IB Mathematics SL Topical Review All numbered problems are required and are to be worked ON YOUR OWN PAPER and turned in as specified on your assignment sheet. You must show your work. On questions marked “no calculator,” you much show sufficient to answer them without the use of a calculator. Diagrams are not necessarily to scale. Give any numerical answers exactly or to three significant figures unless otherwise specified in the problems. The lettered problems marked “Be Able To” are additional practice, and are not required.
Algebra, Functions, and Equations Sequences and Series 1. The Acme insurance company sells two savings plans, Plan A and Plan B. For Plan A, an investor starts with an initial deposit of $1000 and increases this by $80 each month, so that in the second month, the deposit is $1080, the next month it is $1160, and so on. For Plan B, the investor again starts with $1000 and each month deposits 6% more than the previous month. a) Write down the amount of money invested under Plan B in the second and third months. Give your answers to parts (b) and (c) correct to the nearest dollar. b) Find the amount of the 12th deposit for each Plan. c) Find the total amount of money invested during the first 12 months i) under Plan A; ii) under Plan B. 2.
(no calculator) Given that 24, b, c, are the first three terms of an arithmetic sequence, with non-zero common difference, and that 24, c, b, are the first three terms of a geometric sequence, find b and c.
3.
(no calculator) In an arithmetic sequence, the first term is –2, the fourth term is 16, and the nth term is 11 998. a) Find the common difference d. b) Find the value of n.
Exponents and Logarithms 4. Solve for real x:
5.
1 8− x = 4
3
a)
log 8 x = 3–1
e)
1 log 9 81 + log 9 + log 9 3 = log 9 x 9
b)
c)
log 27 x = 1 − log 27 ( x − 0.4 )
d)
2x = 7 x – 1
The number of radioactive atoms N of a particular material present at time t years may be written in the form N = 5000 e–kt, where 5000 is the number of atoms present when t = 0, and k is a positive constant. It is found that N = 2500 when t = 5 years. a) Determine the value of k. b) At what value of t will N = 50?
Binomial Theorem (Pascal’s triangle, combinations) 6.
5
In one of the terms in the expansion of ( x3 − 3 y 2 ) , the powers of x and y will be identical. Find this term, giving your answer in its simplest form.
7.
(no calculator) Find the coefficient of y3 in the expansion of (3 – 2y)5, simplifying your answer as much as possible. 1
9
8.
1 Consider the expansion of 3x 2 − . x
a) b)
How many terms are there in this expansion? Find the constant term in this expansion.
Functions and Graphing 9.
(no calculator) Let f(x) = 2x, and g(x) = a)
10.
(g ° f ) (3);
x , (x ≠ 2). Find x−2
b)
g–1(5).
(no calculator) The diagram shows parts of the graphs of y = x2 and y = 5 – 3(x – 4)2. The graph of y = x2 may be transformed into the graph of y = 5 – 3(x – 4)2 by these transformations. A reflection in the line y = 0 followed by a vertical stretch with scale factor k followed by a horizontal translation of p units followed by a vertical translation of q units. Write down the values of a) k; b) p; c)
q.
Circular Functions and Trigonometry 1.
2.
3.
4.
The diagram at right shows a sector AOB of a circle of radius 15 cm and centre O. The angle θ at the centre of the circle is 2 radians. a) Calculate the length of arc AB. b) Calculate the area of the sector AOB. c) Calculate the area of the shaded region. d) Calculate the perimeter of the shaded region.
A
B
θ O
(no calculator) a) Write the expression 3 sin2 x + 4 cos x in the form a cos2 x + b cos x + c. b) Hence solve the equation 3 sin2 x + 4 cos x + 4 = 0, 0 ≤ x ≤ 90°, showing your work. π 3 (no calculator) Determine the two solutions in the interval [0, π) to the equation sin2x + = , 6 2 giving your answer in terms of π. Show your work. (no calculator) The depth, y metres, of sea water in a bay t hours after midnight is represented by the function 2π y = a + bcos t , where a, b, and k are constants. k The water is at a maximum depth of 14.3 m at midnight and noon, and is at a minimum depth of 10.3 m at 06:00 and 18:00. Write down the values of a, b, and k.
