Mega Math Book 3rd Edition

CIVIL ENGINEERING MATHEMATICS BOARD EXAMINATION REVIEW BOOK

TABLE OF CONTENTS ALGEBRA: SEQUENCE AND SERIES ...................................................................................................

2

ALGEBRA: SEQUENCE AND SERIES SOLUTIONS .............................................................................. 4 ALGEBRA: WORDED PROBLEMS .....................................................................................................

14

ALGEBRA: WORDED PROBLEMS SOLUTIONS ................................................................................

19

ALGEBRA: POLYNOMIALS, PARTIAL FRACTIONS AND INEQUALITIES .................. ................. .... 35 ALGEBRA: POLYNOMIALS, PARTIAL FRACTIONS AND INEQUALITIES SOLUTIONS ................. . 38 ALGEBRA: BINOMIAL EXPANSION ..................................................................................................

49

ALGEBRA: BINOMIAL EXPANSION SOLUTIONS ............................................................................. 50 ALGEBRA: COMPLEX NUMBERS ......................................................................................................

54

ALGEBRA: COMPLEX NUMBERS SOLUTIONS .................................................................................

57

MATRICES AND DETERMINANTS .................................................................................................... 68 MATRICES AND DETERMINANTS SOLUTIONS ............................................................................... 71 PLANE AND SPHERICAL TRIGONOMETRY .....................................................................................

77

PLANE AND SPHERICAL TRIGONOMETRY SOLUTIONS ................................................................ 84 PLANE ANALYTIC GEOMETRY .......................................................................................................

105

PLANE ANALYTIC GEOMETRY SOLUTIONS .................................................................................. 110 SOLID MENSURATION ....................................................................................................................

123

SOLID MENSURATION SOLUTIONS ...............................................................................................

129

SPACE ANALYTIC GEOMETRY .......................................................................................................

150

SPACE ANALYTIC GEOMETRY SOLUTIONS ..................................................................................

152

VECTORS ..........................................................................................................................................

162

VECTORS SOLUTIONS .....................................................................................................................

166

DIFFERENTIAL CALCULUS .............................................................................................................

176

DIFFERENTIAL CALCULUS SOLUTIONS ........................................................................................

184

INTEGRAL CALCULUS .....................................................................................................................

226

INTEGRAL CALCULUS SOLUTIONS ................................................................................................

229

STATISTICS ......................................................................................................................................

247

STATISTICS SOLUTIONS .................................................................................................................

253

PROBABILITY ..................................................................................................................................

258

PROBABILITY SOLUTIONS .............................................................................................................

263

MEGAREVIEW AND TUTORIAL CENTER

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term of the arithmetic progression 6, 10, 14, … – – term of the arithmetic progression 6, 10, 14, … term of the arithmetic progres sion log 7, log 14, log 28, … – log 28, …

ALGEBRA: SEQUENCE AND SERIES 1. Find the nth A. 2 + 4n C. 4 + 2n B. 2 4n D. 4 2n th 2. Find the sum up to the 10 A. 250 C. 240 B. 120 D. 225 3. Find the nth A. 0.845 + 0.301n C. -0.544 + 0.301n B. 0.544 + 0.301n D. -0.544 0.301n 4. Find the sum up to the 10 th term of the arithmetic progression log 7, log 14, A. 18.442 B. 26.397

C. 25.852 D. 21.997

Situation: Logs are stacked so that there are 25 logs in the bottom row, 24 logs in the second row, and so on, decreasing by 1 log each row. 5. How many logs are stacked in the sixth row? A. 21 C. 19 B. 20 D. 18 6. How many logs are there in all six rows? A. 136 C. 137 B. 134 D. 135

–

Situation: The distance a ball rolls down a ramp each second is given by the arithmetic sequence whose nth term 2n 1 in feet. 7. Find the distance the ball rolls during the 10th second. A. 18 ft. C. 19 ft. B. 20 ft. D. 21 ft. 8. Find the total distance the ball travels in 10 seconds. A. 120 ft. C. 110 ft. B. 100 ft. D. 90 ft. Situation: A contest offers 15 prizes. The 1st prize is P 5000, and each successive prize is P 250 less than the preceding prize. 9. What is the value of the 15th prize? A. 1250 C. 1500 B. 1750 D. 1625 10.What 10.What is the total amount of money distributed in prizes? A. P 49,000 C. P 50,750 B. P 48,750 D. P 47,250 Situation: The 4th and 7th terms of an arithmetic sequence are 13 and 25. 11.Find 11.Find the first term. A. 1 B. 3 12.Find 12.Find the common difference. A. 1 B. 4 13.Find 13.Find the 20th term. A. 81 B. 77 2

C. 2 D. 5 C. 5 D. 2 C. 73 D. 83



term of the arithmetic progression 6, 10, 14, … – – term of the arithmetic progression 6, 10, 14, … term of the arithmetic progres sion log 7, log 14, log 28, … – log 28, …

