hyperbola consists of two separate parts called branches. b) The two fixed points are called foci. In the figure, the foci are at  ( c,0). c) The line containing the two foci is called the principal axis. In the figure, the principal axis is the xaxis. St. Louis Review Center, Inc – San Pablo City Branch http://www.slreviewcenter.com | https://www.facebook.com/SLRCspc (049)562-2239 | 0917-584-8106

5


d) The vertices of a hyperbola are the points of intersection of the hyperbola and the principal axis. In the figure, the vertices are at ( a,0). e) The segment whose endpoints are the vertices is called thetransverse axis. In the figure V 'V is the transverse axis. 2

2

2

The line segment with endpoints (0,b) and (0,-b) where b  c  a is called the conjugate axis, and is a perpendicular bisector of the transverse axis. g) The intersection of the two axes is the center of the hyperbola . h) The chord through a focus and perpendicular to the transverse axis is called alatus rectum. In the f)

G’ (-c, 

figure, G' G is a latus rectum whose endpoints are length of

2b

b

2

a

) and

G(-c,

b

2

a

) and has a

2

.

a

3. The Asymptotes of a Hyperbola Shown in the figure belowis a hyperbola with two lines asextended diagonals of the rectangle shown.

y y

 

b a

y

x

b



a

x

P

(0,b)

(-a,0)

O

x

(a,0) (0,-b)

These two diagonal lines are said to be the asymptotes of the curve, and are helpful in sketching the graph of a hyperbola. The equations of the asymptotes associated with the equations of the asymptotes associated with

y2 a2



x2 b2



x2 a2



y2 b2

1 are y 



1 are y 

b b x and y   x . Similarly, a a

a a x and y   x . b b

y

y (6,9)

(-9,6)

F(0,6)

(9,6) 3y



x



0

(0,3) F (-6,0) (-3,0) ’

O

(3,0)

F(6,0)

x x

O (0,-3) 3y

(6,-9)




x



0

F (0,-6) ’

6


The graph of

x2 9



y2 27



1.


The graph of

y2 9



x2 27



1.

7


LICENSURE EXAM FOR TEACHERS (LET) Refresher Course for 2015

CALCULUS PART I

–

CONTENT UPDATE

Theorems Limits of Functions We use theon following theorems to evaluate limits of functions: 1. Uniqueness Theorem: If the limit of a function exists, then it is unique. That is, if lim f ( x )  L1 and lim f ( x )  L2 , then L1  L2 . xa

x a

2.

If m, b  R , then lim( mx  b)  ma  b

3.

If f ( x)  c , a constant, then lim c  c

4.

lim x  a

5.

If lim f ( x)  L1 and lim g ( x)  L2 then

xa

xa

xa

xa

x a

5.1 lim[ f ( x)  g ( x)]  L1  L2 xa

5.2 lim[ f ( x )  g ( x)]  L1  L2 xa

 f ( x)  L1 provided L2  0   g ( x)  L2  If lim f ( x )  L and n  Z then

5.3 lim x a 6.

xa

6.1 lim[ f ( x)]n  Ln x a

6.2 lim

n

x  a1

f ( x) 

n

L with the restriction that if n is even, L  0.

7.

lim f ( x )  L if and only if lim f ( x)  lim f ( x )  L

8.

If r is any positive integer, then 1 8.1 lim r  

x a

xa

x0

8.2 lim x 0

8.3 lim x 0

9.

x

1

xr 1 xr

  r , odd    r , even  

x  a f ( x )  0 and lim x  a g ( x )  c where c Let a  R, lim  R, c  0

9.1

9.2

9.3

9.4 10.

xa

Let

 g ( x)      f ( x)   g ( x)  If c  0 and if f ( x) approaches 0 through negative values of f ( x) , then lim    x  a f ( x)    g ( x)  If c  0 and if f ( x) approaches 0 through positive values of f ( x) , then lim    x  a f ( x)    g ( x)  If c  0 and if f ( x) approaches 0 through negative values of f ( x) , then lim    x  a f ( x)   lim f ( x)   , lim g ( x)   and lim h( x )  c where c is any If c  0 and if f ( x) approaches 0 through positive values of f ( x) , then lim x a

xa

xa

xa

constant, then 1


10.1 lim[ f ( x )  h( x)]   x a

10.2 lim[ g ( x)  h( x)]   xa

10.3 if c  0, lim[ f ( x)  h( x)]   x a

10.4 if c  0, lim[ f ( x)  h( x )]   xa

10.5 if c  0, lim[ g ( x)  h( x )]   xa

10.6 if c  0, lim[ g ( x)  h( x)]   xa

If p  Z , c  R then lim

11.

II.

c



x  

xp

 0 and lim

x  

c xp

0

Theorems on Differentiation In getting the derivative of a function, one or more of the following theorems may be applied: 1. If f ( x)  c, where c is a constant, then f ' ( x)  0 n 2. If f ( x)  x , where n  Z  0, then f ' ( x)  nx n

1

3. If f ( x)  g ( x)  h( x), then f ( x)  g ' ( x)  h' ( x) ,

4. If f ( x)  g ( x)  h( x), then f ' ( x)  g ( x)h' ( x)  h( x) g ' ( x) 5. If f ( x)  cg ( x), where c is a constant, then f ' ( x)  cg ' ( x) 6. If f ( x) 

g ( x) h( x) g ' ( x)  g ( x) h' ( x ) , where h( x)  0 , then f , ( x)  h( x ) h( x)2

 [ g ( x)]r , then f ' ( x)  r[ g ( x)]r 1 g ' ( x) 8. If f ( x)  sin[g ( x)], then f ' ( x)  g ' ( x) cos[g ( x)] 9. If f ( x)  cos[g ( x)], then f ' ( x)   g ' ( x) sin[g ( x)] 2 10. If f ( x)  tan[g ( x)], then f ' ( x)  g ' ( x) sec [ g ( x)] 7. If r is any rational number, f ( x)

2

11. If f ( x)  cot[g ( x)], then f ' ( x)   g ' ( x) csc [ g ( x)] 12. If f ( x)  sec[g ( x)], then f ' ( x)  g ' ( x) sec[g ( x)] tan[g ( x)] 13. If f ( x)  csc[g ( x)], then f ' ( x)   g ' ( x) csc[g ( x)] cot[g ( x)] 14. If f ( x)  e

g ( x)

15. If f ( x)  a

g ( x)

then f ' ( x)  e

g ( x)

where a is any constant not equal to 0, then f ' ( x)  a

16. If f ( x)  ln[g ( x)] then f ' ( x) 

III.

g ' ( x) 1

g ( x)

g ( x)

ln(a) g ' ( x)

g ' ( x)

Formulas for Antiderivatives The following are known formulas for antiderivatives 1. 2. 3.

 dx  x  c  af ( x)dx  a  f ( x)dx where a is a constant.  [ f ( x)  f ( x)]dx   f ( x)dx   f ( x)dx 1

2

1

2

 x n 1   c, n  1 4. If n is a rational number ,  x dx   n  1  ln x  c, n  1 5.  sin udu   cos u  c n

 cos udu  sin u  c 7.  sec udu  tan u  c 8.  sec u tan udu  sec u  c 9.  csc udu   cot u  c 10.  csc u cot udu   csc u  c 11.  tan udu  ln sec u  c 6.

2

2

cot udu  ln sin u  c

12.

 2


13. 14.

 sec udu  ln sec u  tan u  c  csc udu  ln csc u  cot u  c

 u n 1   c , n  1 15.  u du   n  1  ln u  c, n  1 16.  eu du  eu  c n

17.

IV.

1

 a du  ln a a u

u

c

Areas of Plane Regions Below are some formulas for finding the areas of plane regions 1. Area (A) bounded by the curve y  f ( x) , the x-axis and the lines x  a and x  b b

A

 f ( x)dx a

2. Area (A) bounded by the two curves y  f1( x)

and

y  f2 ( x) for all

x  [a , b] b



A  [f2 ( x )  f2 ( x )]dx a

V.

Volumes of Solids of revolution We use the following methods to find the volumes of solids of revolution 1. Disk Method The region bounded by the curve y  f ( x) , the x-axis and the lines x  a and x  b , is revolved about the x-axis. b



V   [f ( x )]2 dx a

2. Ring Method The region bounded by the two curves y  f1( x)

and

y  f2 ( x) ,

where f1(x) > f2(x) for all x  [a , b], is revolved about the x-axis b





V   [ f1( x )]2  [ f2 ( x )]2 dx a

3. Shell Method 3.1 The region bounded by the curve y  f ( x ) , the x-axis and the lines x  a and x  b , is revolved about the y-axis. b



V  2 xf ( x )dx a

3.2 The region bounded by the two curves y  f1( x)

and

y  f2 ( x) , where f1(x) > f2(x) for

all x  [a , b], is revolved about the y-axis b



V  2 x[f1( x )  f2 ( x )]dx a

3

Geometry and Measurement

Mathematics Major

Basic Algebra


Advanced Algebra

Basic Calculus

Trigonometry

Licensure Examination for Teachers Prepared by: Prof. Henry P. Contemplacion

Analytic Geometry

COMPLEX NUMBERS

25

COMPLEXNUMBERS

RADICALS

RADICALS

ALGEBRA

ALGEBRA

ALGEBRA

What must be subtracted from

Find the sum of (a2 – b2 + 2ab)

3x3 –

x2

and (a2 –

x3

x2 –

+

+ 5x – 2 to get

2ab + b2).

2x + 1 ?

2x3 – 2x2 + 7x

–

3

ALGEBRA

If x = -2, y = -1 and z = 2, What is the value of 2x2y3z2?

2a2

1. What fraction of 3 1/4 is ELEM. MATHEMATICS 1 1/3? 2. Find the smallest positive integer that has factors of 9,12 and 18.

-32

3. Give the value: 33 1/3% of 12 ELEM. MATHEMATICS plus 66 2/3% of 18 minus 12 ½% of 16 4. Write 0.243243243243243.... in fraction. 5. How much interest would be paid on a bank loan of P50,000 for 9 months at 12% annual interest?

Number Theory The simplest expression for 340/920 is _____________.

1

Number Theory Find the difference between the largest and smallest prime no. from 1 to 100.

95

Number Theory How many factors of set X are factors of any members of set Y? X = (2,3,4,5,6) Y=(3,4,8,9,10)

4

If x³ is Number an odd number Theory and y³ is an even number, which of the ff. is/are TRUE ? I. II. III. IV.

x³ - y³ is an even number x³ + y³ is an odd number x³ y³ is an even number x³ - y³ + 1 is an odd number

II & III Kevin has collection of match boxes. If NUMBER he counts the match THEORY boxes by twos, threes, fours, fives, or sixes, there is always one left over. If he counts the match boxes by sevens, none remain. What is the fewest number of match boxes Kevin could have?

The green neon lights flash every 4 NUMBER THEORY seconds; the blue, every 5 seconds; the red, every 6 seconds. If all the lights flash together at 8:00 PM, at what time will they flash together again?

8:01 PM Arithmetic and Business Mathematics

A 25-hectare farm land produces 350 cavans of palay. How many cavans can 60-hectare land produce?

Answer: 840

301 Arithmetic and Business Mathematics


What is the smallest number that can be divided by all the numbers 1 to 10?

How much is 30% of 70% of 500?

Answer: 2520


What is the difference between 38 and 83 ?

Answer: 105


A water pipe 10 feet long was cut into 2 with the shorter part measuring 2.5 ft. Find the ratio of the smaller part to the larger part.

Answer: 6049

Answer: 1:3



A bond paper is folded in half, after which it is folded again in half, and again in half. If this folding is done 5 times, how many rectangles on the srcinal sheet of paper will be?

A merchant sold an item for P35,000, in the process earning for himself 15% profit. How much is the profit?

Answer: 32


A graduating class of 160 boys and 180 girls held a seniors’ ball. Approximately what percent attended if 40% of the girls and 30% of the boys were there?

Answer: 35.29%


A large container contains 12 liters when 30% full. How much does it contain when it is 90% full?

Answer: 36 liters


If you are currently earning P22,000 a month, how much will new salary be if you are your to receive a 4% increase next month?

Answer: P22,880

Answer: P5,250


How much would P180,000 amount to in 20 years if it earns 9% per annum?

Answer: P504,000


A book is 2 ½ cm thick. How many copies of this book can be placed in a shelf one meter long?

Answer: 40


Find the least common multiple of the numbers 18, 24 and 54?

Answer: 216


28 students passed while 20% failed in the midterm exam. How many students were tested?

Answer: 35


What is the number 2/3 of whose square root is 10?

Answer: 225


A farmer has enough feeds for 72 ducks for 14 days. If he sells 16 ducks, how long will the feeds last?

Answer: 18


A gas tank of Mr. Reyes car is 5/8 full. If he used 3/5 of this amount travelling from his house to his province, what part of the tank has gas?

Answer: 1/4


How many boys are there if 33 1/3% of 1500 first year high school students are girls?

Answer: 1000


A housewife deposited P150,000 at 6% compounded annually for 5 years. What is the amount in 5 years?

Answer: P200,733.84


In how many years will P50,000 grow to P60,000. If it is invested at 6% compounded monthly?

Answer: 3.05


Successive discounts of 20% and 30% is equivalent to what single discount?

Answer: 44%



A defective ruler was to be found 11.5 in. long. Using this ruler a boy was found to be 4 ft. tall. What is the actual height of the boy?

The average of 6 consecutive no. is 17.5. What is the average of 5 consecutive no.?

Answer: 4ft. 2 in.

FRACTION If 2/5 of x is 20, what is 1/2 of x?

Answer: 17

FRACTION A water tank is 5/6 full. If 1/3 of the water was used for cleaning the car, what part of the tank has water?

25

5/9

FRACTION I used 2/5 of the money in first store and 1/3 in a second store. If I had P56 left how much money did I have at first?

In one class, 1/4 of the students FRACTION

210

RATIO A 10-meter log was cut off at 75 dm. mark. What is the ratio of the smaller piece to the larger piece?

1:3

are honor students. Of these honor students, 2/7 are varsity athletes. If there are 6 athletes in the class how many students are there in the class?

84

RATIO Divide P4340 among A, B and C so that A’s share; B’s share; C’s 1 1 1 share is 2 : 3 : 5 , respectively. How much is the share of each?

2100, 1400, 840


Half of the people in the room left. One-fourth of the remaining started to dance, there were then 12 people who are not dancing. Find the srcinal number of the people in the room.

Answer: 32

PERCENT An employee spends about P3,332 a month. This sum is 70% of his monthly salary. How much does he receive a month?

P4,760 An agency with 3,200 PERCENT employees plans to reduce its staff by 35%. How many employees are to be retained?

2080

PERCENT There are 1400 pupils in Kaiba Central School. Of these students, 12.5% are in Grade VI and 20% themany rest are in are Grade V. of How pupils there in Grade V?

245


What number when increased by 90% of itself equals 133?

Answer: 70

The srcinal price of a pair PERCENT of jogging pants was P300. What was the rate of discount after paying P195?

35%

PERCENT A tank was 50% full of water. 600 liters of water are added making it ¾ full. How many liters does it hold when full?

2400 L

FRACTION/ PERCENT If the length of the rectangle is reduced by 1/5 and width is also reduced by 1/5, what percent of the srcinal area is the new area of the rectangle?

64%


In the Education Dept., 3/5 are BSED students and the rest are BEED students. Of these BSED students, 1/3 are Math majors. If there are 20 Math majors, how many BEED students are there?

Answer: 40


How many hectares are in a rectangular field that is 750 m wide and 800 m long?

Answer: 60 ha


The outer diameter of a metal pipe is 3 ¼ cm and the inner diameter is 2 ¾ cm. Find the thickness of the pipe.

Answer: ¼ cm


The supplement of an angle is three times its measure. What is the supplement?


A mile is about 1.609 km. How many meters is a mile longer than a kilometer?

Answer: 609


The length and width of a rectangle are 3x –2 and 2x + 1, respectively. What is the perimeter of the rectangle?

Answer: 10x – 2


If the perimeter of a rectangle is 72 cm and its length is 6 cm longer than its width. What is its length?

Answer: 21 cm


Rico cuts out the largest circle possible from a square whose side is 10 m. What is the area of the circle?

Answer: 25π or Answer: 135°

78.54 sq.m.


If the width of a rectangle is denoted by w and the length is 4 times the width, what is the perimeter of the rectangle?


Find the measure of an angle if the measure of its supplement is 50 more than twice the measure of its complement.

Answer: 10w

Answer: 50°



The length of an equilateral triangle is 16 cm. Find its area.

Answer: 64


What do you call a regular polygon with 170 diagonals?

Answer: Icosagon

Each interior angle of a regular polygon measures 162°. How many sides does it have?

Answer: 20


If arc AB is 72°, then the measure of angle α?

Answer: 36°





interior of the circle)

Answer: 54°

exterior of the circle)

Answer: 50°


If arc AC has a length of 228° and arc CA 132°, then the measure of angle ABC is ____. (Angles formed by two tangent lines)

Answer: 48°


A rectangular lot has a perimeter of 120 m. If the length of the lot is 20 m. more than its width, what is its area?

Answer: 800 sq. m.


A picture 10 cm x 8 ½ cm is mounted on a piece of hard cardboard. If there is a margin of 2 ½ cm around the picture, what is the perimeter of the cardboard used?