2
5.
A farmer owns a triangular field ABC. One side of the triangle, [AC], is 104 m in length; a second side, [AB], is 65 m in length; and the angle between these two sides is 60°. a) Use the cosine rule to calculate the length of the third side of the field. 3 b) Given that sin 60° = , express the area of the field in the form p 3 , 2 where p is an integer. 104 m The farmer divides the field into two parts by constructing a straight fence, [AD], of length x m, which bisects the 60° angle, as shown in the diagram. A 65x c) Show that the smaller area is given by and obtain a similar 4 expression for the larger area. d) Hence determine the value of x in the form q 3 , where q is an integer. ˆ and sin ADB ˆ ? e) i) What can be said about sin ADC BD 5 = . ii) Use the result of part (i) and the sine rule to prove that DC 8
6.
7.
8.
S is the base of a vertical pole TS. S lies on AB, where A and B are 92.5 meters apart on horizontal ground. ∠TAB = 20˚ and ∠TBA = 30˚. Calculate the length of the pole TS to the nearest tenth of a meter.
D
65 m B
A
The diagrams show two triangles both satisfying the conditions AB = 20 cm, AC = 17 cm, ABˆ C = 50°. Calculate the size of ACˆ B in Triangle 2. B
30° 30°
A
C B
Triangle 1
C
Triangle 2
C
T
A
S
B
In the triangle ABC it is given that BC = 9 cm, CA = 13 cm, AB = 10 cm and D is the midpoint of [AB]. By applying the cosine formula to each of two triangles, or otherwise, find CD.
Be Able To A. B.
θ 2 1 (no calculator) Solve the equation cos = , 0° ≤ θ ≤ 360°. 2 2 (no calculator) Let f (x) = 6 + 6sinx. Part of the graph of f is shown here. The shaded region is enclosed by the curve of f, the x-axis, and the y-axis. a) Solve for 0 ≤ x < 2π. (i) 6 + 6sin x = 6; (ii) 6 + 6 sin x = 0. b) Write down the exact value of the x-intercept of f, for 0 ≤ x < 2π. c) The area of the shaded region is k. Find the value of k, giving your answer in terms of π.
π 2
Let g(x) = 6 + 6sin x − . The graph of f is transformed to the graph of g. d)
Give a full geometric description of this transformation.
e)
Given that
∫
p+ p
3π 2
g ( x ) dx = k and 0 ≤ p < 2π, write down the two values of p. 3
C.
(no calculator) Let f (x) =
3e 2 x sin x + e 2 x cos x , for 0 ≤ x ≤ π. Given that tan
π 1 = , solve 6 3
the equation f (x) = 0.
Statistics and The Normal Distribution 1.
2.
(no calculator) A test which is marked out of 100 is written by 800 students. The cumulative frequency graph for the results of the test is given at right. a) How many students scored 40 marks or less on the test? b) The middle 50% of test results lie between the marks a and b, where a < b. Write down the values of a and b.
The table below represents the weights, W, in grams, of 80 packets of roasted peanuts.
Weight (W) Number of packets
3.
80
85
90
95
100
105
110
5
10
15
26
13
7
4
a) b)
Use the midpoint of each interval to find an estimate for the standard deviation of the weights. Let W1, W2, …, W80 be the individual weights of the packets, and let W be their mean. What is the value of the sum (W1 − W ) + (W2 − W ) + (W3 − W ) + … + (W79 − W ) + (W80 − W ) ?
c)
One of the 80 packets is selected at random. Given that its weight satisfies 80 < W ≤ 110, find the probability that its weight is greater than 100 grams.
The heat output in thermal units from burning 1 kg of wood changes according to the wood’s percentage moisture content. The moisture content and heat output of 10 blocks of the same type of wood each weighing 1 kg were measured. These are shown in the table. Moisture content % (x) 8 15 22 30 34 45 50 60 74 82 Heat output (y) 80 77 74 69 68 61 61 55 50 45 a) i) Write down the correlation coefficient. ii) Which two of the following expressions describe the correlation between x and y? perfect, zero, linear, strong positive, strong negative, weak positive, weak negative b) Write down the equation of the regression line of y on x. Use your regression line as a model to answer the following. c) Interpret the meaning of i) the gradient; ii) the y-intercept. d) Estimate the heat output in thermal units of a 1 kg block of wood that has 25% moisture content. e) Comment on the appropriateness of using your model to i) estimate the moisture content of a 1 kg block of wood with heat output 30 thermal units; ii) estimate the heat output of a block of wood whose moisture content is 54%.