ALGEBRA: SEQUENCE AND SERIES 1. Find the nth A. 2 + 4n C. 4 + 2n B. 2 4n D. 4 2n th 2. Find the sum up to the 10 A. 250 C. 240 B. 120 D. 225 3. Find the nth A. 0.845 + 0.301n C. -0.544 + 0.301n B. 0.544 + 0.301n D. -0.544 0.301n 4. Find the sum up to the 10 th term of the arithmetic progression log 7, log 14, A. 18.442 B. 26.397

C. 25.852 D. 21.997

Situation: Logs are stacked so that there are 25 logs in the bottom row, 24 logs in the second row, and so on, decreasing by 1 log each row. 5. How many logs are stacked in the sixth row? A. 21 C. 19 B. 20 D. 18 6. How many logs are there in all six rows? A. 136 C. 137 B. 134 D. 135

–

Situation: The distance a ball rolls down a ramp each second is given by the arithmetic sequence whose nth term 2n 1 in feet. 7. Find the distance the ball rolls during the 10th second. A. 18 ft. C. 19 ft. B. 20 ft. D. 21 ft. 8. Find the total distance the ball travels in 10 seconds. A. 120 ft. C. 110 ft. B. 100 ft. D. 90 ft. Situation: A contest offers 15 prizes. The 1st prize is P 5000, and each successive prize is P 250 less than the preceding prize. 9. What is the value of the 15th prize? A. 1250 C. 1500 B. 1750 D. 1625 10.What 10.What is the total amount of money distributed in prizes? A. P 49,000 C. P 50,750 B. P 48,750 D. P 47,250 Situation: The 4th and 7th terms of an arithmetic sequence are 13 and 25. 11.Find 11.Find the first term. A. 1 B. 3 12.Find 12.Find the common difference. A. 1 B. 4 13.Find 13.Find the 20th term. A. 81 B. 77 2

C. 2 D. 5 C. 5 D. 2 C. 73 D. 83



14.Insert 14.Insert 5 arithmetic means between -1 and 23. A. 3, 7, 11, 15, 19 C. -1, 3, 7, 11, 15 B. 7, 10, 13, 16, 19 D. 1, 5, 9, 13, 17 15.An 15.An object dropped from a cliff will fall 16 feet the first second, 48 feet the second, 80 feet the third, and so on, increasing by 32 feet each second. What does the total distance the object will fall in 7 seconds? A. 874 ft. C. 748 ft. B. 847 ft. D. 784 ft. 16.A 16.A besieged fortress is held by 5700 men who have provisions for 66 days. If the garrison loses 20 men each day, how many days can the provision hold out? A. 70 days C. 67 days B. 76 days D. 80 days 17.In 17.In a racing contest, there are 240 cars with fuel provision for 15 hours each. Assuming a constantly hourly consumption for each car, how long will the fuel provision last if 8 cars withdraw from the race every hour? A. 72 hours C. 25 hours B. 23 hours D. 20 hours 18.How 18.How many numbers divisible by 4 lie between 70 and 203? A. 33 C. 34 B. 35 D. 36 19.Find 19.Find the sum of the numbers divisible by between 70 and 203. A. 4848 C. 8484 B. 4488 D. 8844 20.Two 20.Two men set out from a certain place going in the same direction. The first travels at a constant rate of 8 kilometers per hour, while the second goes 4 km for the first hour, 4.5 km the second hour, 5 km the third hour, and so on. After how many hours will the second man overtake the first? A. 18 hours C. 17 hours B. 20 hours D. 19 hours 21.Ten 21.Ten balls are placed in a straight line on the ground at intervals of 2 meters. Six meters from the end of the row a basket is placed. A boy starts from the basket and picks up the balls and carries them, one at a time to the basket. How far did he walk all in all? A. 120 m C. 250 m B. 130 m D. 300 m 22.Find 22.Find the third term of a geometric sequence whose first firs t term is 2 and whose fifth term is 162. A. 18 C. 9 B. 6 D. 27 23.A 23.A basketball is dropped from a height of 10m. On each rebound it rises 2/3 of the height from which it last fell. Determine the total distance travelled until it comes to rest. A. 45 m C. 50 m B. 60 m D. 75 m th term. 24. A. 10,682 C. 59,048 B. 177,146 D. 6,560

Find the sum of the geometric progres sion 2, 6, 18, … up to the 10


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term of the arithmetic progression 6, 10, 14, … aa  6a→ firnstte1rmd d  4 → common common dif erence aa  66  4nn144 a  2  4n a  →

ALGEBRA: SEQUENCE AND SERIES SOLUTIONS 1. Find the nth First Solution: Using the formula:

We have:

Second Solution: Since the relationship between and n is linear, hence we can use here the STAT Mode 3-2 A + Bx For Linear Mathematical Model, Model , we only need 2 points to define the function in the form of y = A + Bx. Input: x y 1 6 2 10