Answer: 57 cm


How much larger is the supplement of 54° angle than the complement of a 65° angle?


A room of 10 m by 7m. There is 7.5 m by 5 m carpet in the middle. What percent of the room is uncovered? ?

Answer: 46.4%


If the length of the rectangle is reduced by 1/5 and width is also reduced by 1/5, what percent of the srcinal area is the new area of the rectangle?

Answer: 101°

Answer: 64%



What is the sum of 43 dm and 2.5 m in centimeters?

What is the edge of a cube that has the same volume as a rectangular box with dimensions 18 in by 8 in by 1.5 in?

Answer: 680 cm

Answer: 6 inches

Algebra

What is the value of (16 1/3)(16 1/16)?

Algebra

(16 ½)

Answer: 4

Algebra

Find the numerical value of 36 3/2?

Answer: 216

Algebra

The product of two numbers is 60 and their reciprocal is 4/5, what are the numbers? Answer: 10 and 6

Algebra If Maria can address a box envelopes in 5 hours and Jane can address the same box of envelopes in 10 hours, how many minutes will it take Maria and Jane working together to address all envelopes in the box?

Answer: 200

If (3 100) (4 100)(5 100) = 2 2a 15 50b then a + b = ____.

Answer: 12

Algebra

Given: f(x) = - 2x 2 - 3x, f(-5)= ______.

Answer: - 35

Algebra

If the sum of two numbers is 5 and the difference of the two number is 5, what is the product of the numbers? Answer: 0

Algebra Mario sold five pigs and four goats for P21,000. Selling the animals at the same , Ben sold three pigs and 2 goats for P12,000. What is the selling price of each animal?

Answer: 3000 and 1500

Algebra

Algebra

Find the sum of the first 50 multiples of 4, 4, 8, 12, 16,…,

a50

Answer: 5100

Algebra

In the expansion of (x + the fourth term is:

Simplify the expression

log4 3 * log3 32

Answer: 5/2

Algebra

3y)7

Answer: 945x4y3

Algebra

If y = 3x and 2x + 3y = 22, then y = ____.

Answer: 6

Algebra

If f(x) = - 2x 2 - 3x, then f(-5) = ____.

Answer: - 35

Algebra At an amusement park, Leo bought 3 hamburgers and 4 sodas for a total of P150. While paying the same prices Kaye bought 2 hamburgers and 3 sodas for P105. What is the total cost of 1 hamburger and 1 soda?

Answer: P45

Algebra

If x > 0 and (22x - 1 )(4x + 2) =

If x > 0 and (22x - 1 )(4x + 2) =

(8x + 3)

, then x = _____.

(8x + 3) , then x = _____.

Answer: 3

Answer: 3

Basic Trigonometry

What is the value of sin 75°?

Trigonometry

Determine the length of the arc cut by a central angle of 60° in a circle with radius of 2 cm.

Answer: 2π/3

Answer:

Trigonometry

Trigonometry

An angle of one revolution is equal to an angle whose measure in terms of radians as:

A ladder 18 feet long leans against a building forming angle 60° with the ground. How high up the side of the building?

Answer: 2π

Answer:

Trigonometry From a point level with 1000 ft. away from the base of a monument, the angle of elevation to the top of the monument is 35°. Determine the height in the nearest feet.

Trigonometry If in ΔABC, sin A = 3/5, then what is the value of sin B?

Answer: 700 ft.

Answer: 4/5

Trigonometry In ΔABC, C s a right triangle and tan A = 1. What is the value of sin A + cos A?

Trigonometry

If the ratio of sec x to csc x is 1:4 then the ratio of tan x to cot x is

Answer:

Answer: 1:16

Trigonometry

Find the exact value of tan - 5π/3.

Answer:

Trigonometry

A ladder 30 ft. long leans against a wall, with its foot 8 ft. from the wall. Find the nearest degree which the ladder makes with the ground? Answer: 75°


Find the midpoint of the

segment joining A(-4,5) and B(2,-3).

Answer: (-1,1)


Find the slope of the line joining the points (3,2) and (-1,-2).

Answer: 1

Trigonometry

Find the nearest feet the height of the flagpole which cast a shadow 44.8 ft. long when the angle of elevation of the sun is 36°? Answer: 32 ft.


Find the distance between

A(3,2) and B(-1,4).

Answer:


The midpoint of a line segment M(1,2). One of the endpoints is A(0,3). Find the other endpoint B. Answer: (2,-7)


Find the equation of the line through point (1,-2) of slope 2.

Answer: 2x - y = -4

Basic Calculus

Basic Calculus

Find the derivative of

Differentiate

f(x) = 4x3

f(x) = 6x4 – 5x3 + x2 – 6x +7

Answer: 12x2

Answer: 24x3 – 15x2 + 2x – 6

Basic Calculus

Find the derivative of f(x) = (6x2 + 2x)(3x – 1)

Answer: 54x2 – 2

Basic Calculus

Differentiate y = (2x2 – 4x)2

Answer: y’ = 16x3 – 48x2 +32x

Basic Calculus

Find the derivative of y=

Answer:

Basic Calculus

Evaluate

Answer:

Basic Calculus

Evaluate

Answer:

Basic Calculus

Evaluate

Answer:

Mathematics | Prof. R. Diaz LICENSURE EXAMINATION FOR TEACHERS (LET) Refresher Course for Year 2012 Area: Mathematics DAY 1

PART II – ANALYZING TEST ITEMS 1. How many 1/8’s are there in 2 5/8? A. 21 C. 23 B. 22 D. 24 Solution A: Using diagrams and visualization

There are twenty-one 1 ‘s in 5 ? 2 8 8 Solution B: Divide: 2 5/8 by 1/8 2 5/8 ÷ 1/8 = 21 2. A blue neon light blinks every 4 seconds. A red one blinks every 5 seconds while a green one blinks every 6 seconds. How many times will they blink together in one hour? A. once C. 20 times B. 10 times D. 60 times Solution: To find the number of seconds it takes for the neon lights to blink together, look for the least common multiple of 4, 5 and 6. Find the LCM using the prime factorization method. 2 2 LCM 4: 2 x2 =2 4: 2 5: 6:

5 = 5 2x3=2x3

5: 5 6: 2 x 3 2 2 x 3 x 5 = 60 Multiply all the prime factors in each column with the highest exponent. 2 2 x 5 x 3 = 60 The LCM is 60. It means that the three neon lights blink together every 60 seconds or every minute. Since there are 60 minutes in one hour, the neon lights will blink together 60 times in one hour (Option D). 3. Which of the following is divisible by 3, but not by 9? A. 11, 070 C. 45, 072 B. 20, 103 D. 19, 386 Solution: Find a number whose digital root, or the sum of all its digits, is multiple of 3 but not of 9. A. 11, 070 1 +1+0+7+0=9 B. 20, 103 2+0+1+0+3=6 C. 45, 072 4 + 5 + 0 + 7 + 2 = 18 1+8=9 D. 19, 386 1 + 9 + 3 + 8 + 6 = 27 2+7=9 Options A, C and D are multiples of 3 and 9. Options B is a multiple of 3 but not of 9. The answer is B. 4. The distance between two towns on a given map is 2¾ cm. If ½cm in the map is equal to 6km, find the actual distance between the two towns. A. 18 km C. 36 km B. 33 km D. 42 km

Solution A: ½ cm: 6 km = 2 ¾ cm : N Product of means = Product of extremes 2 ¾ cm x 6 km = ½ cm x N 11 cm x 6 km = ½ cm x N 4 33 cm km = ½ cm x N 2 33  2km = N 2 33 km = N St. Louis Review Center, Inc – San Pablo City Branch http://stlouisreviewcenter-slrc.blogspot.com (049)562-2239 | 0905-914-6145

1

Mathematics | Prof. R. Diaz Solution B: 2 cm = ½ cm + ½ cm + ½ cm + ½ cm ¾ cm = ½ cm + ¼ cm

There are five ½ cm. There is one ¼ cm

5 x 6 km = 30 km (since ½ cm = 6 km) + 3 km = 3 km (since ¼ cm = 4 km) 33 km

The answer is option B. 5. Margo paid P400 for a blouse. If the blouse was sold at 20% discount, what was its srcinal price? A. P80 C. P500 B. P480 D. P540 Solution: Sale price = Original price (N) – discount P400 = N – 0.20 N P400 = 0.80 N 400 =N 0.80 P500 = N The answer is option C. 2 6. The area of a rectangle is 117 cm . Find its perimeter if its length is 13 cm. A. 9 cm C. 44 cm B. 30 cm D. 54 cm Solution: Area of a rectangle = length x width. Since the length is 13 cm and the area is 117 sq. cm. W = A = 117 = 9 cm L 13 Now, we solve for the perimeter. P = 21 + 2w = 2 (13cm) + 2 (9cm) = 26 cm + 18 cm = 44 cm The answer is option C. 7. What is the sum of the first five prime numbers? A. 11 C. 26 B. 18 D. 28 Solution: The first five prime numbers are 2, 3, 5, 7, and 11. Their sum is 28. Remember that 1 is neither a prime nor a composite number. The answer is option D. 8. In a sequence of starts and stops, an elevator travels from the first floor to the fifth floor and then to the second floor. From there, the elevator travels to the fourth floor and then to the third floor. If the floors are 3 meters apart, how far has the elevator traveled? A. 18 m C. 30 m B. 27 m D. 45 m 9. 14.3 is equal to A. 0.143 x 100 C. 143 100 B. 14.3 x 100 D. 0.143 x 1000 10. Which of the following has the greatest value? A. 0.351 C. 3/10 B. 35/100 D. 3/9 11. Which of the following numbers is greater than -3/2? A. -4/3 C. -7/4 B. -3 D. -2 12. Simplify 1/3 + 2/5 – 2/6. A. 1/14 C. 2/3 B. 1/5 D. 2/5 13. Find the product: (2 ½ ) (5/7) (2/5) A. 0 C. 1 B. 5/14 D. 5/7 14. Simplify: 2(-3)2 – (-4) (5) - 2 A. -4 C. 28 B. 14 D. 10 St. Louis Review Center, Inc – San Pablo City Branch http://stlouisreviewcenter-slrc.blogspot.com (049)562-2239 | 0905-914-6145

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Mathematics | Prof. R. Diaz 15. Dante is working on a hill 78 meters above the sea level, while directly underneath. Mark is working in a tunnel 38 meters below the sea level. What is the distance between Dante and Mark? A. 116 cm C. -42 m B. 42m D. -116m 16. How much would 215 boxes of paper clips cost if each box costs P6.75? A. P1,250.75 C. P1,451.25 B. P1,355,25 D. P1,455.75 17. What is the least common multiple of 9.8 and 18? A. 36 C. 72 B. 54 D. 144 18. A group of young people from four countries gathered together for an international workshop: 60 from Manila, 48 from Jakarta, 54 from Kuala Lumpur and 42 from Bangkok. The participants will form discussion groups with equal number of members from each country in each group. What is the greatest number of discussion groups that can be formed? A. 6 C. 18 B. 12 D. 24 19. What are the two digits which when inserted in the blank spaces will make 934__ __ divisible by 8? A. 3 and 4 C. 5 and 0 B. 3 and 0 D. 0 and 0 20. Which of the following is NOT true about the sum of two consecutive odd integers? A. it is even C. it is only divisible by 12 B. it is divisible by 4 D. it is always divisible by 1 21. The ratio of the number of boys to the number of girls in a class is 2:3. If there 40 students in the class, how many boys are there? A. 8 C. 18 B. 16 D. 24 22. Mrs. Jimenez paid P94.50 for 3 ½ dozens of eggs. How much would 2 dozens of such eggs cost? A. 50.50 C. 55.00 B. 54.00 D. 56.00 23. Arthur divides his day into leisure, sleep and work in the ratio 1:2:3. How many hours does he spend working? A. 4 hr C. 10 hr B. 8 hr D. 12 hr 24. It takes 20 men to build a house for 60 days. How many men will needed to build it in 15 days? A. 5 C. 100 B. 80 D. 120 25. Which of these is a correct proportion? A. 3: 5 = 5: 10 C. 11/2 : 2 = 5:7 B. 7:10 = 15:18 D. 4:9 = 2: 4 ½ 26. Express 24% in fraction form. A. ¼ C. 12/51 B. 5/8 D. 6/25 27. How much larger than 20% of 950 is 40% of 600? A. 50 C. 60 B. 54 D. 68 28. Which of the following is true? A. 12% < 0.8 C. 40% > 0.4 B. 30% = 0.03 D. 25% = 0.5 29. In a class of 40 students, 40% are boys. How many are girls? A. 16 C. 22 B. 18 D. 24 30. John took two hours to construct a model boat. He spend 75% of that time reading the instructions. How much time did he actually spend on the construction of the boat? A. 25 min C. 60 min B. 30 min D. 75 min 31. In a examination, 24% of the students failed in math and 33% in Science. If 9% of the students failed in both subjects, find the percent of students who passed in both subjects. A. 43% C. 52% B. 48% D. 81%

St. Louis Review Center, Inc – San Pablo City Branch http://stlouisreviewcenter-slrc.blogspot.com (049)562-2239 | 0905-914-6145

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Mathematics | Prof. R. Diaz st

32. On board an inter-island ship, 25% of the passengers travel 1 class, 30% travel 2 rd class, and 729 travel 3 class. How many passengers are there? A. 1,254 C. 1,560 B. 1,480 D. 1,620

nd

33. The introductory price of a certain item was P12. What is the percent increase in the price of the item if two years later its price is P42? A. 150% C. 250% B. 200% D. 300% 34. The outer diameter of a metal pipe is 3 ¼ cm and the inner diameter is 2 ¾ cm. Find the thickness of the pipe. A. ½ cm C. 1/8 cm B. ¼ cm D. ¾ cm 35. How many pieces of ribbon, each 75 cm long, could be cut from a roll containing 18 meters of ribbon? A. 12 C. 18 B. 16 D. 24 36. Laura jogs at least 1500 meters everyday on a circular track. If the track has a diameter of 80 meters, about how many rounds does Laura jog everyday? A. 15 C. 9 B. 12 D. 6 37. If a peso coin weighs 15 grams, about how much money will you have if you have a kilogram of 1-peso coins? A. P6 C. P660 B. P66 D. P6, 600 38. A right cylindrical can is to hold exactly 3 liters of water. What should the height of the can be if the radius is 10cm? A. 3.25 cm C. 9.55 cm B. 6.4 cm

D. 12.35 cm

39. Which of the following gives the prime factorization of 54? 2 2 3 3 A. 2 3 C. 2 3 3 3 B. 2 3 D. 2 3 40. Apo weighs one third as much as her mother Angela. If Apo weights 43.5 lbs., how much does Angela weight? A. 62.5 lbs. C. 130.5 lbs. B. 95.7 lbs. D. 150 lbs. 41. If the perimeter of a rectangle is 72 cm and its length is 6cm longer than its width. What is its length? A. 14 cm C. 19 cm B. 17 cm D. 21 cm 42. Two numbers are in the ratio 4:3. What are the numbers if their sum is 84? A. 36 and 48 C. 24 and 63 B. 36 and 24

D. 12 and 36

43. A boat travels 8 kph in still water. If it can travel a given distance upstream in 5 hours and the same distance downstream in 3 hours, what is the rate of the current? A. 2 kph C. 6 kph B. 5 kph D. 9 kph 44. What is the value of A. -1 B. 4

x

– 6 if 2(x – 3) + 5 = 19? C. 16 D. 20

45. Lucille is 4 times as old as Nicole. Six years from now, she will be twice as old as Nicole. How old are they now? A.. Lucille 12; – Nicole -3 C. Lucille – 3, Nicole – 12 B. Lucille – 16; Nicole 4 – D. Lucille -4l Nicole – 16


4

Mathematics | Prof. R. Diaz

PART III – ENHANCING TEST TAKING SKILLS

1. CNN Reports the following temperature for 4 cities: Beijing London Chicago Moscow 0 0 0 0 1. C -5 C 0 C -8 C What city is coldest? A. Beijing C. Chicago B. London D. Moscow 2. In a series of card games. Raul starts out with P100 and wins a total of P400. If he later loses P300, wins P50 and loses P250, how much cash does Raul have? A. 0 C. P100 B. P50 D. P150 3. Your score in a game is -12. How many points must you earn to get a score of 18? A. -6 C. 20 B. 6 D. 30 4. If the sum of a certain number and 8 is divided by 5, the quotient is 5. What is the number? A. 5 C. 15 B. 6 D. 17 5. 0.4 x 0.04 x 0.004 equals A. 0.64 C. 0.0064 B. 0.064 D. 0.000064 6. Jeepney fares are computed as follows: P5.50 for the four kilometers plus P0.50 for every additional kilometer thereof. How much should Isagani pay for a ride that covers 11 kilometers? A. P8.00 C. P9.50 B. P9.00 D. P10.00 7. Which of these is greater than 1? A. 1/3 = ¼ C. 1/3 x ¼ B. 1/3 + ¼ D. 1/3  ¼ 8. Which is a set of equivalent fractions? A. 2/3, 4/6, 8/12 C. 4/5, ¾ , 7/10 B. 2/3, 4/6,7/10 D. ¾, 7/10, 4/5 9. Robert ate ¾ pan of a cake on the table. Karen ate 2/3 of what is left. What fraction of the cake was left? A. 1/12 C. 1/3 B. ¼ D. ½ 10. The distance from C to D is 24 km. The distance from B to C is 2/3 of the distance from C to D. The distance from A to B is 3/8 of the distance from B to C. What is the distance from A to B? A. 4 km C. 6 km B. 5 km B D. 8 km A C

