ENGINEERING MATHEMATICS Conversion Problem 1 What is the temperature in degree Celsius of absolute zero? A. -32 B. 0 C. 273 D. -273 Problem 2 (ME April 1996) How many degrees Celsius is 100 degrees Fahrenheit? A. 37.8 °C B. 2.667 °C C. 1.334 °C D. 13.34 °C Problem 8 The angle of inclination of the road is 32. What is the angle of inclination in mils? A. 456.23 B. 568.89 C. 125.36 D. 284.44 Problem 9 An angle measures x degrees. What is its measure in radians? A. 180° x / π B. π x / 180° C. 180° π / x D. 180° π x Problem 10 (ECE November 1995) Express 45° in mils. A. 80 mils B. 800 mils C. 8000 mils D. 80000mils Problem 3 (ECE November 1997) A comfortable room temperature is 72 °F. What is this temperature, expressed in degrees Kelvin? A. 263 B. 290 C. 295 D. 275 Problem 11 (ME April 1997) What is the value in degrees of π radians? A. 90° B. 57.3° C. 180° D. 45° Problem 4 255 °C is equivalent to: A. 491 °F B. 427 °F C. 173.67 °F D. 109.67 °F Problem 12(CE May 1993) How many degrees is 3200° mils? A. 360° B. 270° C. 180° D. 90° Problem 5 At what temperature will the °C and °F readings be equal? A. 40° B. -40° C. 32° D. 0° Problem 13 (ECE November 1995) An angular unit equivalent to 1/400 of the circumference of a circle is called: A. mil B. grad C. radian D. degree Problem 6 (ME October 1994) How many degree Celsius is 80 degrees Fahrenheit? A. 26.67 B. 86.4 C. 33.33 D. 16.33 Problem 14 (EE October 1994) Carry out the following multiplication and express your answer in cubic meters: 3cm×5mm×2m. A. 3 x 10-3 B. 3 x 10-4 C. 8 x 10-2 D. 8 x 102 Problem 7(EE October 1990) What is the absolute temperature of the freezing point of water in degree Rankine? A. -32 B. 0 C. 428 D. 492 Problem 15 (ME April 1994) Add the following and express in meters: 3m + 2cm + 70mm A. 2.90 m B. 3.14 m C. 3.12 m D. 3.09 m Problem 16 One nautical mile is equivalent to: A. 5280 ft. B. 6280 ft. C. 1.256 statute mile D. 1.854 km Problem 24 (ME April 1999) The prefix nano is opposite to: A. mega B. tera C. hexa D. giga Problem 17 (ME October 1991) How many square feet is 100 square meters? A. 328.10 B. 956.36 C. 1075.84 D. 156 Problem 25(ME October 1996) 10 to the 12th power is the value of the prefix: A. giga B. pico C. tera D. peta Problem 18 A tank contains 1500 gallons of water. What is the equivalent in cubic liters? A. 4.256 B. 5.865 C. 6 D. 5.685 Exponents and Radicals Problem 19 (ME October 1994) How many cubic feet is equivalent to 100 gallons of water? A. 74.80 B. 1.337 C. 13.37 D. 133.7 Problem 20 (ME April 1998) How many cubic meters is 100 gallons of liquid? A. 0.1638 cu. meters B. 1.638 cu. meters C. 0.3785 cu. meters D. 3.7850 cu. meters Problem 21 (ME October 1995) The number of board feet in a plank 3 inches thick, 1 foot wide, and 20 feet long is: A. 30 B. 60 C. 120 D. 90 