4
4.
The lifespan of a particular species of insect is normally distributed with a mean of 57 hours and a standard deviation of 4.4 hours. a) The probability that the lifespan of an insect of this species lies between 55 and 60 hours is represented by the shaded area in the following diagram. This diagram represents the standard normal curve.
a 0
b
i) ii)
b)
5.
6.
Write down the values of a and b. Find the probability that the lifespan of an insect of this species is a) more than 55 hours; b) between 55 and 60 hours 90% of the insects die after t hours. i) Represent this information of a standard normal curve diagram, similar to the one given in part (a), indicating clearly the area representing 90%. ii) Find the value of t.
The mass of packets of a breakfast cereal is normally distributed with a mean of 750 g and standard deviation of 25 g. a) Find the probability that a packet chosen at random has mass i) less than 740 g; ii) at least 780 g; iii) between 740 g and 780 g. b) Two packets are chosen at random. What is the probability that both packets have a mass which is less than 740 g? c) The mass of 70% of the packets is more than x grams. Find the value of x. The graph shows a normal curve for the random variable X, with mean µ and standard deviation σ. y
0
A 12
x
It is known that p(X ≥ 12) = 0.1. a) The shaded region A is the region under the curve where x ≥ 12. Write down the area of the shaded region A. It is also known that p(X ≤ 8) = 0.1. b) Find the value of µ, explaining your method in full. c) Show that σ = 1.56 to an accuracy of three significant figures. d) Find p(X ≤ 11). Be Able To A. An unbiased coin is tossed twice. A random variable X is defined as follows: X = 1 if both tosses are heads, otherwise X = 2. a) What is the mean of X? b) What is the standard deviation of X?
5
B.
(no calculator) The population P is the set of numbers {–3, 3, a, b}, and has a mean of 0 and a standard deviation of 17 . Given that b > a, determine the values of a and b.
C.
(no calculator) Some data are reported as a set of scores, and are presented in the table below. Score 6 7 8 9 10 11 12 13 14 Frequency 2 5 13 16 20 18 15 7 4 a) Draw a histogram on millimeter-square graph paper to display the distribution of scores. b) i) Find the median score. ii) Find the inter-quartile range.
D.
Given the following frequency distribution, find a) the median; b) the mean; c) the standard deviation Number (x) 1 2 Frequency (f) 5 9
3 16
4 18
5 20
6 7
E.
The speeds of cars at a certain point on a straight road are normally distributed with mean µ and standard deviation σ. It is known that 15% of the cars travelled at speeds greater than 90 km h–1 and 12% of them at speeds less than 40 km h–1. Find µ and σ.
F.
Reaction times of human beings are normally distributed with a mean of 0.76 seconds and a standard deviation of 0.06 seconds. a) The graph below is that of the standard normal curve. The shaded area represents the probability that the reaction time of a person chosen at random is between 0.70 and 0.79 seconds.
a
0 b
i) ii)
b)
Write down the value of a and of b. Calculate the probability that the reaction time of a person chosen at random is a) greater than 0.70 seconds; b) between 0.70 and 0.79 seconds. Three percent (3%) of the population have a reaction time less than c seconds. i) Represent this information on a diagram similar to the one above. Indicate clearly the area representing 3%. ii) Find c.
Vectors 1.
6 4t + 4 , b= , t ∈, have equal lengths. Find the two −4 2t − 5
(no calculator) The two vectors a = possible values of t.
2.
6 −3 Two vectors a = −1 and b = 4 and a point P with coordinates (–3, 2, 2) are given. Q and R are 5 1 points such that PQ = a and RP = b .
a) b)
Find the coordinates of Q and R. Find ∠QPR in the triangle PQR, giving your answer correct to the nearest degree. 6
3.