D

11. Bebot buys a refrigerator for P7 920. She pays 1/3 of the price in cash and the balance in 12 equal monthly installments. How much will she pay each month? A. P300 C. P400 B. P350 D. P440 12. Mang Oskar delivers 210 kg of vegetables to two distributors. If 2/5 of the vegetables goes to Manang Linda, how many kilograms of vegetables does the other distributor receive? A. 84 C. 126 B. 116 D. 132 13. A 210-cm stick is cut so that one part is ¾ of the other. How long is the shorter segment? A. 120 C. 80 B. 90 D. 70 14. For the rest of the day, Bus A passes by Station 3 every 45 minutes, while Bus B passes by Station 3 every 1 hour and 15 minutes. What time will both bells ring at the same time again? A. 9.15 C. 10:45 B. 10:30 D. 11:45 15. Find the GCF of 12, 32 and 56. A. 2 C. 6 B. 4 D. 56 St. Louis Review Center, Inc – San Pablo City Branch http://stlouisreviewcenter-slrc.blogspot.com (049)562-2239 | 0905-914-6145

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Mathematics | Prof. R. Diaz 16. Two bells ring at 8 A.M. For the rest of the day, one bell rings every half hour whereas the other rings every 45 minutes. What time will both bells ring at the same time again? A. 9:00 AM C. 9:30 AM B. 9:45 AM D. 10:00 AM 17. Which of the following numbers give a remainder of 2 when divided by 5 and a remainder of 1 when divided by 4? A. 7 C. 17 B. 12 D. 31 18. What number can be inserted in the blank space so that 59___172 is divisible by 6? A. 1 C. 3 B. 2 D. 7 19. Jeff had fewer than 100 blocks. When he made four equal rows, he had three left over, and with nine equal rows, he had none left over. How many blocks did Jeff have? A. 54 C. 72 B. 63 D. 91 20. How many prime numbers are less than 31? A. 9 C. 12 B. 10 D. 13 21. If one prime factor of 42 is 3, what are the other prime factors? A. 2 and 3 C. 2 and 7 B. 3 and 5 D. 3 and 7 22. A kilogram of dried squid costs P240. Jigs wants to buy 400 grams only. How much will it cost? A. P96 C. P120 B. P105 D. P130 23. If there are 18 mangoes and 45 bananas in a fruit basket, what is the ratio of the mangoes to the bananas? A. 2:5 C. 2:3 B. 4:7 D. 3:7 24. A 3½ meter ribbon is cut into four pieces in the ratio 1:2:3:4. Find the length of the longest piece. A. 35 cm C. 100 cm B. 70 cm D. 140 cm 25. What one number can replace x in 1: x = x:64? A. 2 C. 6 B. 4 D. 8 26. In a certain municipality the real estate tax is P5.00 for every P1,000 of the assessed valuation of the property. Find the assessed valuation on property if a tax of P1250 is levied on it. A. P62.50 C. P250.000 B. P6,250 D. P500,000 27. A tank of water supplies the needs of 5 children for 6 days. How long will the water last if there were only 3 children? A. 3.6 days C. 8 days B. 5 days D. 10 days 28. A man gives 10% of his monthly earnings to the church. Another 70% is spent on food, electricity, water and other bills. After paying all these, he still ahs P2,000. How much does he earn in a month? A. P5,000 C. P10,000 B. P8,000 D. P12,000 29. Jake deposited P10,000 at 5% annual interest. How much will it gain in 2 ½ years? A. P1,250 C. P1,500 B. P1,350 D. P1,750 30. In a candy jar, there are 15 lemons, 12 chocolates, and 3 mints. What percent of the candies are mints? A. 10% C. 12.5% B. 12% D. 15% 31. The cellphone model 5110 sold for P4,000 in 2001. Two years later, the same cellphone model sold for P2,800. What was the percent decrease of the price? A. 15% C. 30% B. 20% D. 35% 32. Based on the graph, how much of the P12,500 monthly income of the Margate family goes to food? A. P3,125 C. P5,025 B. P5,625 D. P6,225 rent 25% bills 30% food 45%

33. The perimeter of a rectangular lawn is 37 meters. If it is 8 meters wide, how long is the lawn? A. 9 ½ m C. 10 ½ m B. 10 m D. 11 m 2 34. The area of a square is 225 cm What is its perimeter in centimeters? A. 15 C. 56 St. Louis Review Center, Inc – San Pablo City Branch http://stlouisreviewcenter-slrc.blogspot.com (049)562-2239 | 0905-914-6145

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Mathematics | Prof. R. Diaz B. 44 D. 60 35. A pail contains 5 liters of water. If this will be placed in bottles that should contain 450 milliliters each, how many bottles are needed? A. 10 C. 12 B. 11 D. 13 36. If the volume of a cube with side 6 is equal to the volume of a rectangular solid with length 12 and width 9, then the height of the rectangular solid is A. 2 C. 6 B. 4 D. 8 2 37. The volume of a cylinder is given by the formula V = Лr h. The approximate volume of a cylinder with radius 5 cm and perpendicular height 8 cm is: 3 3 A. 62.8 cm C. 628 cm 3 3 B. 125.6 cm D. 714 cm 38. Dave cuts out the largest circle possible from a square whose edge is 10cm. What is the area of the circle? A. 15.7 cm2 C. 78 cm2 2 2 B. 31.4 cm D. 94.2 cm 39. At the beginning of the month, the school tank was 8/9 full. After 1 week, it was 2/9 full. If the tank’s capacity is 27 cubic meters, how much water was used in 1 week? A. 24 cu.M C. 20 cu.M B. 22cu.M D. 18 cu.M 40. When a certain number is subtracted from 176, the result is the same as when 21 is added to 4 times the number. What is the number? A. 11 C. 31 B. 22 D. 52 41. Suppose we write fractions as ordered pairs. Then 3/5 would be written as (3,5), and 5/7 as (5,7). Following the same rule, what then would be the sum of (6,11) and (2,11)? A. (4,11) C. (4,22) B. (8,11) D. (8,22) 42. If the width of a rectangle is denoted by y and the length is 4 times the width, what is the perimeter of the rectangle? A. 5y C. 15y B. 10y D. 20y 2 43. If x = y , how would x change if y is doubled? A. treble C. quadrupled B. double D. get squared 44. Arman is 5 years older than An. Three years ago, he was twice as old as An. How old is An now? A. 6 C. 13 B. 8 D. 15 45. If Justine can run 3 km in an hour, how far can she run in 45 mins? A. 1.75 km C. 2.5 km. B. 2.15 km D. 2.25 km. 46. Which property of real numbers does the statement (a + b) + c = a + (b + c) illustrate? A. Associative C. Commutative B. Closure D. Distributive 2 47. If 2a + 2b + 3c = 16 and c = 2, what is the value of (a+b) ? A. 5 C. 25 B. 10 D. 625 48. Which of the following is a function? A. {(1,2), (2,3), (3,4)} C. {(x, y) / x = 4} 2 2 B. {(1,3), (1,4), (1,5)} D. {(x, y)/ x + y = 1} 49. Andrea can encode a 100-page thesis in 24 hours, while Ani can do the same task in 16 hours. How many hours will it take them to finish the job if they work together? A. 10 C. 12 B. 11 D. 13 50. If Justine can run 3 km in an hour, how far can she run in 45 mins? A. 1.75 km C. 2.5 km. B. 2.15 km D. 2.25 km.


7

LICENSURE EXAMINATION FOR TEACHERS (LET) Refresher Course for Year 2012 Area: Mathematics DAY 2

PART II – ANALYZING TEST ITEMS

Directions:Choose the best answer from the choices given. Write the corresponding letter of your choice. 1. Which of the following has a definite length? A. ray B. line C. point D. line segment 2. Which of these triangles cannot be isosceles? A. Acute B. scalene C. obtuse D. right 3. A diameter of a circle is also a ________. A. Radius B. secant C. chord D. tangent 4. In angle TOY, what is the vertex? A. T B. O C. Y D. cannot be determined 5. Refer to the figure. Given: m  2 = 55° and m  3 = 80. Find m  4.

3

6. 7.

2

1

4

A. 95° B. 105° C. 115° D. 135° MAZE is a parallelogram. If m  A = 67°, then m  Z = _____. A. 113° B. 90° C. 67° D. 23° Which two properties of the rhombus do not hold true for other parallelograms? A. Diagonals are perpendicular and congruent. B. Diagonals are congruent and bisect each other. C. Diagonals are perpendicular and bisect each other. D. Diagonals are perpendicular and each bisects the angles of the parallelogram. For item 8, refer to this figure: r s t

1 2

8. 9.

5

3

4 6

5

8

7

If lines r and s are parallel, which of the following pairs of angles are congruent? A.  4 and 5  B.  4 and 3  C.  1 and 8 D.  3 and  5 Three different triangles are placed side by side as shown. What is the sum of the angles of the polygon formed? A. 300° B. 360° C. 540° D. 720°

10. The supplement of an angle is three times its measure. What is the supplement? A. 22.5° B. interior 45° C.pentagon? 67.5° D. 135° 11. What is the measure of each angle of a regular A. 108° B140° C. 180° D. 540° 12. The angles of a triangle are in the ratio 2: 3: 5. What is the largest angle? A. 36° B. 54° C. 90° D. 99° 13. The graph shows the number of socks, belts, handkerchiefs, and neckties sold by a store in one week. 150

s m te i 100 f o r e b 50 m u N 0

Items Sold by Store


1

The names of the items are missing from the graph. Socks were the item most often sold, and fewer neckties than any other item were sold. More belts than handkerchiefs were sold. How many belts were sold? A. 80 B. 90 C. 120 D. 140 14. A pair of dice is thrown. What is the probability of throwing a double number (e.g. 3, 3)? A. 1/3 B. 1/6 C. 1/12 D. 1/36 15. In how many ways can the first, second, and third place winners be chosen from a group of 10 contestants? A. 6 B. 72 C. 720 D. 10! 16. In how many ways can 5 people be seated around a circular table ? A. 600 B. 120 C. 60 D. 24 17. In how many ways can two swimmers be chosen from a group of six swimmers? A. 15 B. 30 C. 45 D. 90 18. A coffee shop offers 4 flavors of coffee: mocha, cappuccino, vanilla, and latte. Coffee may be served hot or cold and it comes in tall, regular and short sizes. In how many ways can the costumer order his/her coffee? A. 3 ways B. 8 ways C. 24 ways D. 36 ways 19. Find the number of distinct permutations of the letters in the word ‘ANNUAL’? A. 180 B. 270 C. 360 D. 720 20. Mario had three test scores of 78, 76, and 74, while Kristyn had scores of 72, 82, and 74. How did Mario’s average (mean) score compared with Kristyn’s average score. A. Mario’s was 1 point higher. C. Both averages were the same. B. Mario’s was 1 point lower. D. Mario’s was 2 points lower. For items 21-23 Mr. Macho gave a mathematics test to a group of students. The distribution of the scores is as follows: SCORES FREQUENCY 21-25 2 16 - 20 5 11 - 15 12 6 - 10 4 1- 5 1 21. How many pupils took the test? A. 25 B. 24 C. 22 D. 12 22. What is the size of the interval of the frequency distribution? A. 2 B. 3 C. 4 D. 5 23. What is the computed mean value? A. 13.6 B. 14.1 C. 12 D. 15.4 24. Which of the following classes performed best and with the least variability in the test? Classes Mean Standard Deviation I-Hope 35.7 3.16 I - Sincerity 36.4 3.53 I - Love 35.7 3.87 I - Charity 36.4 3.72 A. I-Hope B. I-Sincerity C. I-Love 25. What is the standard deviation of the scores 6, 7, 9, 11, 13 and 14? A. 2.94 B.3.15 C. 3.26


2

D. I-Charity D. 3.48

PART III – ENHANCING TEST TAKING SKILLS

Directions:Choose the best answer from the choices given. Write the corresponding letter of your choice. 1. Which of these angles is closest to 45°?

A. 2. 3. 4.

B.

6.

D.

An angle is 65.5°. What is its complement? A. 24.5° B. 25. 5° C. 114.5° D. 124.5° Given: RS = 3, RT = 11, and ST = 8. Which of these points is between the other two? A. R B. S C. T D. any point What is the altitude of this figure? B C D

E A.

5.

C.

F

G B.

C.

C.

Which refers to a pair of lines that meet at a right angle? A. Parallel B. perpendicular C. intersecting Which of the following is a convex polygon? A. B. C.

D. concentric D.

7.

Which of the following quadrilaterals best describes a square? A. Its diagonals are perpendicular to each other. B. It is an equiangular rhombus. C. Its diagonals are congruent. D. It has four right angles. For item 8-9, refer to this figure: r t

1 2

3

4 6

5

s

8

7

8.

If lines r and s are parallel, which of the following pairs of angles are supplementary? A. 1 and 5  B.  5 and 7 C.  3 and  6 D.  3 and  7 9. If lines r and s are parallel and m  2 = 38°, what is the measure of 6? A. 38° B. 52° C. 68° D. 142° 10. If ABC is an isosceles triangle with a right angle at B, then A. is the hypotenuse. C. AC = BC B.

BCA = 45°.

D. AB = ½ AC

11. Which of the following statements is NOT true? A. Every square is a rectangle. B. Every square is a rhombus. C. Every rhombus is a square. D. Every rhombus is a parallelogram. 12. What is the sum of the measures of the interior angles of a regular heptagon? A. 1260° B. 1080° C. 900° D. 112.5° 13. The figure below shows a spinner with 12 sectors. When someone spins the arrow, it is equally likely to stop on any sector.


3

1/2 of the number of sectors are blue, 1/12 are green, ¼ are orange, and 1/6 are red. If a person spins the arrow, on which color sector does the spinner have the least probability to stop? A. Blue B. green C. orange D. red 14. What is the probability of getting a multiple of 3 when a die is tossed? A. 1/6 B. ¼ C. 1/3 D. ½ 15. In a candy jar, there are 15 lemons, 12 chocolates and 3 mints. If a candy is picked at random, what is the probability of getting a mint? A. ½ B. 1/3 C. 1/6 D. 1/10 16. The mean score of a set of 40 tests is 81. Find the sum of the 40 test scores. A. 2 250 B. 2 280 C. 2 820 D. 3 240 17. A show manufacturer conducted a survey of the average shoe size among women to help him decide what shoe size to put on display. What measure of central tendency would be most appropriate for his purpose? A. Mean B. median C. mode D. any of the three 18. In how many ways can 5 basketball players be chosen from a group of 9 players? A. 126 B. 212 C. 15 120 D. 362 880 19. How many 4-digit numbers can you write using the digits 1, 2, 3 and 4 if the digits may appear more than once in each number? A. 22 B. 24 C. 64 D. 256

20. In how many ways can 5 people be seated in a row of 5 vacant chairs if they can sit anywhere? A. 120 B. 90 C. 64 D. 25 21. The following polygon shows a shop’s monthly sales of umbrellas for the year.

How many more umbrellas were sold in June than in November? A. 45 B. 35 C. 30 D. 20 For items 22-24: The scores of 10 students who took the qualifying exam for a major mathematics competition are as follows: Student A B C D E F G H I J Score 18 32 10 15 27 18 23 28 29 30 22. What is the median score? A. 18 B. 20 C. 23 D. 25 23. What is the mode of the scores? A. 18 B. 20 C. 23 D. 25 24. What is the range of scores? A. 18 B. 22 C. 23 D. 25 25. Joey aims to get an average of 90 in 5 unit tests. The results of the first four unit tests are as follows: 89, 92, 86, and 91. What should his grade in the fifth unit be for his average to be at least 90? A. 91 B. 92 C. 93 D. 94


4

1.

2.

3.

4. 5.

6.

SUPPLEMENTARY PROBLEMS (MATHEMATICS) Which of the following number is divisible by 45? A. 300,000,000,450 C. 600,000,000,045 B. 100,200,600,090 D. 400,450,000,000 th th On its anniversary, the canteen offers a free sandwich for every 4 customer and a free softdrink for every 6 customer. After 75 customers, how many had received both free sandwich and softdrink? A. 30 B. 18 C. 12 D. 6 What is the difference in the elevation between the top of a mountain 51 meters above sea level and a location 28 meters below sea level. A. 23 m B. 33 m C. 79 m D. 89 m A pack of P50-bills is numbered from RV628 to RV663. What is the total value of the pack of bills, in pesos? A. 35 B. 36 C. 1750 D. 1800 2 2 3 Simplify: [ 5  81  3 – 5  3 + 2] (4 – 2 ) A. 15 ¼ B. 4 C. –6 D. – 15¼

If a student works on his assignment 3

1 3

hours a day, what part of the day does he spend doing his assignment?