Problem 22 Which of the following is correct? A. 1horsepower = 746kW B. 1horsepower = 0.746watts C. 1 horsepower = 0.746kW D. 1 horsepower = 546 watts Problem 26 Solve for x: x = -(1/-27)-2/3 A. 9 B. 1/9 C. -9 D. -1/9 Problem 27 Solve for a in the equation: a = 64x4y A. 4x+3y B. 43xy C. 256xy D. 43x+y Problem 28 Simplify 3x - 3x-1 - 3x-2 A. 3x-2 B. 33x-3 C. 5×3x-2 D. 13×3x Problem 29 Which of the following is true? A. √−2 × √−2 = 2 B. 24 = 4√6 C. √10 = √5 + √2 D. 55+55+55+55+55=56 Problem 30 Solve for x: x=√18 − √72 + √50. A. -2√2 B. 2√2 C. 4 D. 4√3 Problem 31 Problem 23 (ME October 1996) The acceleration due to gravity in English unit is equivalent to? A. 32.2 ft/sec2 B. 3.22 ft/sec2 C. 9.81 ft/sec2 D. 98.1 ft/sec2 Solve for x: √𝑥 − √1 − 𝑥 = 1 − √𝑥. A. -16/25 & 0 B. 25/16 & 0 C. -25/16 & 0 D. 16/25 & 0 Problem 32 3 3 3 Simplify √2𝑥 4 − √16𝑥 4 + 2√54𝑥 4 . 3 A. 5 √𝑥 4 3 B. 2 √5𝑥 4 3 C. 5√2𝑥 4 3 D. 2√𝑥 4 Problem 33 Solve for x: 3x5x+1 = 6x+2 A. 2.1455 B. 2.1445 C. 2.4154 D. 2.1544 Problem 34 (𝑎−2 𝑏3 )2 Simplify 𝑎2 𝑏−1 . A. 𝑎−2 𝑏7 B. 𝑎−2 𝑏5 C. 𝑎−6 𝑏7 D. 𝑎−6 𝑏5 Problem 35 (3x)x is equal to: 2 A. 3𝑥 B. 3xxx C. 3xx D. 32x Problem 36 Solve for x: 37x+1 = 6561. A. 1 B. 2 C. 3 D. 4 Problem 37 If 3a = 7b, then 3a2/7b2 = A. 1 B. 3/7 C. 7/3 D. 49/9 Problem 40 If 33y = 1, what is the value of y/33? A. 0 B. 1 C. Infinity D. 1/33 Problem 41 (ME April 1998) Find the value of x that will satisfy the following expression: √𝑥 − 2 = −√𝑥 + 2 A. x = 3/2 B. x = 18/6 C. x = 9/4 D. none of these Problem 42 (ME April 1998) 𝑒 −3 is equal to: A. 0.048787 B. 0.049001 C. 0.049787 D. 0.048902 Problem 43 B to the m/nth power is equal to: A. Nth root of b to the m power B. B to the m+n power C. 1/n square root of b to the m power D. B to the m power over n Problem 44 (ECE April 1993) Find x from the following equations: 27x = 9y 81y3-x = 243 A. 2.5 B. 2 C. 1 D. 1.5 Problem 45 (ECE April 1990) 𝑎𝑛 Solve for a if (am)(an) = 100000 and 𝑚 = 10 𝑎 A. 15.85 B. 10 C. 12 D. 12.56 Problem 38 Solve for U if U = √1 − √1 − √1 − ⋯ A. 0.723 B. 0.618 C. 0.852 D. 0.453 Problem 39 (ME April 1996) If x to the ¾ power equals 8, then x equals: A. -9 B. 6 C. 9 D. 16 Problem 46 (ECE April 1991) 1 (𝑥 2 𝑦 3𝑧 −2 )−3(𝑥 −3 𝑦𝑧 3 )2 Simplify A. B. C. D. 5 (𝑥𝑦𝑧 −3 )2 . 1 𝑥2 𝑦 7 𝑧3 1 𝑥2 𝑦 7 𝑧6 1 𝑥2 𝑦 5 𝑧3 1 𝑥5 𝑦 7 𝑧3 Problem 47(ECE April 1991) Simplify the following: 7a+2 – 8(7a+1) + 5(7a) + 49(7a-2). A. -5a B. 3 C. -7a D. 7a Problem 48 𝑥𝑦 −1 𝑥 2 𝑦 −2 Simplify(𝑥−2 𝑦3 )4 ÷ (𝑥−3 𝑦3)3 A. xy3 𝑦 B. 𝑥3 C. D. 