5 7 (no calculator) Given that p = 12 and q = −2 , 8 −4
a) b) c) 4.
write down the square of the length of the vector q; calculate the scalar product p ⋅ q; find the value of rational number t such that the vector p + tq is perpendicular to the vector q.
Three of the coordinates of the parallelogram STUV are S(–2, –2), T(7, 7), and U(5, 15). a) Find the vector ST and hence the coordinates of V. b) Find a vector equation for the line (UV) in the form r = p + λd, where λ ∈ . c)
1
Show that the point E with position vector is on line (UV), and find the value of λ for E. 11 a
The point W has position vector , a ∈ . 17 d)
(i)
If EW = 2 13 , show that one value of a is –3 and find the other possible value of a.
(ii)
For a = –3, calculate the angle between EW and ET .
1 In question 5, the vector km represents a displacement due east, and the vector 0 displacement due north. 5.
0 1 km represents a
(no calculator) The diagram shows the path of the oil-tanker Aristides relative to the port of Orto, which is situated at the point (0, 0). The position of Aristides is given by the x 0 6 vector equation = + t at a time y 28 −8 t hours after 12:00. a) Find the position of the Aristides at 13:00. b) Find (i) the velocity vector; (ii) the speed of the Aristides. c) Find a Cartesian equation for the path of the Aristides in the form ax + by = g. 18 Another ship, the cargo-vessel Boadicea, is stationary, with position vector km. 4 d) Show that the two ships will collide, and find the time of collision. 5 To avoid collision, the Boadicea starts to move at 13:00 with velocity vector km h–1. 12 x 13 5 e) Show that the position of the Boadicea for t ≥ 1 is given by = + t . y −8 12 f) Find how far apart the two ships are at 15:00.
7
Be Able To A. In this question, a unit vector represents a displacement of 1 metre. A miniature car moves in a straight line, starting at the point (2, 0). After t seconds, its position, (x, x 2
0.7
y), is given by the vector equation = + t . y 0 1 a) How far from the point (0, 0) is the car after two seconds? b) Find the speed of the car. c) Obtain the equation of the car’s path in the form ax + by = c. Another miniature vehicle, a motorcycle, starts at the point (0, 2), and travels in a straight line with constant speed. The equation of its path is y = 0.6x + 2, x ≥ 0. Eventually, the two vehicles collide. d) e)
Find the coordinates of the collision point. If the motorcycle left point (0, 2) at the same moment the car left point (2, 0), find the speed of the motorcycle. −3
2
2
B.
If a = , b = , and c = , find the scalar constants λ and µ such that λa + µb = c. 1 4 −3
C.
Two crews of workers are laying an underground cable in a north-south direction across a desert. At 06:00, each crew sets out from their base camp which is situated at the origin (0, 0). One crew is in a 18
Toyundai vehicle and the other in a Chryssault vehicle. The Toyundai has velocity vector km 24 36
-1 h-1, and the Chryssault has velocity vector km h . −16 a) Find the speed of each vehicle. b) (i) Find the position vectors of each vehicle at 06:30. (ii) Hence, or otherwise, find the distance between the vehicles at 06:30. c) At this time (06:30) the Chryssault stops and its crew begin their day’s work, laying cable in a northerly direction. The Toyundai continues traveling in the same direction at the same speed until it is exactly north of the Chryssault. The Toyundai crew then begin their day’s work, laying cable in a southerly direction. At what time does the Toyundai crew begin laying cable? d) Each crew lays an average of 800 m of cable in an hour. If they work non-stop until their lunch break at 11:30, what is the distance between them at this time? e) How long would the Toyundai take to return to base camp from its lunchtime position, assuming it traveled in a straight line and with the same average speed as on the morning journey? (Give your answer to the nearest minute.)
Probability and the Binomial Distribution 1.
(no calculator) Out of 50 men in a room, 10 are left-handed and 6 are deaf. Two of them are both left-handed and deaf. If a man in the room is chosen at random, find, giving your answers as fractions in their simplest forms, the probabilities that: a) he is neither deaf nor left-handed; b) he is not deaf given that he is not left-handed.
2.
(no calculator) If P(A) = a)
P(A ∪ B)
1 2 and P(B) = and the two events are independent, calculate 3 5 b) P(A′ ∪ B′).
8
3.