A. 1/8 B. 1/7 C. 5/36 D. 15/28 Teodoro spent his monthly salary as follows: 3/5 for food and allowances, 1/3 for his child’s education and house rental. If his monthly salary is P15, 000, how much would he left at the end of the month? A. P 1,000 B. P2,000 C. P5,000 D. P 14,000 8. Isko and Jose run for President for their organization. Isko got 1/3 of the votes. If Jose got 300 votes, how many students voted for Isko? A. 900 B. 200 C. 150 D. 100 9. Gregorio has obtained the following grades: 1.4, 1.7, 1.8 and 2.5. What must be his fifth grade so that his average is 1.7? A. 2.1 B. 1.9 C. 1.5 D. 1.1 10. Out of the 20 numbers, 6 were 2.5’s, 4 were 3.25’s and the rest were 2.2’s. Find the arithmetic mean of the numbers. A. 2.5 B. 2.65 C. 10 D. 22 11. Diego bought X number of notebooks at P23.00 each, Y pad papers at 18.45 each, and Z ballpens at P8.25 each. If he gave an amount of P1000 to the cashier, how much change did he receive? A. P 434.25 C. 1000 – [(23.00)(X) + (18.45) (Y) + (8.25) (Z)] B. P 334.25 D. none of these 7.

12. Marcelo sells a real estate. He receives a monthly salary of P10,000 plus a commission of 1/5 % of his net sales for that month. Find his gross pay for a month during which his net sale is one million pesos. A. P 2,000 B. P 12,000 C. P 200,000 D. P 210,000 13. There are 20 million Filipinos who are qualified voters. If 25% of the population are qualified voters, how many are not qualified voters? A. 80 million B. 60 million C. 15 million D. 5 million 14. A pair of shoes srcinally priced at P600 is being sold at a discount of 30%. How much would you pay if you buy that shoes? A. P 30 B. P180 C. P 420 D. P 570 15. A book with a selling price of P120 is sold at 40% discount. What is its srcinal price? A. P 48 B. P72 C. P 200 D. P 300 16. A store owner advertises a polo-shirt srcinally sold for P200 for P170 only. What rate of discount is he giving? A. P 30 B. P15 C. 30% D. 15% 17. Rhoda bought an article for P400 and sold it for P500. What rate of profit did she enjoy in that deal? A. P100 B. 100% C. 25 % D. 20% 18. The price of an item is increased by 70% and t hen offered at 40% discount. What happened to the srcinal price? A. There is an increase of 30%. C. There is an increase of 2 %. B. There is an increase of 28%. D. There is a decrease of 32%. 19. An item is offered discount. it is offered at 30% discount. If the new selling price is P112, whatatis20% the first srcinalLater, price? A. P162 B. P200 C. P224 D. P1866.67 20. To have a 25% profit, the vendor should sell the item at P80.00. How much is his profit? A. P20.00 B. P60.00 C. P16.00 D. P64.00 21. Celia deposited P50,000 in a bank that pays a simple annual interest of 14

1 5

%. How much

money will she have in the bank after five years. A. P 85,500 B. P35,500 C. P36,250 D. P86,250 22. How much should Antonio invest so that his money earns P2,250 deposited at 6% for 9 months? A. P 50,000 B. 37,500 C. P 135 D. P 101.25 23. A photographer wishes to enlarge a picture 18 cm long and 12 cm wide so that it will be 36 cm wide. How long will the enlarged picture be? A. 54 cm B. 72 cm C. 24 cm D. 6 cm 24. If 8 secretaries can type 800 pages in 5 hours, how long would it take for 12 secretaries to type 800 pages at the same rate? A. 7 1/2 hours B. 3 1/3 hours C. 10 hours D. 2 1/2 hours St. Louis Review Center, Inc – San Pablo City Branch http://stlouisreviewcenter-slrc.blogspot.com (049)562-2239 | 0905-914-6145

5

25. The ratio of cows and carabaos in the field is 4:9. If there are 468 cows and carabaos in the field, how many are carabaos? A. 52 B. 117 C. 144 D. 324 26. A defective meter stick actually measures 95 cm. If a cloth is measured to 5 m long using this meter stick, what is the actual length of the cloth? A. 500 cm B. 495 cm C. 4 m and 85 cm D. 4 m and 75 cm 27. If in a map, 2 cm represents 7 km, then how far are two towns from each other if they were 18 cm apart in the map? A. 14 km B. 27 km C. 63 km D. 126 km 28. Which of the following is the longest? A. 6540 mm B. 934 cm C. 9.25 m D. 0.009 km 29. Which of the following sets of numbers can be sides of a right triangle? A. 4, 5, 6 B. 6, 6 2 , 3 C. 3, 6 3 , 2 D. 2, 2, 3 30. The length and width of a rectangle are (3x – 2) and (2x + 1). What is its perimeter? 2 2 A. 5x-1 B. 10 x - 2 C. 6x –x -2 D. 6x –x +2 31. The area of a square is 32x. Which of the following could be the value of x? A. 2 B. 6 C. 3 D. 4 32. If the area of one circle is twice of another circle, what is the ratio of the area in percent of the smaller to larger circle? A. 70% B. 25% C. 75% D. 50% 33. One hectare is equivalent to 10,000 square meters. How many hectares are in a rectangular field which is 750 m wide and 800 m long? A. 120 B. 56 C. 60 D. 40 34. An aquarium is 40 cm high. Its length is twice its height, while its width is half its height. How much water is needed to fill in the aquarium in cu. cm.? A. 64,000 B. 32,000 C. 16,000 D. 8,000 35. What is the volume of a cube if one face has a perimeter of 36 cm? A. 729 cu. cm. B. 216 cu. cm. C. 46,656 cu. cm D. 5,832 cu. cm 36. How much water can be filled with a cone whose diameter is 24 cm and whose height exceeds 2/3 of the radius by 2 cm? A. 480  cu. cm. B. 540  cu. cm. C. 660  cu. cm. D. 900  cu. cm. 2 37. Which of the following is a factor of quadratic equation x -2x-24=0 A. x-4 B. x + 2 C. x+6 D. x+4 38. Michael is 15 years older than his brother Rafael. However, y years ago Michael was twice as old as Rafael. If Rafael is now x years old and x > y, find the value of x - y. A. 13 B. 14 C. 15 D. 16 39. If x + A. 6 =39, then 3x + 1 = __________. B. 10

C. 9 40. Monet (X) is taller than Rizza (Y) but shorter than Joy (Z). Which is TRUE? A. Z < X < Y B. X < Z < Y C. Y < X < Z o 41. Refer to the figure. Given: m  2 = 50° and m  3 = 80 . Find m  4.

D. 34 D. Y < Z < X

3 4

2

1

5

A. 30° B. 50° C. 80° D. 130° 42. RODA is a parallelogram. If m  O = 68°, then m  D = _____. A. 68° B. 12° C. 112° D. 22° 43. The supplement of an angle is three times its measure. What is the supplement? A. 22.5° B. 45° C. 67.5° D. 135° 44. Jose is 5 years older than Tess. Three years ago, he was twice as old as Tess. How old is Tess now? A. 15 B. 13 C. 8 D. 6 45. If x + y = 2a and x – y = 2b, what is the value of xy? 2 2 2 2 2 A. ab B. a + b C. (a – b) D. a – b 46. In a political science survey, voters are classified into six income categories and five education categories. In how many different ways can a voter be classified? A. 11 B. 25 C. 30 D. 45 47. The number of permutations of the word “probabilities” is A. 13 B. 2!3! C. 13! D. 13!/(2!3!) 48. The number of ways seven visitors can be seated on a round table is A. 6! B. 7! C. 6!7! D. 6!/2 49. In a certain town, 40% of the people have brown hair, 25% have brown eyes, and 15% have both brown hair and brown eyes. A person is selected at random from the town. If he has brown eyes, what is the probability that he does not have brown hair? A. 3/8 B. 2/5 C. ½ D. none of these 50. A die is tossed. If the number is odd, what is the probability that it is prime? 2

A.

3

3

B.

4


1

2

C.

5

6

D.

3

2013 REVIEW MATERIALS FOR MATH MAJOR

1. A student finishes the first halfof the exam in 2/3 of the times it takes him to finish the second half. If the entire exam takes him an hour, how many minutes does he spend on the first half of the exam? A. 20 B. 24 C. 30 D. 36 2. If a is an odd integer and b is an even integer, which of the following is an odd integer? 2 A. b3a B. 1a – C. 2a b – D. a + 3b 3. Steve, Karen and John want to buy a Php 1 350 stereo. If Karen agrees to pay twice as much as Karen John, and much must pay?Steve agrees to pay three times as much as Karen, how A. Php 300 B. Php 400 C. Php 500 D. Php 600 4. If a and b be true? I. II. III. A. I only

are integers and the sum of ab and b is odd, which of the following could a and b are both odd a is even and b is odd a is odd and b is even B. II only

C. III only

D. I and II

5. Simplify : { 7 + [ 5 x ( 8 + 4 ÷ 22 ) ÷ 3 ] – 5 } A. 6 B. 12 C. 17

D. 21

6. When x is divided by 7, the remainder is 6. When x is divided by 6, the remainder is 0. Which of the following is the possible value for x? A. 42

B. 46

7. If x/y = 4/3 and x/k = ½, then k/y = ? A. 1/6 B. 3/8

C. 79

D. 90

C. 8/3

D. 3/2

8. If e : f : g = 1 : 5 : 6 and g = 18, then f– e = ? A. 5 B. 6 C. 9

D. 12

9. It takes 27 women to clean the building in 15 days. At this rate, how many women will be needed to clean the same buildings in 9 days? A. 32 B. 45 C. 55 D. 60 10. Henry divides his 5 hours to study Math, English, and Science in the ratio 5 : 2 : 3. How many minutes does he spend studying Math? A. 1 hour B. 1.5 hour C. 2 hour D. 2.5 hour 11. A. Which inconsistent 2x =of2xthe following is B.an 2(x-3) = -2(1+x)equation? C. 3 – 2x = -2x – 1 D. 2/3x + 3 = - 7/2x

–  ? 

12. Which of the following is equal to A.

 

B.

 

C.

  

D. -

–   

13. Which of the following sets of ordered pairs defines relation as a function? A. {(2,4), (3,4), (4,5)} C. {(-2, 4), (-2, 6), (3,6)} D. {(π, -1), (π , 1) (-1, 1)} B. {(2, -5), (2, 5), (3, 10)}


14. The domain of the function f(x) = A.

{x│x ≠ 0}

 is 

B. {x│x ≠ 5}

C. {x│x ≠ 5/2}

D. {x│x ≠ - 5/2}

15. The system of the equations x + y = 2 and 2x +2y = 8 is an example of A. Independent sytem C. Inconsistent system B. Dependent sytem D. Consistent system 16. If n is a perfect square number, what is the next larger perfect square number? A. n2+ 2n + 1 B. n 2 + n + 1 C. n+2 1 D. n + 2    17. A man, 32 years old, has a son 8 years of age. In how many years so that the man will be twice as old as his son? A. 16 years B. 24 years C. 32 years D. 48 years 18. How many gallons of milk with 5% butterfat must be mixed with 90 gallons of milk with 1% butterfat to obtain the mixture of milk 2% butterfat? A. 10 gallons B. 20 gallons C. 30 gallons D. 40 gallons 19. The length of a rectangle is two times its width. If the length is decreased by 5 cm and the width is increased by 5 cm, the area is increased by 75 sq. cm. Find the dimensions of the srcinal rectangle. A. 28 cm & 56 cm B. 25 cm & 50 cm C. 22 cm & 44 cm D. 20 cm & 40 cm 20. If x + y = 7 and x – y = 3, what is 2x + y? 18 A. – B. 8 C. 10 21. What is the value of x, if log x = 1? A. 0 B. 10

D. 12 D. – 1

C. 1

22. One of the diagonal of the rectangle is 10 cm long. The rectangle’s length is 8 What is the area of the rectangle? 2 A. 24 cm2 B. 36 cm C. 48 cm 2 D. 80 cm 2 23. What is the value of x if 16x + 2 = 642x A. 7/4 B. 2

–1

cm.

?

24. What are the roots of x2 = 8x – 15? A. {3, 5} B. {-3, 5}

C. ¼

D. – 2

C. {-3, -5}

D. {3, -5}

25. If 2y – 6 varies directly as x + 2 and inversely as z– 3, and if y = 1 when x = 2 and z = 11, What is y when x = 4 and z = 9? 1 A.

–

B. 0

C. 1

D.

–2

26. The 7th term of the geometric sequence is 405 and 10th term is 10935. What is the common ratio of the sequence? A. 1 B. 2 C. 3 D. 4 27. What is the value of k so that x– 1 is a factor of 7x4 – 8x3 - 9x2 + 6x + k? A. 9 5 B. – C. 4 – D. 4 28. Find the measure of an angle if the measure of its supplement is 39 more than twice the measure of its complemnent . A. 38 B. 39 C. 60 D. 120


29. Which of the following statements is true about parallel lines? A. They form a right angle C. They are skew B. They do not intersect D. None of these 30. In an Isosceles triangle, the vertex angle is half the measure of base angle. What is the measure of the angle? A. 72 B. 60 C. 36 D. 30 31. A 10 m long steel pipe shrank 0.05% when exposed to below freezing temperature. By how many centimeters did the pipe shrank? A. 5 32. What is 30% of A.

B. 0.005

C. 0.5

D. 0.05

 ? 

 

B.

 

C.

 

D.

 

33. Which of the following times gives approximately 90° angle between the two hands of a clock? A. 11:40 B. 9:00 C. 7:20 D. 6:15 34. One thousand two hundred(1200) boys enrolled in a school year this year. Four years ago 1500 boys enrolled. What is the percent of decrease in enrolment? A. 10% B. 8% C. 25% D. 20% 35. It takes a teacher 20 minutes to check 25 papers and take a rest of 5 minutes after 25 papers before starting another set. At this rate, how many minutes will it takes the teacher to correct 75 test papers? A. 75 B. 60 C. 70 D. 80

36. Simplify the expression A.

 

   B. 

C.





D. -4

37. The equation y = x4 – 5x2 + 4 has A. three (3) real roots and one imaginary root B. one (1) real root and three imaginary roots C. two (2) real roots and two imaginary roots D. four (4) real roots 38. If 16 is 4 more than 3x, then what is the value of 2x– 5? A. 4 B. 5 C. 2

D. 3

39. If 2x + 13 represents an odd number, what must be the next consecutive odd number? A. 3x + 15 B. 2x + 15 C. 2x + 14 D. 3x + 13 40. Let G(x) = 1 – x2. What is G()? A. -6 B. 8

C. 48

D. 50


41. What term will make 36x2 – 25y4 a perfect square trinomial? 2 A. 5 + y2 B. 12xy C. 60xy D. 30xy 42. What is the 20th term of the arithmetic series 7, 10, 13....? A. 46 B. 23 C. 64

D. 32

3 – 6x2 + 5x + 7 is divided by 43. Which of the following is the remainder when 3x x – 2? A. 12 B. 21 C. 3 D. 17

44. In the standard plane (x,y), coordinate plane, the y-coordinate of every point on the line is the same as its corresponding x-coordinate. What is the slope of the line? A. -2 B. 1 C. 3 D. 2 45. On a circle with center C, CD⊥AB = 24 and CD = 5. Find the radius of the circle A. 19 C. 17 A D B B. 7 D. 13 C

46. The measure of each interior angle of a regular polygon is 160 deg. How many sides does it have? A. 29 B. 24 C. 12 D. 15 47. If two points of a line lie in a plane, then I. all points of the line lie in the plane II. the line pierces the plane exactly two points III. the plane contains all the points of the line IV. the intersection contains only two points A. I only B. II and IV C. II only

D. I and III

48. The sum of the measures of the 3 angles of a triangle is 180 degree. If the angles are in the ratio 1:2:3, what is the measure of the largest angle? A. 90° B. 60° C. 30° D. 45° 2 49. Determine the foci of the ellipse whose equation is 16x + 9y2 – 144 = 0.

A. (4,3)

B. (0,0)

C. (0, +)

D. (+,0)

50. Find the equation of a line with slope 2 that passes through the midpoint (3,-2) and (4,7). A. 4x + 2y - 9 = 0 B. 4x – 2y – 9 = 0 C. 4x – 2y + 9 = 0 D. 4x + 2y + 9 = 0 51. Which of the following has a graph which is a circle with center at (0,0) and having a radius of 4? A. x2 + y=2 4 B. y+ x 2= 4 C. x 2/2 + y2/2 = 4 D. (x+ y)2 = 16 52. There are 4 different designs of wrapping paper and 4 colors of ribbon. In how many different ways can a gift be wrapped? A. 14 B. 12 C. 8 D. 16


53. Suppose tan ø is in third quadrant. What is the numerical value of cos ø? A. -3/5 B. -4/5 C. 4/5 D. 5/4 54. Which of the following is the asymptote in the graph of y=tan x? π A. x = π B. x = 2 C. x = 3 π D. x =

/2

π

55. What are the roots of the equation (x2 – 4) (x – 1) = 0? A. -4,+1,-1 B. -1,+1,-2 C. +2,-2,+1

D. +1,-1,+2

56. If log3 x =2 and log2 y=3, what is xy? A. 72 B. 9

D. 27

57. Evaluate 

 A. +C

 

B.

 + C 

C. 81

C. 2

+C

58. What is the range of the scores 73, 89, 81, 95 and 87? A. 23 B. 4.4 C. 12

D.