𝑥3 𝑦 1 𝑥3 𝑦 Problem 49 Simplify the following: A. B. C. D. √5−√3 . √5+√3 4+√15 4-√15 8+√8 8-√8 Problem 50 𝑚 𝑛 Which of the following is equivalent to √ √𝑎 𝑚 A. √𝑎𝑛 𝑛 B. √𝑎𝑚 C. √𝑎𝑚𝑛 𝑚𝑛 D. √𝑎 FUNDAMENTALS IN ALGEBRA Problem 51 (ME Board) Change 0.222… common fraction. A. 2/10 B. 2/9 C. 2/13 D. 2/7 Problem 52 (ME Board) Change 0.2272722… to a common fraction. A. 7/44 B. 5/48 C. 5/22 D. 9/34 Problem 53 (ME Board) What is the value of 7! or 7 fatorial? A. 5040 B. 2540 C. 5020 D. 2520 Problem 54 (ME October 1994) The reciprocal of 20 is: A. 0.50 B. 20 C. 0.20 D. 0.05 Problem 55 If p is an odd number and q is an even number, which of the following expressions must be even? A. p+q B. p-q C. pq D. p/q Problem 56 (ECE March 1996) MCMXCIV is a Roman Numeral equivalent to: A. 2974 B. 3974 C. 2174 D. 1994 Problem 57 (ECE April 1998) What is the lowest common factor of 10 and 32? A. 320 B. 2 C. 180 D. 90 Problem 58 4xy – 4x2 –y2 is equal to: A. (2x-y)2 B. (-2x-y)2 C. (-2x+y)2 D. –(2x-y)2 Problem 59 Factor x4 – y2 + y – x2 as completely as possible. A. (x2 + y)(x2 + y -1) B. (x2 + y)(x2 - y -1) C. (x2 -y)(x2 - y -1) D. (x2 -y)(x2 + y -1) Problem 60 (ME April 1996) Factor the expression x2 + 6x + 8 as completely as possible. A. (x+8)(x-2) B. (x+4)(x+2) C. (x+4)(x-2) D. (x-8)(x-2) Problem 61(ME October 1997) Factor the expression x3 + 8. A. (x-2)(x2+2x+4) B. (x+4)(x2+2x+2) C. (-x+2)(-x2+2x+2) D. (x+2)(x2-2x+4) Problem 62 (ME October 1997) Factor the expression (x4 – y4) as completely as possible. A. (x+y)(x2+2xy+y) B. (x2+y2)(x2-y2) C. (x2+y2)(x+y)(x-y) D. (1+x2)(1+y)(1-y2) Problem 63(ME October 1997) Factor the expression 3x3+3x2-18x as completely as possible. A. 3x(x+2)(x-3) B. 3x(x-2)(x+3) C. 3x(x-3)(x+6) D. (3x2-6x)(x-1) Problem 64 (ME April 1998) Factor the expression 16 – 10x + x2. A. (x+8)(x-2) B. (x-8)(x-2) C. (x-8)(x+2) D. (x+8)(x+2) Problem 65 Factor the expression x6-1 as completely as possible. A. (x+1)(x-1)(x4+x2-1) B. (x+1)(x-1)(x4+2x2+1) C. (x+1)(x-1)(x4-x2+1) D. (x+1)(x-1)(x4+x2+1) Problem 66 What are the roots of the equation (x-4)2(x+2) = (x+2)2(x-4)? A. 4 and -2 only B. 1 only C. -2 and 4 only D. 1, -2, and 4 only Problem 67 If f(x) = x2 + x + 1, then f(x) – f(x-1) = A. 0 B. x C. 2x D. 3 Problem 68 Which of the following is not an identity? A. (x-1)2 = x2-2x+1 B. (x+3)(2x-2) = 2(x2+2x-3) C. x2-(x-1)2 = 2x-1 D. 2(x-1)+3(x+1) = 5x+4 Problem 69 (ME October 1997) 𝑥+3 4𝑥 2 𝑥+9 Solve for x: 4 + 𝑥−3 − 𝑥2 −9 = 𝑥+3 . A. -18 = -18 B. 12 = 12 or -3 = -3 C. Any value D. -27 = -27 or 0 = 0 Problem 70 (ME October 1997) Solve the simultaneous equations: 3x – y = 6; 9x – y = 12. A. x = 3; y = 1 B. x = 1; y = -3 C. x = 2; y = 2 D. x = 4; y = 2 Problem 71 (ME April 1998) Solve algebraically: 4x2 + 7y2 = 32 11y2 – 3x2 = 41 A. y = 4, x = ±1 and y = -4, x = ±1 B. y = +2, x = ±1 and y = -2, x = ±1 C. x = 2, y = 3 and x = -2, y = -3 D. x = 2, y = −2 and x = 2, y = -2 Problem 72 (CE May 1997) Solve for w from the following equations: 3x – 2y + w = 11 x + 5y -2w = -9 2x + y - 3w = -6 A. 1 B. 2 C. 3 D. 4 Problem 73 When (x+3)(x-4) + 4 is divided by x – k, the remainder is k. Find the value of k. A. 4 or 