A fair coin is tossed eight times. Calculate a) the probability of obtaining exactly 4 heads; b) the probability of obtaining exactly 3 heads; c) the probability of obtaining 3, 4 or 5 heads.
4.
When two standard six-faced dice are tossed, T is the total score obtained. a) Evaluate P(T > 8). b) If the two dice are tossed twice, what is the probability that T exceeds 8 exactly once?
5.
(no calculator) Draw a Venn diagram to show a universal set U and intersecting subsets A and B. a) Shade the area in the diagram which represents the set B ∩ A′. n(U) = 100, n(A) = 30, n(B) = 50, n(A ∪ B) = 65. b) Find n(B ∩ A′). c) An element is selected at random from U. What is the probability that it is in B ∩ A′?
6.
(no calculator) Bag A contains 2 red balls and 3 green balls. Bag B contains 4 red balls and 2 green balls. A person reaches into one of the bags and takes out two balls. If bag A is chosen, the probabilities for the different outcomes are P(2 red balls) = a)
1 10
P(2 green balls) =
3 10
P(1 red ball and 1 green ball) =
6 10
Calculate the probabilities for the same three outcomes if bag B is chosen.
In order to decide which bag to choose, a standard die with six faces is rolled. If a 1 or 6 is rolled, bag A is chosen. If a 2, 3, 4, or 5 is rolled, bag B is chosen. The die is rolled and then two balls are drawn from the selected bag. b) Calculate the probability that two red balls will be selected. c) Given that two red balls are obtained, what is the conditional probability that a 1 or 6 was rolled on the die? 7.
(no calculator) The events B and C are dependent, where C is the event that “a student takes Chemistry,” and B is the even “a student takes Biology.” It is known that P(C ) = 0.4, P ( B C ) = 0.6, P ( B C ′ ) = 0.5 . a) Copy and complete the tree diagram. Chemistry Biology B 0.4
C B′ B C′ B′
b) c) 8.
Calculate the probability that a student takes Biology. Given that a student takes Biology, what is the probability that the student takes Chemistry?
A factory makes calculators. Over a long period, 2% of them are found to be faulty. A random sample of 100 calculators is tested. a) Write down the expected number of faulty calculators in the sample. b) Find the probability that three calculators are faulty. c) Find the probability that more than one calculator is faulty. 9
Be Able To A. When Alexis plays Boris at tennis, Alexis wins with a probability of 3/4. If they play each other seven times, find the probability that Alexis wins five times and Boris twice. B.
(no calculator) An unbiased coin is tossed 4 times. What is the probability that the fourth toss is a head for the second time?
C.
(no calculator) Two unbiased dice are thrown and the total score is observed. X is the event that the total score is even, and Y is the event that the total score is a factor of 12. a) Find P(X). b) Find P(Y). c) Find P(X ∪ Y).
D.
A and B are independent events with p(A) = 1⁄15 and p(B) = 1⁄10. Calculate the following, giving your answers as fractions in lowest terms. a) p(A ∩ B) b) p(A ∪ B) c) p((A ∩ B)|(A ∪ B))
E.
An unbiased ten-faced die has the numbers 1, 2, ... , 10 on the faces. The die is thrown 10 times. a) What is the probability of obtaining a 6 on the fourth throw? b) What is the probability of obtaining a 6 exactly four times in 10 throws?
F.
(no calculator) A random variable X represents the sum of the digits in a randomly chosen integer between 100 and 999 inclusive. Find, by careful consideration of all possible cases, the probability that X = 4.
G.
(no calculator) A box contains 22 red apples and 3 green apples. Three apples are selected at random, one after the other, without replacement. a) The first two apples are green. What is the probability that the third apple is red? b) What is the probability that exactly two of the three apples are red?
H.
(no calculator) A box contains 35 red discs and 5 black discs. A disc is selected at random and its colour noted. The disc is then replaced in the box. a) In eight such selections, what is the probability that a black disc is selected i) exactly once? ii) at least once? b) The process of selecting and replacing is carried out 400 times. What is the expected number of black discs that would be drawn?
I.
An integer is chosen at random between 1 and 100 inclusive. Find the probability that it is divisible by 3 but not by 5.
10