 +C 

D. 22

59. John borrowed P25,000 from Robert at an interest rate of 12%. How much interest should John pay after 2 years? A. P7,000 B. P6,400 C. P2,400 D. P6,000 60. A collection agency was able to collect 7/8 of P20800 account from a client and charged 10% for the services which includes expenses. How much commission did the agency receive? A. P2080 B. P182 C. P18,200 D. P1820 61. If ∆ABC is equilateral, then it is A. Scalene B. Right

C. Isosceles

D. Equiangular

b

62. In the adjoining figure, if a = 40° and b = 38°, then d = ___________. A. 40° B. 38° C. 78° D. 102°

a

c d

(5x -32)°

63. In the adjoining figure, the quadrilateral is a parallelogram. What is the value of x? A. 86°

B. 75°

64. What is m

C in the figure at the right?

C. 60°

(2x + 118)°

E

D. 50°

D 30°

65° 45°

A A. 45°

B. 50°

C. 65°

65. Find the value of x to make the two lines cut by a transversal parallel. A. 7.33° B. 13.6° C. 11°

60°

B

D. 70° l

(8x – 4)°

D. 10° m

(2x + 48)°

C


66. Which of the following statement is true about a circle? i. The radii are congruent ii. A diameter is twice the radius iii. A secant line intersect the circle in one point iv. A tangent line intersects the circlesin two points A. i and ii B. iii and iv C. i, ii and iii

D. i, ii and iv

67. In a circle, a central angle measures 70°. What is the measure of the intercepted arc? A. 140° B. 70° C. 90° D. 35° 68. If P is inscribed in a semi circle, thenm A. 30° B. 45°

P is C. 60°

D. 90°

69. The base of a cylinder is a circular region of diameter 8. If the altitude is 8, what is its volume? A. 128π B. 64π C. 16π D. None of these 70. A drainage tile is a cylindrical shell 21 cm long. The inside and outside diameters are  4.5 and 5.1 cm. Find the volume of clay needed to make tile. ( Use forπ )  3 A. 95 cm3 B. 72.1 cm C. 31 cm 3 D. 22.95 cm 3 71. What is the value of cos 120°?   A.  B.   

C.

72. What is the value of sin2 30° cos2 30°?  A. B. 1  73. Solve for the equation sin  = 0.5 for 0 ≤    A. , B. 





 

D.

D. 

C. -1

 ≤ 2π? C.

 

  ,  

D.

 

 

74. What is the expansion of cos (60° -  )? A.

  cos  + sin  



 

B. cos  +

     sin  C. cos   sin  D. cos  + sin     

75. The expression cos 3 cos  sin3 sin is equivalent to   A. cos 4 B. cos 2 C. sin 2

D. sin 4



76. Which of the following statements are true? i. tan 225° =1 ii. csc(-390°) = -2  iii. sec 510° =  A. i and iii only B. ii and iii only C. i, ii and iii

D. I and ii only

 77. For which of these values of x would sin x = ?  A. 510° B. 840° C. 870°

D. 660°

78. In the right triangle ABC, what is the value of sin A?   A. C.     B. D.  

C 4

A

6



B


79. InXYZ, m A. 30°

X = 90° and cot Z = 1. What is the measure of angle Z? B. 50° C. 45° D. 60°

2 80. Refer to the given in problem 79. What is the value of 1 + sec Z?  A. B. 2 C. 4 D. 3 

For questions 81 – 82, refer to ABC.

B



81. What is the measure ofθ ? A. 19.5° C. 48.6° B. 19.5° or 160.5° D. 48.6° or 131.4°

12 cm c

82. What is the length of c? A. 2.03 m C. 18.2 cm B. 15.7 cm D. 27.4 cm

30° A

For questions 83 – 84, refer to XYZ. 83. What is the length of XZ? A. 40.5 cm C. 14.7 cm B. 27.3 cm D. 6.4 cm 84. What is m A. 31.5° B. 82.1°

8 cm Y 25°

15 cm

X? C. 59.1° D. 78.4°

C

13 cm

Z X

85. Which trigonometric function has a graph which neither intersects neither the y-axis nor the x-axis? A. sec x B. csc x C. tan x D. cot x 2 86. Find the conic section represented by equation 100x = 4 + 81y2 A. Ellipse B. Hyperbola C. Parabola D. Circle

87. The directrix y2 = -12x is A. x = 3 B. y = 3

C. x = -3

D. y = -3

88. Find the equation of a parabola with focus at (-1, 4) and directrixx = 7. – 3)2 = -16(x – 4) A. (x – 4)2 = -16(y 3) – C. (y – 3)2 = -16(y – 4) B. (y – 4)2 = -16(x 3) – D. (x 89. Which of the following is true? A.    B.   90. Solve for x: x =  A. 

C.      D. 55 + 55 + 55 + 55 + 55 = 56

  B.  C. 4

D.



91. Find the value of x that will satisfy the following expression:    A. x = 3/2 B. 18/6 C. 9/4 D. 4/9 -2

3 2

92. Simplify: (a b ) 2 -1 a b A. a2/ b7

B. a2/b5

C. a6/b7

D. a6/b5




93. Simplify the following: A.   

      B.

  

C.   

D.   

94. If the ratio of sec x to csc x is 1:4, then the ratio of tan x to cot x is ___________ A. 1 : 4 B. 1 : 1 C. 4 : 1 D. 1 : 16 95. Mario sold five pigs and four goats for 21,000 pesos. Selling the animals at the same price, Ben sold three pigs and two goats for 12, 000 pesos. What is e selling price of each animal? A. =P P 3,000; 4,000; goat B. Pig Pig = goat = = P1,000 P1,500

C. D. Pig Pig = =P P 3,500; 2,500; goat goat == P2,500 P2,000

96. The product of two numbers is 60. If the sum of their reciprocals is 4/15, what are the numbers? A. 12 and 5 B. 10 and 6 C. 15 and 4 D. 20 and 3 97. Find the equation of the line whose slope is 4 and passing through the point (2, -3). A. 4x – y –11 = 0 B. 4x – y + 11 = 0 C. 4x + y – 11 = 0 D. 4x + y + 11 = 0 98. What is the equation of the line passing through the points (3,1) and (-5, 4)? A. 3x + 8y + 17 = 0 B. 3x – 8y + 17 = 0 C. 3x – 8y – 17 = 0 D. 3x + 8y – 17 = 0 99. Let sin A = 3/5 and sin B = -4/5 where A and B have terminal sides in QII and QIII, respectively. What is sin (A-B) equal to? – 7/25 A. 7/25 B. 1 1 C. – D. 100. A rectangular lot is 5 meters wide and 12 meters long. How long is a path that cuts diagonally across the lot? A. 5 m B. 13 m C. 15 m D. 17 m 101. Jose sold five pigs and four goats for Php 21,000. Selling the animals at the same price, Pedro sold three pigs and two goats for Php 12, 000. What is e selling price of each animal? A. P = 4000 & G = 1000 C. P = 2500 & G = 2000 D. P = 2000 & G = 2500 B. P = 3000 & G = 1500 102. A dog trainer sets 5 bones 2 meters apart in a straight line starting from where he stands. He trains the dog to pick up the 1st bone and bring it back to him, then go for the 2nd bone and bring it back to him, and so on till he gets back all the 5 bones. By the time the dog returns the 5th bone to the trainer, how many meters will the dog have walked? A. 30 m B. 40 m C. 60 m D. 80 m 2

2

103. If x < o and (3x-2) = 25, then 2x + 1 is _____. 1A. – B. 2 C. 7/3

D. 3

104. What must be the value of x in the ratio (2x–3): (x+1) is equal to 4:7? A. 1.5 B. 2.5 C. 4 D. 7 105. Evaluate 6m – [5n + (2m-5)]. – n 1– A. 3m 5 – B. 3m

C. 2m

–3n + 5

D. 4m

– 5n + 5


106. If log3 x = 2 and log2y = 3, what is xy? A. 81 B. 72 107. Multiply (xn +2) (x2n – 2xn + 4) – A. X3n + 8xn +8 B. X 3n– 8xn8

C. 27

D.9

3n C. X + 8

D. X

3n

–8

108. Factor completely the expression 4p2q (x + y) + 16pq2(x + y). A. 4p (qx + y ) (4q + p) C. 4pq (x+y) (p+4q) B. B.4p2q (x + y) (1 + q) D. pq (x+y) (4p + 16q) 109. The difference of two polynomials is 6m + 5n – 5p, if second polynomial is 2m– 3n + 3p, what is the first polynomial? A. 4m + 8n -7p B. 8m + 2n -3p C. –m – 8n + 7p D.– 8m – 8n – 3p 110. In a farm of cows and chickens, the head count is 25, but the leg count is 70.How many chickens are there in the farm? A. 10 B. 11 C. 12 D. 15 111. What is log2 8 + log4 16 – log5 25? A. 6 B. 3

4 C. –

112. If the first derivative of f(x) is 3x2 + 2x, what is f(x)? A. 3x + 32x B. x 3 + x2 C. 6x3 + 2x2

–2

D.

D. 3x

3

+ x2

113. Which of the following statements id TRUE? A. All squares are rectangles C. All rectangles are squares B. All rhombuses are rectangles D. All parallelograms are trapezoids 114. What is midpoint of the segment joining (3, 8) and (-5, 4)? A. (8,4) B. (2, 4) C. (2, 12)

D. ( -1, 6)

115. Which of the following is the slope of a line whose equation is 5x – 4y = 3? ¾ A. – B. 4/5– C. 3/5 D. 5/4 116. Find an equation of the line through (2, 3) and the point common to x + y = 1 and 2x + y = 5 ? A. 3x + y –9 = 0 B. 3x + y+ 9 = 0 C. x + y –1 = 0 D. x – y+1 = 0 117. Which function DOES NOT intersect the x-axis? A. y = tan x B. y = cot x C. y = sec x

D. y = csc x

118. What is the reference angle of 120 degrees? A. 90° B. 60° C. 45°

D. 30°

119. Find the value of x in log2X = 5 A. 10 B. 25

D. 125

C. 32

120. If  is an angle of a right triangle and cos = 3/5 sin is ____. A. 2/5 4/5 B. – C. 1 – D. 4/5

MATH MAJORS LET SEPTEMBER 2015 (INTRODUCTION TO ANALYTIC GEOMETRY AND CALCULUS) 1. Find the distance between (-2, -1) and (4, 3). A. 22

B. 23

C. 211

D. 213

2. If (1, -2) is equidistant with (4,4) and (x, 4), find x. A. (2,4)

B. (-2,4)

C. (2, -4)

D. (-2,-4)

3. Find the midpoint of the line segment with endpoint (4,3) and (-2,1). A. (-1,-2)

B. (-1, 2)

C. (1, -2)

D. (1,2)

4. The midpoint of a line segment is (2, -1) andone endpoint of the segment is(-1, -3). Find the coordinates of the other segment. A. (1, 5)

B. (-1, -5)

C. (5,1)

D. (-5, -1)

5. The point (0,1) is the midpoint of the line segment joining (x,4) a nd (-3, y), find x and y. A. (1,3)

B. (1,-3)

C. (3,1)

D. (3,-1)

6. Find the slope of the line that passes through (1,2) and (3,4) A. 0

B. 1

C. -1

D. ∞

C. -1

D. ∞

C. -1

D. ∞

7. Find the slope of the line y = 1. A. 0

B. 1

8. Find the slope of the line x = - 2. A. 0

B. 1

9. Find the equation of a line passing through the points (-2,1) and (3,4). A. 5x + 3y = -11

C. 5x - 3y = -11

B. 5x - 3y = 11

D. 5x + 3y = 11

10. Find the equation of the line joining the point (2, -1) and midpoint of the line segment from (5, 3) to (1, -1). A. 2x + y + 5 = 0

C. 2x + y - 5 = 0

B. 2x - y + 5 = 0

D. 2x - y - 5 = 0

11. Find the equation of the line passing through point (3/2 , -3) and having slope– ¼. A. 2x + 8y 21–= 0

C. 2x

B. 8x + 2y + 21 = 0

–

3y – 21 = 0

D. 8x – 2y + 21 = 0

12. What is the equation of the line whose x-intercept is– 5 and y – intercept is 3? A. 5x + 3y 15–= 0

C. 5x

–

3y + 15 = 0

B. 3x + 5y 15–= 0

D. 3x

–

5y + 15 = 0

– 3y = 6. 13. Find the equation of a line passing through the point (0,1) and parallel to the line 2x

A. 2x + 3y - 3 = 0

C. 2x - 3y - 3 = 0

B. 2x + 3y + 3 = 0

D. 2x - 3y + 3 = 0

– y/2 = 1. 14. Find the equation of the line passing through (-2, 2) and perpendicular to the line x/2

A. x + y = 1

C. x + y = 0

B.y =x 1 –

D. x

–

y=0

15. What is the distance from point (– 1, 3) to the line 3x – 4y = 10? 5

A.

–

C. 5

25

B.

–

D. 25

16. Find the equation of the circle whose center is at the srcin and which passes through the point (2, 1). 2 A. x= + 5 y2

2

C. x

B. x2 + y2 = 5

+ y2 = 25

D x2 + y2 = 2

2

17. The equation of the circle is 4x + 4y

–

–

5

16 = 0, find its radius.

A. 4

C. 12

B. 8

D. 16

18. Determine the equation of a circle with center at (1,– 2) and passes through the point (5,– 5). A. (x B. (x

– –

2 1)2 + (y – = 2)25 2

2

1) + (y + =2)25

C. (x

–

1)2 + (y – 2)2 = 5

D. (x

–

1)2 + (y + 2)2 = 5

19. Find the equation of the circle with center at the srcin and tangent to the line 4x + 3y = 10. A. x=2 4 + y2 2

2

B. x= 8 +y

20. Express the equation x2 + y2 A. (x

–

2

2

2

2

–

2x

C. x

2

–

y2 = 4

D. x

2

–

y2 = 8

–

1) + (y – =2)2

4y + 1 = 0 in standard form of a circle. C. (x

B. (x + 1) + (y – =2)2

–

D. (x

–

2

2

2

2

1) + (y + 2) = 4 1) + (y – 2) = 4

21. What is the general form of the equation of a circle with center at (3, 1) and tangent to the line 4x + 3y + 5 = 0? A. x2 + y2 2

–

6x + 2y6–= 0

2

–

C. x

2

+ y2 + 6x + 2y – 6 = 0

2

–

2

–

–

B. x + y + 6x 2y6 = 0 D. x + y + 6x 2y 6 = 0 22. Find the equation of the parabola having its focus at (2,0) and directrix x =– 2. = A. 8y x2 2

=B. -8yx

C. y

2

= -8x

D. y

2

= 8x

23. What is the length of the latus rectum of the parabola having its vertex at the srcin and focus at (– ½, 0)? A. 1

C. 4

B. 2

D. 8

24. Find the equation of the parabola with directrix y =– 4 and focus at (0,4). =A. 16y x2

C. y

2

= –16x

D. y

2

= 16x

2

=B. -16y x

25. Given the equation of the parabola A. V(– 4, 3) and F( 4,–2)

+ 4y + 8x = – 4, find the vertex and focus.

x2

C. V(4,

B. V(4, 3) and F(4, 2)

–

D. V(– 4,

26. Determine the foci of the ellipse in the equation A. (+ 2, 0)

( +2)2

27. Given the equation of the ellipse B. (1,

2)–

–

3) and (– 4,

2

2

4

+

C. (0, + 12)

B. (+ 4, 0)

A. 2,1) (–

3) and F(4, – 2)

9

+

(−1)2 4

C. (2,

–1)

16

–

2)

= 1. D. (+ 12, 0)

= 1, find the coordinates of the center. D. (–1,2)

28. Find the equation of the ellipse with vertices at –( 2, 1) and (4, 1) with eccentricity 2/3. A. 9x + 5y

2

2

–

10x – 18y –31 = 0

C. 9x

2

+ 5y + 10x – 18y – 31 = 0

2

B. 5x2 + 9y2

–

10x – 18y –31 = 0

D. 5x

2

+ 9y2 + 10x – 18y – 31 = 0

29. Find the center of the ellipse in the equation 2x + 4y2 A.2,1) (–

B. (1,

2)–

C. (2,

–

2x + 16y + 13 = 0

–1)

D. (–1,2)

30. What do you call the segment passing through the center and perpendicular to the transverse axis? A. Asymptote

C. Major axis

B. Minor axis

D. Conjugate axis

31. Given the equation of the hyperbola A. (+ 4, 0)

B. (+ 5, 0)

2 16

−

2

= 1, find the coordinates of the foci.

9

C. (0, + 3)

D. (+ 3, 0) 2

2

32. Determine the vertices of the hyperbola in the equation 9x – 4y = 36. A. (+ 4, 0) B. (+ 5, 0) C. (0, + 3) D. (+ 3, 0) 33. Find the center of the hyperbola in the equation 2x – 4y2 + 4x + 24y = 28. – A. (– 2,3)

B. ( 2,–3)

C. (2, 3)

D. (2,

–

3)

34. Which of the following represents a parabola? 2

A. + 5xy = 10 2

B. x

2

2xy = + yy+3

–

2

C. 3y 2

D. y

+ 8y = 5x + 2

–

xy = 4x + 9

35. Determine which of these equations represents a circle? 2

– 10 A. 4y x =

2

C. 2x

2

2

B. 3x = 8 –+4y 4y

2

D. 4x

+ 4x + 4 = 10 – 6y

–

2

2y

2

= 4 + 4y

36. Which of these set notations represents the half-open interval notation? A. { xI a < x a or x > b }

37. In the following set of ordered pairs, identify which set is a function. A. { (x,y), (x,z), (y,z)}

C. { (1,2), (2,3), (3,4)}

B. {(3,1), (3,2), (3,3)}

D. {(0,-1), (-1,3), (-1,4)

2

38. If f(x) = x

–

2 and g(x) = x + 1, find (f o g)(x)

2

A. x+ 1

2

B. x

1– 2

39. Given the function f(x) = x A.

 

B.