2 B. 2 or -4 C. 4 or -2 D. -4 or -2 Problem 74 Find k in the equation 4x2 + kx + 1 = 0 so that it will only have one real root. A. 1 B. 2 C. 3 D. 4 Problem 75 Find the remainder when (x12 + 2) is divided by (x – √3). A. 652 B. 731 C. 231 D. 851 Problem 76 (CE November 1997) If 3x3 – 4x2y + 5xy2 + 6y3 is divided by (x2 – 2xy + 3y2), the remainder is A. 0 B. 1 C. 2 D. 3 Problem 77 (CE November 1007 & May 1999) If (4y3 + 8y + 18y2 – 4) is divided by (2y + 3), the remainder is: A. 10 B. 11 C. 12 D. 13 Problem 78 (ECE April 1999) Given f(x) = (x+3)(x-4) + 4 when divided by (x-k), the remainder is k. Find k. A. 2 B. 3 C. 4 D. -3 Problem 79 (EE March 1998) The polynomial x3 + 4x2 -3x + 8 is divided by x-5. What is the remainder? A. 281 B. 812 C. 218 D. 182 Problem 80 Find the quotient of 3x5 – 4x3 + 2x2 + 36x + 48 divided by x3 – 2x2 + 6. A. -3x2 – 4x + 8 B. 3x2 + 4x + 8 C. 3x2 – 4x – 8 D. 3x2 + 6x + 8 Problem 81 If 1/x = a + b and 1/y = a-b, then x-y is equal to: A. 1/2a B. 1/2b C. 2a/(a2 – b2) D. 2b/(a2 – b2) Problem 82 If x-1/x = 1, find the value of x3 – 1/x3. A. 1 B. 2 C. 3 D. 4 Problem 83 If 1/x + 1/y = 3 and 2/x – 1/y = 1. Then x is equal to: A. ½ B. 2/3 C. ¾ D. 4/3 Problem 84 5𝑥 Simplify the following expression: 2𝑥2 +7𝑥+3 − 𝑥+3 2𝑥+1 2𝑥 2 −3𝑥−2 A. B. C. D. + 𝑥2 +𝑥_6. A. B. C. D. 4𝑦 2 D. is equal to: Problem 86 Simplify: (a+1/a)2-(a-1/a)2. A. -4 B. 0 C. 4 D. -2/a2 Problem 87 (ECE November 1996) The quotient of (x5 +32) by (x+2) is: A. x4 – x3 + 8 B. x3 +2x2 – 8x + 4 C. x4 – 2x3 + 4x2 – 8x + 16 D. x4 + 2x3 + x2 + 16x + 8 Problem 88 (ME April 1996) Solve the simultaneous equations: y - 3x + 4 = 0 y + x2/y = 24/y A. x = (-6 + 2√14)/5 or (-6 - 2√14)/5 y = (2 + 6√14)/5 or (-2 + 6√14)/5 B. x = (6 + 2√15)/5 or (6 - 2√15)/5 y = (-2 + 6√14)/5 or (-2 - 6√15)/5 C. x = (6 + 2√14)/5 or (6 - 2√14)/5 y = (-2 + 6√14)/5 or (-2 - 6√14)/5 D. x = (6 + 2√14)/5 or (6 - 2√14)/5 y = (-6+ 2√14)/5 or (-6 + 2√14)/5 Problem 89 (CE May 1996) Find the value of A in the equation. (𝑥 3 +2𝑥 2+5𝑥) A. B. C. D. 2 -2 -1/2 ½ 𝐴 𝐵(2𝑥+2) 𝑥 −7𝑥+12 6 2 𝑥−4 6 𝑥−4 6 𝑥−4 6 𝑥−4 − 𝑥−3 7 + 𝑥−3 5 − 𝑥−3 5 + 𝑥−3 Problem 92 (ECE April 1998) The arithmetic mean of 80 numbers is 55. If two numbers namely 250 and 850 are removed what is the arithmetic mean of the remaining numbers? A. 42.31 B. 57.12 C. 50 D. 38.62 ¾ 4/3 2/3 3/2 (𝑥 2 +4𝑥+10) A. C. Problem 85 If 3x = 4y then Problem 91 (ME October 1996) 𝑥+2 Resolve 2 into partial fraction. B. 2/(x-3) (x-3)/5 (x+3)/(x-1) 4/(x+3) 3𝑥 2 Problem 90 𝑥+10 𝐴 𝐵 Find A and B such that 𝑥2 −4 = 𝑥−2 + 𝑥+2 A. A = -3; B = 2 B. A = -3; B = -2 C. A = 3; B = 2 D. A = 3; B = 2 𝐶 = 𝑥 + (𝑥2 +2𝑥+5) + (𝑥2 +2𝑥+5) Problem 93 (ECE April 1998) The arithmetic mean of 6 numbers is 17. If two numbers are added to the progression, the new set of number will have an arithmetic mean of 19. What are