– –

Press AC. Then press Shift 1 5 (Reg Regression) Then select 1: A and then select 2: B A = 2; B = 4 Therefore, y = A + Bx

a  2  4n term of the arithmetic progression 6, 10, 14, …  1dn s  2a22a6 n2101 ss  240 1 012 41010 s  → y  A  B x  Cx 

2. Find the sum up to the 10 th First Solution: By using the formula of the sum of an arithmetic progression of n terms:

Second Solution: From the formula, it shows that the relationship between n and is in quadratic form, so we can use STAT MODE 3-3 A + Bx + Cx2 For Quadratic Mathematical Model, we need 3 points to define the function in the form of . Input: x y 1 6 2 6 +10 = 16 3 6 + 10 +14 = 30 For the sum of the first 10 terms: Find the value of y which corresponds to the value of x = 10. Hence .

10ŷ  240

4



term of the arithmetic progres sion log 7, log 14, log 28, … aa  laog 7nn1d1log2 aa  llooggg7g77 nlnlogo2g2 log 2 2 a  0.544544  0.0.301n 301n

3. Find the nth First Solution: Simplifying:

Second solution: Go to MODE 3-2: Input:

x 1 2

lloog14g7 y

– – a  0.544544  0.0.301n 301n

Press AC. Then press Shift 1 5 (Reg Regression) Then select 1: A and then select 2: B A = 0.544; B = 0.301 Therefore, y = A + Bx

log 28, …

4. Find the sum up to the 10 th term of the arithmetic progression log 7, log 14, First Solution: By using the formula of the sum of an arithmetic progression of n terms:

1dn s  2l2a22alo g7 n2101 ss  21.997 1 012 log21010 → y  A  B x  Cx  lloog7g7log14 log7log14log28 10ŷ  21.997

s

Second Solution: From the formula, it shows that the relationship between n and is in quadratic form, so we can use STAT MODE 3-3 A + Bx + Cx2 For Quadratic Mathematical Model, we need 3 points to define the function in the form of . Input: x y 1 2 3 For the sum of the first 10 terms: Find the value of y which corresponds to the value of x = 10. Hence .


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Situation: Logs are stacked so that there are 25 logs in the bottom row, 24 logs in the second row, and so on, decreasing by 1 log each row. 5. How many logs are stacked in the sixth row? First Solution: The relationship between the order of rows and number of logs follows an arithmetic progression. Let a1 be the number of logs in the bottom row, that is 25; a2 = 24; and a3 =

23, so that the arithmetic sequence is 25, 24, 23, … aa  25,a d n11d aa  2025 6 11 Therefore, the number of logs in the sixth row is:

Second Solution: Go to MODE 3-2. Input to the x-column the order of rows of logs and to the y-column the number of logs in that particular row, that is: x 1 2

y 25 24

Press AC. To solve for the number of logs in the sixth row, press

6ŷ  20

.

6. How many logs are there in all six rows? First Solution: By using the formula of the sum of an arithmetic progression of n terms:

  n2 1dn s  22a25 ss  135 6 12 16 → y  ABxCx 2524  49 252423  72 6ŷ  135

s

Second Solution: From the formula, it shows that the relationship between n and is in quadratic form, so we can use STAT MODE 3-3 A + Bx + Cx2 For Quadratic Mathematical Model, we need 3 points to define the function in the form of . Input: x y 1 25 2 3 For the sum of the first 6 terms: Find the value of y which corresponds to the value of x = 6. Hence .

6



–

Situation: The distance a ball rolls down a ramp each second is given by the arithmetic sequence whose nth term 2n 1 in feet. 7. Find the distance the ball rolls during the 10th second. First Solution: Let: an be the distance traveled by ball in a certain interval at any nth second n = nth second The distance traveled in the first second is 2n 1; hence ; ; .

– a  21 – 1   1 foot a  22 1  3 feet d  31  2 fe t a  a  n 1d aa  191feet1 012

Therefore, using the formula: We have

Second Solution: Go to MODE 3-2: Input:

Press AC. Then press

10ŷ  19 feet

x 1 2

y 1 3

.

8. Find the total distance the ball travels in 10 seconds. First Solution: By using the formula of the sum of an arithmetic progression of n terms:

    2 a  n 1 d n  s  21 21 01210 ss  100 feet 2 → y  ABxCx 13  4 135  9 10ŷ  100 feet

s

Second Solution: From the formula, it shows that the relationship between n and is in quadratic form, so we can use STAT MODE 3-3 A + Bx + Cx2 For Quadratic Mathematical Model, we need 3 points to define the function in the form of . Input: x y 1 1 2 3 For the sum of the first 10 terms: Find the value of y which corresponds to the value of x = 10. Hence .


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Mega Math Book 3rd Edition

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