40. Evaluate lim →2

 +2

A. 4

–

C. {x/x

R, x > –1}

B. 12x

2

 

D. 0

− 1 , the domain is defined by ϵ

R, x > 1}

D. {x/x ϵ R, x > –1}

42. What is the first derivative of 3

D.

C. 2

A. {x/x R,ϵx > 1}

A. 12x

D. x

.

41. Given the function y = 

ϵ

+1

 

C.

B. -4

B. {x/x

–2x

5x + 4, find f(-2/3).

 

( 2 − 4)

2

C. x

4

 −4 = 3 ?   C.

12

3

D.

12

4

+ 2x – 1

43. Find the first derivative of y =2 . A.

1

B.

2 2 

1

C.

2 

2

D.

2 

2  2

2

44. Determine the first derivative off(x) = (5x + 6)(2x). 2

2

A. 10x+ 12

45. What is the first derivative of f(x) = A.

12

B.

(2 −4  )2

4

+C



C. 30x + 12 3 − 5

2

– C A. 2x + 3x 4x +

2 3

3

2

1

3

1

+C

14

D.

(2 −4  )2

C.

−4 

−14 2 (2−4)

3

+C

D. – 4x

− 4) ? 2

3

D.

2

 2

C.

C. 4x

x + 4x x +–C

48. Evaluate

A.

3

2

+3+C 3

x +

2 3

2

x

–

4x + C

.

1

B. − 2 + C 

C. −

1



+C

D. 2x + C

4

49. Find the integrals of 3 9 2  . A. 12

B. 15

C. 18

D. 21

2 ( 2 1

50. Evaluate the integrals of  A. 1/6

+ 10

−4

47. What is the integral of(2 2 + 3

B.

2

 2 .

– + C B. 8x

3

D. 12x

?

2 −4

−12 (2 −4  )2

46. Evaluate the integral of

A.

2

B. 20x + 12

B. 5/6

 −  + 1) . C. 1 5/6

D. 2

MATH MAJORS LET SEPTEMBER 2015 (BASIC TRIGONOMETRY AND ADVANCED ALGEBRA)

3  1. Rationalize and simplify

4  2

B. a12

A. 2x

.

C. 4a

3  2

D.

2. Half the product of 16 and a number is increased by 3is 27. Find the number. A. 2

B. 3

C. 4

D. 8

3. What is the value of x in the equation (7 – 5i) + x = 4i – 6? A. 1/8

C. I 3–

B. 1 + 9i

– 13 + 19i

D.

4. Lito can mow the lawn in 5 hours, while Ronie can mowsame lawn in 4 hours. Working together, how long will it take them to mow the lawn? A. 4 hrs.

B. 5 hrs.

C. 2 2/9 hrs.

D. 4 ½ hrs.

5. A chemist needs to mix 20 liters of 40% salt solution with some 70%solution to get a mixture that is 50% salt. How many liters of the 70% solution should be used? A. 10

B. 15

C. 20

D. 25

6. Ken is 2 years older than Ben, In8 years, Ben will be 4/3 as old asKen is now. Find the age of Ben? A. 15

B. 16

C. 18

D. 20

7. If y varies directly as x, and y = 15 when x = 30. Find y when x = 8. A. ½

B. 2

C. 4

D. 8

8. The ratio of 4 less than a number to 26more than that of a number is 1 isto 3. What is the number? A. 15

B. 19

C. 21

D. 24

9. Which trigonometric function hasa graph which intersects neither they-axis nor the x-axis? A. sec x

B. csc x

C. tan x

D. cot x

10. Solve for x in log (x – 4) + log (x – 3) = log 30 9A. –

B. 9

C. 7

D. 12

11. A quadrantal angle is an angle in standard position whose terminal side coincides with either xaxis or the y-axis. Which of these is a quadrantal angle? π/5 A. 8π/3 B. 2 C. 225° D. 630° 7!

12. How much is A. 7

5!

6!

÷

5!

?

B. 14

C. 35

D. 42

2  +3

13. The inverse function of y = 3  +2

A. y =

4

4

is

2  +4

B. y =

3

3 −4

C. y =

2

4 −3

D. y =

2

14. Which of the following is the asymptote in the graph of y = tan x? A. x = π

B. x = 2π

C. x = 3π

D. x = π/2

15. A circle has a radius of 18 cm. Find the arc length cut by a central angle equal to 3π/8 radians. A. 6.75 cm

B. 21.20 cm

C. 54 cm.

D. 56.55 cm

16. Determine the length of an arc cut by a central angle of 60° in a circle with a radius of 2 cm. A. π/2

B. 2π/3

C. 3π/2

D. 2π

17. A wheel of radius 80 cm rolls along the ground without slipping and rotates through an angle of 45°. How far does the wheel move? A. 20π cm

B. 18π cm

C. 15π cm

D. 12π cm

18. Which of the following is equal to sin 60 deg? A. sin (– 420°)

B. sin π/6

C. sin 780°

– 30°)

D. sin (

19. An angle of one revolution is equal to an angle whose measure in terms of radians as: A. π/2

B. π

C. 2π

D. 3π

20. In what quadrant do secant and cosecant of an angle have the same algebraic sign? A. III and IV

B. II and IV

C. I and III

D. I and II

21. What is the exact value of sin 75°?

2 −6 A.

3 −6 B.

4

3 −2 C.

4

6+2 D.

4

4

3

22. Solve for x: 2 − 5 = 5 A. 25

B. 40

C. 50

D. 65 2

23. What will be the solutions for the quadratic equations 2x + 3x + 3 = 0? A. Two distinct rational roots

C. Two distinct irrational roots

B. One repeated rational solution

D. Two distinct imaginary solutions

24. How many solutions does 3x– 4y = 8 and 2x + 8y = 9 have? A. One unique solution, consistent and independent B. No solution, inconsistent C. Infinite solutions, consistent and dependent D. One imaginary solution, indeterminate 25. Find the area of a triangle with vertices (2,– 3), (– 1, – 2 ) and (5, 3). A. 10

B. 10. 5

C. 11.25

D. 12

26. Find the area of a sector with radius 5 m and central angle of 85 degrees. A. 15.8 m B. 16.25 m C. 18.5 m D. 20.75 m 27. In a circle of radius 8 cm, find the length of the arc intercepted by a central angle with measure of 3π/4. A. 6 π

π

B. 8

π

C. 9 3

π

D. 10

2

28. Determine the remainder when f(x) = 2x – 5x + x + 7 is divided by x – 2. A. 0

2B. –

C. 5 25

29. Find the remainder when y A. 1

B. 5

D. 12

7

– y + 5 is divided by y + 1.

C. 5 –

D. 10

30. In the function y = 3/2 cos 2x, what is the amplitude? A. 3

B. 2

C. π

D. 3/2

31. What is the period of the function y = 3/2 sin x/2? A. 3 15

32. Simplify: i

+ 3i

B. 2

C. π

D. 3/2

B. 2i

C. 4i

D. – 4i

21

A. – 2i

33. What is the sum of (5 – 3i) and (– 6 + 5i) ? A. – 1 + 2i

B. 1 – 2i

C. 2 + 2

i

D. – 2 – 2i

34. Find the conjugate of (3 + 5i). A. (5 + 3i)

B. (5 – 3i)

2

C. (3 – 5i)

D. (9 – 25i )

35. What is the absolute value of 2– 3i ? A. 5

B. 13

C. 1

D. 3

36. A ladder 18 feet long leans against a building, forming of 60° with the ground. How high up the side of the building? A. 92

B. 3

C. 3

3

D. 93

37. From a point level with 1000 ft. away from the base of a monument, the angle of elevation to the top of the monument is 35°. Determine the height of the monument in the nearest feet. A. 350 ft.

B. 500 ft.

C. 700 ft

D. 750 ft.

38. In a restaurant, a person can choose from 7 viands, plain, garlic or java rice, 2 kinds of beverages and 3 kinds of dessert. In how many ways can this person choose what to have if he is to order one from each group? A. 52

B. 126

C. 104

D. 252

39. In how many ways can you arrange the letters of the word “Statistics”? A. 2100

B. 4200

C. 8400

D. 16800

40. Find the number of license plates possible in a city using only 3 letters, if none of the letters can be repeated. A. 512

B. 729

C. 15600

D. 17576

41. In how many ways can 7 children be seated in around in a circular table? A. 5040

B. 720

C. 360

D. 120

42. Find the odds in favor of and against rolling a sum of 6 with a pair of dice. A. 5/36

B. 31/36

C. 31/5

D. 5/31

43. A marble is chosen at random from a bag containing 6 white marbles, 4 black marbles and 12 red marbles. What is the probability that it will be red marble or white marble? A. 9/11

B. 6/11

C. 11/9

D. 3/11

44. The muddy river has a current of 3 kph. A motorboat takes as long to go 12 kilometers downstream as to go 8 kilometers upstream. Find the speed of boat in still waters. A. 12

B. 15

C. 18

D. 21

45. If the same number is added to both the numerator and denominator of a fraction ¾, the result is 5/6. Find the number. A. 2

B. 3

C. 4

D. 5

th

46. Find the 15 term of an arithmetic sequence with a1 = 7 and a9 = 19. A. 3

5B. –

C. 32

D. 37

47. A university had an enrolment of 8500 students in the year 2008. Each year the enrolment increased by 350 students. What was the enrolment in the year 2013? A. 10600

B. 10350

C. 10000

D. 6350

48. What is r for the geometric sequence with a2 = 24 and a5 = 64? A. 3

B. 1/3

C. 2

D. ½

49. A certain substance decomposes and loses 20% of its weight each hour, If the srcinal quantity of the substance is 500 grams, how much remains after 8 hours? A. 17.12 g

B. 20 g

C. 80 g

D. 83.37 g

4

rd

50. Find the 3 term of (5a – 2b) . 2 2

A. 600a b

B. 1000

3

a b

C. 600

3

a b

D. 100

2 2

a b

MATHEMATICS 1 MULTIPLE CHOICE 1. If 16 ½ feet equals 1 rod, how many inches are there in 4 rods? a. 2376 c. 198 b. 792 d. 66 2. A 10 cm long steel pipe shrank 0.05% when exposed to below freezing temperature. By how many centimeters did the pipe shrink? A. 5 C. 0.5 B. 0.005 D. 0.05 3. Which of these numbers is greater than ¼ ? A. ½ B. 1 .04 4. What is 30% of 25/18? A. 5/108 B. 5/12

C. D. 1/8 1/ 0.04 C. 25/54 D. 25/6

5. A meter stick was cut into 2 at the 25 cm. mark. What is the ratio of the smaller piece to be larger piece? A. 3:4 C. 1:3 B. 2:5 D. 4:5 6. Which of the following times gives approximately a 900 angle between the two hands of a clock? A. 11:40 C. 7:20 B. 9:00 D. 6:15 7. A package of food can feed 3 children of 2 adults. If in a home there are 5 packages of this food, and there are 12 children to feed, how many adults can be fed? A. 8% C. 10% B. 12% D. 6% 8. One thousand two hundred (1200) boys enrolled in a school this year. Four years ago there were 1, 500 boys enrolled. What is the percent decrease in enrollment? A. 10% C. 8% B. 20% D. 25% 9. What is the ratio of 3/10 to 5/8? A. 37/40 C. 25/12 B. 3/16 D. 12/25 10. It takes a teacher 20 minutes to check 25 papers and to starting take a another rest of set. 5 minutes after 25 before At this rates, howpapers many minutes will it take the teacher to correct 75 test papers? A. 75 C. 70 B. 60 D. 80 11. An arithmetic teacher is putting together her final examination of the 50 problems 46% are on fractions, 20% are on decimals, 18% are on percent, and 16% are on basic computation. Totally, how many problems are on fractions and percent? A. 18 C. 32 B. 45 D. 64

12. Simplify the expression 5! – 4! 6! A. 2/3 C. 7 ½ B. 2/15 D. – 4 13. A detective ruler was found to be 11.5 inches long. Using this ruler, Lourdes was found to be 4 ft. tall. What is the actual height of Lourdes? A. 4 ft. 12 in C. 4 ft. 11.5 in B. 3 ft. 4 in D. 3 ft. 10 in. 14. The equation y = x 4 – 5x2 + 4 has A. Three (3) real roots and one imaginary root B. one (1) real root and three imaginary roots. C. two (2) real roots and two imaginary roots D. four (4) real roots 15. If x > 1, which of the following decreases as x decreases? I. x + x2 II. 2 x - 2x III __1___ x+1 A. III only C. II only B. I only D. I and II only 16. If 16 is 4 more than 3x, then what is the value of 2x 5? A. 4 C. 2 B. 5 D. 3 17. If m and n are integers and mn - 10, which of the following CANNOT be a value of m + n? A. 25 B. 29

C. 50 D. 52

18. If 2x + 13 represents an odd number, what must be the next consecutive odd number? A. 3X + 15 C. 2X + 14 B. 2X + 15 D. 3X + 13 19. Let G (x) = 1 - x 2. What is G (7 )? A. 6 C. 48 B. 8 D. 50 20. How much is 7!/ 5! Divided by 6!/5! A. 35 C. 42 B. 7 D. 14 21. What term will make 36 x 2 + 25 y4 a perfect square trinomial? 2 A. 5 + y2 C. 60xy B. 12xy D. 30xy 22. What must be multiplied to make it rational? A. C. 2 2

B. 2

2

3 2–

2

3+2

D. 3 + 2

to

2

2

th

23. What is the 20 term of the arithmetic series 7,

10, 13….?

A. 46 B. 23

C. 64 D. 32

24. Which of the following is the remainder when 3x3 – 6x2 + 5x + 7 is divided by x – 2? A. 12 C. 3 B. 21 D. 17

25. Let matrix A = 3 -4

5 7

and matrix B = 2 - 1 6 8

Which of the following matrices shows A + B? A.

5 6 -2 1

C.

5 2

D.

30. On a circle with center C, CD CD= 5. Find the radius of the circle A. 19 C. 17 B. 7 D. 13 A

-5 9 48 -29

AB, AB= 24 and

D

B

C

{

B.

4 15

1 -10

6 -1

26. Two numbers have a sum of 53. Three times the smaller number is equal to 19 more than the large number. What are the numbers? A. 19 and 34 C. 18 and 35 B. 17 and 36 D. 16 and 37 27. In the standard (x, y), coordinate plane, the y – coordinate of every point on a line is the same as its corresponding x – coordinate. What is the slope of the line? A. -2 B. 1 C. 3 D. 2 28. In ACB shown below, sin A= ½ and BC = 20 meters Which of the following are the measures of  B, AB and AC ___ ___ 0

A. m angle B =B30 , AB __ = 40, AC ___ = 34.6 B. m angle B = 600, AB = 40, AC = 34.6 ___ ___ C. m angle B = 450, AB = 50, AC = 24.8 ___ ___ D. m angle B = 600, AB = 30, AC = 36.6

31. The measure of each interior angle of a regular polygon is 165 deg. How many sides does it have? A. 29 C. 12 B. 24 D. 15 32. If two points of a line in a plane, then I. II.

all points of the line in the plane. The line pierces the plane in exactly two points III. The plane contains all the points of the line IV. The intersection contains only two points. a. I only c. II only b. II and IV d. I and II 33. In circle O , xy is a diameter O X = 8.5, and YZ = 15. What is the area of XYZ in square unit? A. 127.5

C. 120

B. 60 D. 40 34. What is the measure of angle x if is formed by a chords and a tangent and a major arc intercepted is 280 deg? A. 140 degrees C. 100 degrees B. 20 degrees D. 40 degrees

B

C A 29. Given the marked figure at the right DB = EC. We can prove that ABC = AED by ______________. A. SAA B. SSS

B

C. ASA D. SAS

A

C

D

35. The surface area of a cube is 72 sq.cm. how many as much is the volume whose surface area istimes 144 sq.cm? A. 12 C. 8 B. 9 D. 4 36. An equilateral triangle 8 cm. on each side is divided into smaller equilateral triangle 2 cm. on each side. What is the maximum number of such triangles that can be formed? A. 16 C. 24 B. 20 D. 36

E

45. Which of the following has a graph which is a circle with center at (0,0) and having a radius of 4?

37. ABD and CBD are right angles, AB =

2

9, CD = 5 and AD = 15, what is the length of BC? A. 7 C. 13 B. 15 D.12 A

2

A. x2 + y=2 4

C.

x

+

y

2 B. y + x2 = 16

=4 2

D. (x + y)2 = 16

46. Determine the center of the ellipse whose equation is 4x2 + 9y2 + 48x – 72y + 144 = 0 A. (4, -6) C. (-8, 12) B. (-6, 4) D. (6, -4) 47. What are the coordinate of the center of a circle whose equation is 4x2 + 4y2 – 20 + 4y – 55 = 0?

C C 38. The sum of the measures of the 3 angles of a triangle is 180 degree. If the angles are in the ratio 1:2:3, what is the measure of the largest angle? 0 A. 300 C. 90 0 0 B. 60 D. 45 39. Determine the foci of the ellipse whose equation is 4x2 + 9y – 48x + 72y + 144 = 0. A. (10, -4) C. (12, -4) √20, -4) B. (6, 0) D. (6 + 40. At what point is the function 3x + 2____ discontinuous? X2 – 6X + 9 A. X = - 2 3 B. X = -3

C. X = 9 D. X = 3

a. 1, 2 5 2 b. -5, 1 2 2

2

2sc. 5, - 1

d. -1, - 5 2 2

48. Determine the center of the circle whose equation is x2 + y2 + 8x – 12y + 34 = 0. a. (4, - 6) c. (-8, 12) b. (-4, 6) d. (8, -12) 49. If f(x) = x(x+1)(x+2), then f (n) – f (n equal to ____________? 3 A. n (n + 1) C. n (n + 1) (n – 3) 3 3 B. 3n(n + 1) D. n (n + 1)

–

1) is

41. Find an equation of the line with slope 2 that passes through the midpoint of the line segment connecting (3, -2) and (4, 7). A. 4x + 2y –9 = 0 C. 4x – 2y + 9 = 0 B. 4x – 2y –9 = 0 D. 4x + 2y + 9 = 0

50. A parabolic arch has a height of 25 ft. and a span of 40 ft. How high is the arch 8 ft. from each side of the center? A. 19 ft C. 21 ft. B. 22 ft. D. 20 ft

42. The graph of the function y = x 2 – 2x + 1 meets the x – axis at: A. y = 0 C. y = 1 B y=2 D. y = -1

51. Which of the following equations has a slope

43. What is the area bounded by y = x 2, x = 1, and x = 3 in the x – axis? A. 28 C. 7 3 B. 26 D. 22 3 3 44. The inverse function of y =

A. y =

2x + 3 4

3x + 2 4

C.y =

2x + 4

D. y =

is

3x -4

B. y=

4x - 3

equal to ½? A. 4x + 3y = 12 B. x –2y = 4

C. ½ y = 2x + 7 D. 3y + 6x = 5

52. What is the equation of the parabola whose focus is at (2, -3) and the equation of the directrix is x = 6? A. (y + 3)2 = -2 (x – 4) C. (y + 3) 2 = -8 (x – 4) B. (x – 4)2 = 2 (y + 3) D. (y + 3) 2 = 8(x – 2) 53. areof4 ribbon. differentIndesigns of wrapping andThere 4 colors how many differentpaper ways can a gift be wrapped? A. 14 C. 8 B. 12 D. 16 54. If 2/11 is the probability that an event will happen, what is the probability that it will not occur? 3 2 A. 9/13 C. 7/11 B. 2/13 D. 9/11 55. In a single throw of a pair of dice, what is the probability that a double (2 of the same number) will appear? A. 1/72 C. 1/16 B. 1/36 D. 1/6

56. In how many ways can a 5-member committee of 3 women and 2 men be chosen from a group consisting of 5 women and 4 men? A. 20 B. 30 C. 120 D. 60 57. A snack is made up of a sandwich and softdrink. In how many ways can you choose your snack from ham, bacon and egg sandwiches and four types of softdrinks? A. 15 B. 12 C. 10 D. 6 58. A box contains 7 blue ball pens, and 3 red ball pens. Three ball pens are drawn from the box one after the other. Find the probability that the first two are blue and the third is red. A. 1/360 B. 7/40 D. questions 1/24 59. A student C. is 3/40 to answer 8 out of 10 in an examination. How many ways can he do it if he must answer the first three questions? A. 10 B. 45 C. 21 D. 35 60. Suppose it is known that 65% of the scores in a distribution I greater than 90, this means ______________. A. 10 is at the 65th percentile B. 90 is at the 65th percentile C. 90 is at the 35th percentile D. 35 is at the 90th percentile 61. If 0 deg =?

≤ Ө ≥ 900 and 4 cos2 Ө - 3 = 0, then Ө A. 45 degrees B. 60 degrees C. 30 degrees D. 0 degrees

62. Suppose tan Ө is in the third quadrant. What is the numerical value of cos Ө ? A.3/5 – B. -4/5 C. 4/5 D. 5/4 63. Starting at 12 noon of Monday up to Tuesday 3:00 pm, how many degrees does the hour hand of a wall clock travel? A. 810 degrees B. 450 degrees C. 210 degrees D. 90 degrees 64. Evaluate tan 3π 4 A. 2 B. 1

C. -1

D.

3

65. What kind of triangle is determined by the points M(0,1),R(4,0) and Q(3,4)? A. Scalene B. Isosceles C. Right D. Equilateral 66. Which of the following is equal to sin π cos π? 8 8 A. 2 sin π C. –sin π 4 8 π ) B. sin π D. sin (4 4

67. If Cos Ө is divided by Cot Ө, what will be the resulting function?

A. Csc Ө C. Tan Ө

Ө D. Sec Ө

B. Sin

68. If y =Arc sin ½ which of the following is a value of y? A. π B. π C. π D. π 2 6 4 3

69. The sum of the interior angles of a polygon is 2160 deg. How many sides does the polygon have? A. 16 B. 14 C. 22 D. 20 70. The value of tan – 21π is __________? 8 A. 1

B. -1

C.

2+1

D. -

2–1

71. What is true about the graph of y = sin2 x + cos2 x? I. It is a line parallel to the x – axis. II. Its y – intercept is 1. III. It is curve. A. II and III only C. I and II only

B. III only D. II only

72. Find the length of the circular arc which substends an angle of 450 at the center of a circle with a radius of 8. A .3 π B. π C. π D. 2 π 2 2 73. Which of the following is an asymptote in the graph of y=tan x? A. x = π B. x = 2 π C. x = 3 π D. x = π 2 74. What are the roots of the equation (x 2-4) (x-1) = 0? A. -4, +1, -1 B. -1, +1, -2 C. +2, -2 + 1 D. +1, -1, +2 75. A railroad toy is laid out in circulation form. What diameter should be used if the track is to change direction by 32 degrees in a distance of 50 cm? A. 179.2 cm B. 179.32 cm C. 179.62 cm. D. 179.14 cm 76. If log3 x=2 and log2 y = 3, what is xy? A. 27 B. 9 C. 81

D. 72

77. A circle has a radius of 18 cm. find the arc length cut by a central angle equal to 3 π radians 8 A. 21.20 cm C. 56.55 cm

B. 54 cm D. 6.75 cm

78. Which of the following is equal to sin 60 deg? A. sin (- 4200) B. sin π 6 0 C. sin 7800 D. sin (-30 ) 79. An angle of one revolution is equal to an angle whose measure in terms of radians as: A. π B. π C. 2 π D. 3 π 2 80. If each base angle of an isosceles triangle is four times the vertex angle, what is the measure of the vertex angles? 0 0 A. 900 B. 20 0 C. 40 D. 36

81. If tan

sec Ө? A. 5

Ө = 5/3 and Ө is in Quadrant II, what is

34 34

C.

B. 34/9

-3

34 34

D. .

34 3 82. In what quadrants do the secant and co-secant of an angle have the same algebraic sign A. III and IV B. II and IV C. I and III D. I and II 83. What is the exact value of sin 75 degrees? A. 2 4

B.

6

C.

3 –6 4

D.

36+

3

4

85. Evaluate

3

C.

3 4

D.

3 2

√x

C. 2 √x + C

B. ½ √x + C

D. 1

+C

2 √x

86. The velocity at time t, of a moving body is given by v=at, where c is a constant. If the body’s coordinates is S. at time t=0, what is the distance s as a function of t? 2 A. S = at+ So B. S = ½ at 2 C. S = ½ at 2 + So D. S = ½ at + C 87. What is the maximum area of a rectangle which can be enclosed by a wire 48 m long? A. 320 sq.m B. 576 sq.m C. 560 sq.m D. 144 sq.m 88. Find the curve whose slope at the point (x,y) is 3x2 if the curve is also required to pass through the point (1,1) 2 A. x3 B. y=3x +1 C. y = x23 – D. x 1 – 1

C. x2 + y=2 1 25 4

A. 1/3 x + ½ x + 1 C.+x22x

–

98. Evaluate:

2

Ө

sin Ө π _

π tan Ө

4 A. ½ B. √3 C. √2 D. √2 2 3 2 99. Evaluate: f(x) = cos (2x3 + 5) A. -6x2 sin (2x+3 5) C. –sin (2x3 + 5) 2 B. –sin (6x ) D. cos (6x2) 100. What is the value of the first derivative of the function y = x2 + 3x + 5 at x = -1? A. 3 B. 1 C. 5 D. 1 101. Evaluate: lim(3x3 – 4x + 1) x 2 A. 209 B. 15 C. 17 D. 33 102. Evaluate ⌠(2x + 3)dx A. 2x+2 C 3 C. 2(2x + 3) + C 4

B. x

2

+ 3x + C

2

D. x + 3x + C 2

– 5)4 + C 8

95 and 87?

8

2 2

lim

103. Find the first derivative of y = x 3 -12x + 11 A. 3x2 –12 B. x 2 -12 C. 3x-12 D. 2x3 + 12 104. A spherical balloon is being inflated so that its volume is increased at the rate of 5 cu. m/min. At what rate in (m/min) is the diameter increasing when the diameter is 6 m? A .5 B. 36 π C. 5 D. 864 π 72 π 8π 3 105. What is the range of these scores: 73, 89,81,

91. Evaluate ⌠(x2 – 5)3 dx A. (x2 – 5)2+ C B. (x 2 – 5)2 + C D. (x

+ y2 = 1 9 1

97. If y=x3 – 2x, what is y? A. 3x2+ 2x B. ½ x 2 – x 2 C. 3x - 2 D. 3x4 x2

2

90. Which of the following quadratic functions will have a graph which opens downward? A. y = 1 – (3x – x2) C. y = 5x (x -1) + 3 B. y = -4 (2 – x2) + 5x D. y = 7x – (5x2 + 1)

4 C. (x2 – 5)+ C

2

C. 1/3 cos 2 x + c D. cos3 x + sin x + c 3 95. What is the slope of y = 3x2 – 2x at (2,8)? A. 0 B. 4 C. 2/3 D. 10 96. Find the points where the tangent is parallel to the a – axis A. (3, -7) B. (-3, 7) C. (-1, -3) D. (-1, 3)

B. 3x = 2x – 1 D. x 2 – x =1 2

D. x

+ y2 = 1 9

93. A stone is dropped into a still pond. Concentric circular ripples spread out and the radius of the disturbed region increases at the rate of 16 cm/sec. at what rate does the area of the disturbed region

89. Find dy in x2 – x2 + x -1 dx 3 2 2

2

4

A. 1/3 cosx2 + c B. 1/2x + c

⌠dx

A. √x + C

B. x

94. Evaluate ⌠sin 3xdx

2

84. What is tan 2A if sin A = ½ ? B.

2 A. x2 + y= 1 16 9

2 radius is 4 cm? increase A. 16when π cmthe /sec B. 256 π cm2/sec C. 128 π cm2/sec D. 64 π cm2/sec

4

A. 1/3

92. Write this equation of an ellipse in standard form: 9x2 + 4y2 = 36

A. 23 B. 4.4 C. 12 D. 22 106. This mean of a given distribution is 70. if each score in this distribution is increased by 5, when the mean of the new distribution will be: A. 80 B. 75 C. 90 D. 65 107. Which of the following is LEAST influenced by extremely large or extremely small values? A. Range C. Standard deviation B. Quartile deviation D. Average deviation 108. If we add the same constant C to each item in a set of data what will happen to the standard deviation of the new set of data? A. The standard deviation of the srcinal data is multiplier c times The standard deviation of the srcinal data plusB.a constant c C. The standard deviation remains unchanged. D. The standard deviation of the srcinal data is doubled 109. Susan got the following MATH scores: 78, 80, 77, 85, 90 What is her mean Score? A. 76 B. 82 C. 80 D. 78 110. The scores of a student in three test 19, 17 and th 15. what must be his/her 4 score to make an average of 17? A. 18 B. 16 C. 15 D. 17 111. What is the value of = 4, x4 = 5? A. 14 B. 26

∑ 2x, if x1 = 2, x2 = 3, x3 C. 54

D. 28

112. The lower boundary of a class interval is 29.5 and the interval width is 5. The upper boundary is A. 30 B. 35 C. 34.5 D. 35.5 113. John borrowed P25, 000 form Robert at an interest rate of 12%. How much interest should John pay after 2 years? A. P 7000 B. P6, 400 C. P2,400 D. P6,000 114. A loan of P10, 000 was repaid at the end of 14 moths. What amount was repaid if 12% annual rate of interest was charged? A. P12, 500 B. P12, 000 C. P11, 100 D. P11, 400 115. Melissa wants to invest her P50, 000. Threefourth was invested at 6% simple interest and ¼ at 5% interest 2 years 3 months. What for is the finaland amount at 2 years and 3 months? A. P56, 468.75 C. P57,468.75

B. P58, 468.75 D. P59, 468.75

116. Mr. Santos is employed on a salary plus commission basis. His total earning last month amounted to P18, 500 for which P16, 000 represented his basic monthly salary. If his total sales from that moth amounted to P125, 000, what percent of commission did he receive? A. 5% B. 10% C. 20% D. 2%

117. Which will earn more interest, P6,000 invested at 8% for 100 days, P6000 invested at 5% for 270 days, P5,000 invested at 12% for 120 days or P5000 invested at 10% for 150 days? A. P6000 invested at 8% B. P5000 invested at 12% C. P6000 invested at 5% D. P5000 invested at 10% 118. Lucia borrowed P8000 form the bank February 5, 2006 and paid it back on July 15, 2006 at 6% interest. How much exact amount did Lucia pay on the agreed date? A. P8, 610.41 B. P8, 210.41 510.41 agency D. P8,able 410.41 119.C.AP8, collection was to collect 7/8 of P20, 800 account from a client and charged 10% for services which includes expenses. How much commission did the agency receive? A. P2080 B. P182 C. P18, 200 D. 1820 120. The proceeds of a P20, 000 investment discounted for 2 years were P15,000. What was the discount rate? A. 16.7% B. 25% C. 12.5% D. 33%

MATHEMATICS 2

1. Find the GCF of 120, 80 and 140. a. 4 b. 12 c. 10 d. 20 2. When Joseph sorts his collection of marbles into groups of 2, 3, 4 or 8, there is always one marble left. What is the smallest number of marbles Joseph have? a. 35 b. 28 c. 25 d. 21 3. Two brothers and their young sister are to divide an inheritance worth P300,000 in the ratio of 5:6:4, with the girl getting the least share. How much is the share of the sister? a. P20,000 b. P80,000 c. P100,000 d. P40,000 4. Which of the following statements is FALSE? a. If a number is a multiple of 9, then it is also a multiple of 3 b. If a number is divided by 9, then it is divisible by 3 c. If 9 is a divisor of a number, then 3 is also a divisor of the number d. If a number is divisible by 3, then it is divided by 9. 5. Find 1 2/5 x (1 ¼ ÷ 7/16) a. 1/5 b. ¾ c. 4 d. 4/5 6. Which of the following numbers is the difference of two consecutive prime numbers less than 41? a. 8 b. 2 c. 9 d. 6 7. Employees in a firm are entitled to 1 – day leave for every 20 working days. If in 2007, this firm had 280 working days, how many days of vacation leave are the employees entitled to? a. 20 days b. 15 days c. 24 days d. 14 days 8. What is the perimeter of a square whose area is 256 m2? a. 1024 m b. 64 m c. 32 m d. 16 m 9. How much bigger is 36 than 63 ? a. 27 b. 235 c. 513 d. 729 10. Three brothers inherited a cash amount of P 120,000 and they divided it among themselves in the ratio of 5:2:1. How much more is the largest share than the smallest share? a. P15,000 b. P30,000 c. P60,000 d. P75,000 11. The simplest expression 240 is ____________. 420 a. 4 b. 1 20 2 c. 1 d. 2 12. 40% of 35 is what percent of 140? a. 29% b. 28% c. 10% d. 14% 13. The initial temperature of an object was 27°C. After exposing the object to different surrounding media, its temperature decreased by 7°C, then increased by 10°C and then finally decreased by 25°C. What was the final temperature of the objects? a. 15°C b. 20°C c. 5C d. 5°C

14. A tank that holds 400 gallons of water can be filled by one pipe in 15 minutes and emptied by another in 40 minutes. How long would it take to fill the tank if both pipes are functioning? a. 28 min b. 21 min c. 24 min d. 23 min 15. Linda paid P 360.00 for 12 notebooks but she was given 3 additional notebooks for free. In effect, what rate of discount did she enjoy? a. 50 % b. 40% c. 18% d. 20% 16. A clock is 4 minutes ahead every 8 hours. If the clock is set correctly at 8:00 a.m. Monday, what time will be shown in this clock at 8:00 p.m. Friday? a. 8:48 am b. 8:54 am c. 9:08 am d. 8:56 am 17. What is the LEAST positive integer that has 6.8 and 10 as factors? a. 240 b. 80 c. 120 d. 300 18. Jose walks M miles in H hours. At the same rate how many miles will he walks in J hours? a. J b. M c. MJ d. MH MH H-J H J 19. What numbers is NOT exactly divisible by 8? a. 7304 b. 5000 c. 3584 d. 5218 20. The bus stops are equally spaced along a bus route. The distance from the first to the third stops is 600 meters. How far is the distance from the first stop to the last stop? a. 2100 meters b. 2400 meters c. 900 meters d. 2700 meters 21. Three out of every 24 flashlights turned out by a particular factory are found to be effective. If the factory turns out 1248 flashlights in a week, how many are defective? a. 72 b. 18 c. 416 d. 156 22. If ab + cd = 12ad, and ad = O, then b + c = __________ d a a. ab + cd b. 12 c. 12a + 12d d. 12 ac 23. What must be the value of x in the arithmetic progression x – 7, x – 2, x + 3 so that its 10th term will be 40? a. 3 b. 2 c. 1 d. 4 24. The roots of the equation (x – 5)(x – 3) = 0 are __________ a. 3 or -5 b. 3 or 5 c. -3 or 5 d. -3 or -5 25. The average of m and n is 9, and p – 12. What is the average of m, n, and p? a. 7 b. 21 c. 10 d. 18 26. An angle of 7 π radians is equal to ___________ a. 115° b. 225° c. 105° d. 89° 27. What is the value of x in ½ + 2/3 – 1/x – 1/6? a. 2 b. 3 c. 1 d. 4

28. M varies inversely as N. If M = 25 when N = 2, what is M when N = 5? a. 16 b. 10 c. 5 d. 2 29. Which of the following expressions always gives an odd number? a. (N – 1) (N –5) b. 5N – 1 c. (2N + 1)2 d. N2 -1 30. Three circles are tangent to each other extremely. What is the perimeter of the triangle formed by connecting the centers if the areas of the circles are 9π, 16πand 25 π respectively? a. 24cm b. 12cm c. 50cm d. 25cm 31. A falling body strikes to the ground with a velocity V, which varies directly as the square root of the distance d it falls. If the body that falls 100 feet strikes the ground with a velocity of 80ft./sec.. with what velocity will a ball dropped from a height of 550 ft. strike the ground? a. 150ft/sec b. 190ft/sec. c. 170ft/sec d. 188ft/sec 32. Which of the following is true identity? a. sin2 θ = cos2 θ b. cos 2 θ + 1 = sin2 θ c. tan 2 θ = sec2 θ – 1 d. cos 2 θ – 1 = sin2 θ 33. The same number is subtracted from the numerator and denominator of the fraction 8/11. If the resulting fraction is equivalent to 2/3, what is the number subtracted? a. 5 b. 2 c. 3 d. 4 4 34. What is √256? a. 5.4 b. 4 c. 16 d. 6 35. Given this table of numbers relating x to y, what is x when y is 45? X 0 4 8 12 16 y 3 9 15 21 27 36. Madel spent one – sixth of her money in one store. In the next store she spent three times as much as she spent in the first store, and had P80.00 left. How much money did she have at the start? a. P240.00 b. P360.00 c. P252.00 d. P380.00

40. Which statement is TRUE? I. 1 m is longer than 1 km II. I mm is longer than I in III. 1 gallon is heavier than 1 lb IV. 1 gallon is more than 1 liter a. I only b. I and IV c. I and III d. III and IV 41. If 3x = 81 and 4y = 64. What is (x-y)? a. 17 b. 1 c. 0 d. -1 42. Two buses leave the same station at 9:00 pm one bus travels north at the rate of 30kph and the other travels east at 40 kph. How many kilometers apart are the buses at 10 pm? a. 100 km b. 70 km c. 50km d. 140km 43. Which of the following is a factor of x 4 – 4x 3 – 6x2 + 3x + 10? a. (x2) – b. (x + 5) c. (x + 1) d. (x – 5) 44. If X is an even number and Y is odd number, which of these expressions will always give an even number? a. X+Y b. XY c. X ÷ Y d. X – Y 45. Which of these is NOT factorable? a. 6x2 – 5x6– b. 6x 2 – 13x + 6 2 c. 6x – 7x6– d. 6x 2 – 35x – 6 46. What is the value of x in this equation? 1 + 2 - 1 = 1 2 3 x 6 a. 4 b. 2 c. 1 d. 3 47. If 16 is 4 more than 3x, then what is the value of 2x – 5? a. 2 b. 4 c. 5 d. 3 48. What is the area of the shaded portion as shown in the figure? 4

2

37. Which gives the quantity “the time it takes to red

a book that is x page long at a rate of y ages per

hour”?

a. y/x hours b. xy hours c. x/y hours d. (x+ y) hours 38. If x/y = 4/z, then 16y2 equals a. x2 z2 b. x2 c. x3 z d. x2z2 16 16 39. Marlyn copied x + y2 instead of (x + y) 2. Compute the amount of her error if x = 8 and y = 3. a. 12 b. 0 c. 104 d. 96

b. π c. 8 4 π d. a. 8 -8 – 2π π 49. Which of these could be the measures of the angles of an isosceles triangle? a. 60°, 60°, 80° b. 32°, 32°, 116° c. 51°, 98°, 51° d. 45°, 45°, 100° 50. A box is 12 in. in width, 16 in. in length, and 6 in. in height. How many square inches of paper would be required to cover it on all sides? a. 900 b. 192 c. 720 d. 360 51. What is the total length of fencing needed to enclose a rectangular area 46 feet by 34 feet? a. 48 yd b. 26 yd.1 ft c. 53 1/3 yd d. 52 yd 2ft

52. If the area of a square is 81x 2, what will represent the perimeter of the square? a. 18x2 b. 81x c. 9x2 d. 36yd 2ft 53. The vertices of a triangle are (2.1), (2.5), and (5.1). What is the area of a triangle? a. 10 b. 8 c. 5 d. 6 54. Four small squares are put together to form a bigger square. If the perimeter of the big square is 256, what is the side of each of the smaller squares? a. 24 b. 16 c. 32 d. 8 55. These segments have lengths a, b and c, with a  b  c. Under what conditions could these segments be used to form a triangle? a. a + b > b b. b + c > a c. a + c c 56. The measure of each interior angle of a regular polygon is 165 degrees. How many sides does it have? a. 12 b. 15 c. 24 d. 29 57. What happens to the circumstance of a circle if the length of the radius is doubled? a. It becomes 3 times longer b. It becomes 4 times longer c. It is doubled d. Itofisthe halved 58. Which following equations is parallel to 3x – 2 = 4 and has a y – intercept of -3? a. y = 2x3– b. 2x – 3y = 6 c. 2x –3y = 4 d. 3x – 2y = 6 59. Which line in the figure has an underlined slope?

2 1 3 4 a. 3 b. 4 c. 2 d. 1 60. All of the quadrilaterals have two pairs of opposite sides parallel EXCEPT a. Rectangle b. Trapezoid

61. Thec. Square area of a circle d.isRhombus 49π. What is its circumstances in terms of π? a. 49 π b. 14 π c. 28 π d. 98 π 62. What is the slope of the line that passes through the points (-1,2) and (3, - 4)? a. 3/2 b. 3 c. 2 d. -3/2 63. A rectangular bin 4 feet long, 3 feet wide and 2 feet high is solidly packed with bricks whose dimensions are 8 inches, 4 inches and 2 inches. How many bricks are there in the bin? a. 54 b. 848 c. 648 d. 320

64. P is 7 less than the square of the sum of the quotient of x and y and the product of x and y. Which equation expresses this relationship? a. P = (7 – x/y + xy) 2 c. P = 7 (x/y + xy)2 2 b. P = (x/y + xy – 7) d. P= (x/y + xy) 2 - 7 65. What is 135° in radians? a. 3π/4 b. 5 π/8 c. 2 π d. 3 3 4π 66. A square is inscribed in a circle with radius r. Find the area of the shaded region. r

*

a. (2 – π)r2 b. (π – 2) r2 2 c. π – 2 r d. πr2 – 2 67. If the first derivative of f (x) is 3x 2 + 2x, what is f(x)? a. 3 x+3 2x b. x 3 + x2 c. 6 x3 + 2 x2 d. 3 x 3 + x 68. What are the endpoints of the major axis of 9 (x – 3)2 + 25 (y – 2)2 – 225? a. (1, - 2) and ( - 7, -2) c. (-1, 2) and (7,2) b. (0, 2) and (6,2) d. (-2, 2) and (8, 2) 69. What is the equation of a straight line with a gradient – 1 and y – intercept 4? a. 2y = x + 4 b. 4y = x + 1 c. y = 2x + 5 d. y = -x + 4 70. If f(x) = 5x3 – x2 + 3, find the second derivative of f(x) 2 a. 45x2x – b. 30x –2 2 3 c. 3x2x – d. 15x – 2xy 71. Evaluate: lim √ 3x2 – 4 is ___________ x

2

4a. – b. 4 c. 2 d. √ 10 72. What is the second derivative of y = (x 3 + 2)2? a. 2x2 (x3 2) – b. 6x (x 3 + 2) c. 6 (x5 + 3x2) d. 6x2 ( 5x2 + 8) 73. The cost C (x) in the thousand of pesos of producing x thousand math textbooks is given by C (x) = 30 + 20x – 0.5x2. What is the marginal cost of producing 12,000 textbooks? a. P 12,000 b. P16,000 c. P10,000 d. P8,000 74. Determine the equation of the line parallel to 3x – 8y + 11 – 0 and passing through (8,10) a. 8x – 3y –34 = 0 b. 3x – 8y + 56 = 0 c. 3x – 8y – 104 = 0 d. 3x + 8y + 56 = 0 75. What is the equation of the line with an x – intercept of 5 and y – intercept of 2? a. 5y + 2x = 4 b. 2y + 5x = 25 c. 2y + 5x = 10 d. 5y + 2x = 10 76. A particular car run 100 km on x liters of gasoline. If the car travels a fixed route of 20km per day, how many liters of gasoline will the car consume in 365 days? a. 73x b. x/73 c. 36x d.x/36

77. One end of a line segment is P(1,2). If the midpoint of the segment is (4, 3), find the coordinate of the other endpoint of the segment. a. (6,9) b. (6,7) c. (5,8) d. (7,8) 78. What is the equation of the line through (4,1) and parallel to the line through (7,3) and (5,-1)? a. x –4y = 0 b. 3x + y = 13 c. 2x y– = 7 d. 5x – 6y = 14 79. What is the equation of a circle with center (- 5, 7) and a diameter that is 20 cm? a. (x +5)2 + ( y – 7)2 b. (x – 5)2 + (x + 7)2 = 100 c. (x – 5)2 + (y + 7)2 d. (x+ 5)2 + (x – 7)2 – 100 80. Find the first derivative of (x 2 + 2x) (x3) 4 a. x4 + 6x3 b. 2x + 2x3 4 c. 5x4 + 8x3 d. 3x + 6x3 81. What is the equation of the line with slope – 6/11 and passing through C(4,9)? a. 6x + 11y – 75 = 0 b. 11x + 6y + 10 = 0 c. 6x – 11y + 123 = 0 d. 6x + 11y – 123 = 0 82. How long is the diameter of x 2 – y2 – 10x + 14y = -10? a. 128 units b. 16 units c. 8 units d. 64 units 83. In the graph of y = 2 – 2x2 – 4x, what is the highest point? a. (1, - 4) b.(4, - 1) c. (- 4, 1)) d. ( - 1, 4) 84. What are the coordinates of the point of intersection of the tangent with slope equal to 2 the curve y = - x 2 – 4x + 7? a. ( - 3, 13) b. (- 3, 28) c. (-3, 10) d. (3,4) 85. The derivative of y = 2x 3 – 5x2 + 3x + 2 is _______ a. y1 = 6x2 – 10x3– c. y 1 = - 6x2 + 10x – 3 b. y1 = 6x2 6x – +3 d. y 1 = 6x2 – 10x + 3 86. Which of the following quadratic functions will have a graph which opens downward? a. y = 1 – (3x – x2) b. y = 5x (x – 1) + 3 c. y = 7x – (5x2 + 1) d. y = -4 (2 – x2) + 5x 87. What is the median of the following set of numbers: 11,8,3,2,3,2,14,13,8,7,8,6? a. 8.5 b. 7 c. 7.8 d. 5 88. The scores of eight boys in an IQ Test are as follows: 100,120,101.107.116, and 101. What is the 75 percentile? a. 120 b. 118 c. 116 d. 117 89. A pair of dice is thrown. If the sum of the dots on the two dice is 6, what is the probability that one of the dice is 2? a. 1/18 b. 2/5 c. 1/6 d. 11/36 90. If majority of the students in Statistics class was able to answer most of the questions given during the examination, the score is expected to be ____________ a. bimodal b. normally distributed c. negatively skewed d. positively skewed

91. Thirty people were asked a question answerable by yes or no and maybe, their responses were recorded. Which measure of central tendency is MOST appropriate for the recorded data? a. Range b. Mode c. Mean d. Median 92. Determine the length of the latus rectum of the parabola. 5y2 + 20y – 6x + 20 = 0 a. 6 b. 3/5 c. 3/10 d. 6.5 93. Francisco received the following scores on tests in Calculus: 73,89,81,95, and 87. What is his mean score? a. 84 b. 85 c. 89 d. 87 94. How many ways can 3 books be selected from a group of 7 books? a. 42 b. 35 c. 210 d. 21 95. The president of the senior class is authorized to appoint a committee of 5 from 4 elected males and 8 elected females. How many different committees are possible if 3 elected males are to be members? a. 32 b. 112 c. 120 d. 220 96. How many different signals, each consisting of 7 flags hung in a vertical line, can be formed from 4 identical red flags and 3 identical green flags? a. 7!/(4!3!) b. (7 – 4)! c. 7! d. 4!3! 97. How many distinct permutations can be made from the letters of the word INFINITY if all letters are be taken? a. 3540 b. 4512 c. 3360 d. 3124 98. Four married couples have reserved eight seats in a row for a basketball game. In hw many ways can they be seated if each couple is sit together? a. 350 b. 384 c. 320 d. 300 99. A family has 3 children. What is the probability that all children are boy? a. 3/8 b. ½ c. 1/8 d. 1/6 100. A card is selected at random from a box of 20 cards numbered 1 to 20. What is the probability that the number is multiple of 3? a. 7/20 b. 3/20 c. 5/20 d. 3/10 101. How much interest would be collected on a loan of P25,000 borrowed for 7 months at 10% per year? a. P 1,485.33 b. P 1,548.33 c. P1,458.33 d. P 1,584.33 102. A store is giving a 15% discount off marked price of electrical items, what is the discounted price of a water dispenser marked at P3,000? a. P3,150 b. P2, 550 c. P 2,850 d. P 3,450 103. If sin A = ½, what is tan 2A? a. √ 3 /4 b. √ 3 c. √ 3/2 d. 1/3

104. In a sale, Janice paid P 270.00 for a skirt srcinally marked P450.00. What was the discount rate given? a. 20% b. 60% c. 50% d. 40% 105. In sin2 x + y = m, and cos2 x+ y = n, find y. a. m + n + 1 b. m – n – 1 2 2 c. m –n + 1 d. m + n – 1 2 2 106. Atty. Bravo collected P 10,750 from Mrs. Rivera after deducting commission, the principal received P

8, 492.35. What was the attorney’s rate of

commission? a. 27% b. 12% c. 21% d. 79% 107. In one Sophomore class, 1/3 of the students are honor students, 2/7 are varsity athletes. If there are four athletes in the class, how many students are there in the class? a. 42 b. 54 c. 73 d. 62 108. How much money was borrowed at 15% simple interest for 8 months if the interest paid was P3,500? a. P36,500 b. P36,000 c. P35,000 d. P37,200 109. Which of these numbers CANNOT be the hypotenuse of a right triangle whose sides are whole numbers? a. 10 b. 13 c. 26 d. 4 110. A man sold his car for P180,000. This means losing 10% of the price he paid for it. What was the cost of the car when he bought it? a. P195,000 b. P190,000 c. P200,000 d. P210,000 111. What is the area of a square lot whose diagonal

is 6√2 m?

a. 28 sq.m b. 72 sq.m c. 12 sq.m d. 36 sq.m 112. Find the value of tan α if sin α =- 5/13 and tan α > 0 a. 5/13 b. 5/12 c. 1 d. 12/13 113. A businessman borrowed P50,000 from a rural bank at 14% compound interest compounded semiannually, how much did he pay at the end of two years? a. P61,252.15 b. P57,245 c. P53,407.15 d. P65,539.80 114. An equilateral triangle is drawn with one vertex at the srcin and the two other vertices on the circle x2 + y2 = 25. Find the area of the triangle a. 12√2 b. 25√3 4 c. 5√3 d. 25 115. Jocelyn has P7,200 in the time deposit in a bank which she receives 12% interest each year. How much interest will she receive in five years? a. P7,200 b. P4,320 c. P4,800 d. P5,200

116. The interest of a loan of P6, 350.00 is P158.75. If the loan is to be paid after 150 days, what is the rate of interest charged? a. 5% b. 7% c. 6% d. 4% 117. Which of these is equal to 1 + tan2 x? 2 a. sec x 2 b. cos x 2 2 c. cot x d. sin x 118. What is the second derivative of y = (x 3 + 2)2? a. 6 (x5 + 3x2) b. 2x2 (x3 + 2) 3 c. 6x2 (x+ 2) d. 6x (x 3 + 2) 119. Transform an angle of 190 deg to radian measure a. 20π/18 radians b. π / 18radians c. 3.39 radians d. 19 π radians 18 120. If sin A = √2, what is sin 2A? 2 a. ½ b. 2 c. 2√2 d. 1

LET (Math Major) Reviewer Combined 195Pages 16mb

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