the two numbers if their difference is 4? A. 21, 29 B. 23, 27 C. 24, 26 D. 22, 28 Problem 94 If 2x – 3y = x + y, then x2 : y2 = A. 1:4 B. 4:1 C. 1:16 D. 16:1 Problem 95 If 1/a :1/b : 1/c = 2 : 3 : 4, then (a + b + c) : (b + c) is equal to: A. 13:7 B. 15:6 C. 10:3 D. 7:9 Problem 96 Find the mean proportional to 5 and 20. A. 8 B. 10 C. 12 D. 14 Problem 97 Find the fourth proportional of 7, 12, and 21. A. 36 B. 34 C. 32 D. 40 Problem 105 Find the value of a in the equation loga2187 = 7/2. A. 3 B. 6 C. 9 D. 12 Problem 98 (ECE November 1997) If (x+3):10 = (3x – 2) : 8, find (2x -1) A. 1 B. 2 C. 3 D. 4 Problem 106 If log 2 = x and log 3 = y, find log 1.2. A. 2x + y B. 2xy/10 C. 2x + y - 1 D. xy - 1 Problem 99 Solve for x: -4 < 3x - 1 < 11. A. 1 < x < -4 B. -1< x < 4 C. 1 < x < 4 D. -1 < x < -4 Problem 107 Problem 100 Solve for x: x2 + 4x > 12. A. -6 > x > 2 B. 6 > x > -2 C. -6 > x > -2 D. 6 > x > 2 Logarithms, Binomial Theorem, Quadratic Equation Problem 101 𝑙𝑜𝑔10 𝑥 If 1−𝑙𝑜𝑔 = 2, what is the value of z? 2 10 A. B. C. D. ¼ 25 4 5 Problem 102 (EE October 1992) Solve for x: log 6 + x log 4 = log 4 + log (32 + 4x) A. 1 B. 2 C. 3 D. 4 Problem 103 Which of the following cannot be used as a base of a system of logarithm? A. e B. 10 C. 2 D. 1 Problem 104 If log5.21000 = x, what is the value of x? A. 4.19 B. 5.23 C. 3.12 D. 4.69 𝑙𝑜𝑔𝑥 𝑦 𝑙𝑜𝑔𝑦 𝑥 A. B. C. D. is equal to: xy/yx y log x – x log y (y log x)/ (x log y) 1 Problem 108 If 10ax+b = P, what is the value of x? A. (1/a)(log P-b) B. (1/a) log ( P-b) C. (1/a) P10-b D. (1/a) log P10 Problem 109 Find the value of log(aa)a. A. 2a log a B. a2 log a C. a log a2 D. (a log a)a Problem 110 Solve for x: x = logb a ×logc d × logd c A. logb a B. loga c C. logb c D. logd a Problem 111 Find the positive value of x if logx 36 = 2. A. 2 B. 4 C. 6 D. 8 Problem 112. Find x if logx 27 + logx 3 = 2. A. 9 B. 12 C. 8 D. 7 Problem 113 Find a if log2 (a+2) + log2 (a-2) = 5 A. 2 B. 4 C. 6 D. 8 Problem 114. Solve for x if log5 x = 3. A. 115 B. 125 C. 135 D. 145 Problem 115. Find log P if ln P = 8. A. 2980.96 B. 2542.33 C. 3.47 D. 8.57 Problem 116 If log8 x = -n, then x is equal to: A. 8n B. 1/8-n C. 1/8n D. 81/n Problem 117 If 3 log10 x – log10 y = 0, find y in terms of x. 3 A. y = √𝑥 B. y = √𝑥 3 C. y = x3 D. y = x Problem 118 Which of the following is correct? A. -2 log 7 = 1/49 B. log7 (-2) = 1/49 C. log7 (1/49) = -2 D. log7 (1/49) = 2 Problem 119 (ME April 1996) Log of nth root of x equals log of x to the 1/n power and also equal to: log(𝑥) A. 𝑛 B. n log (x) 𝑙𝑜𝑔(𝑥)1/𝑛 C. 𝑛 D. (n-1) log (x) Problem 120 (ME April 1996) What is the natural logarithm of e to the xy power? A. 1/xy B. 2.718/xy C. xy D. 2.718xy Problem 121 (ME April 1997) What expression is equivalent to log x – log (y + z)? A. log x + log y + log z B. log [ x/(y + z) ] C. log x –log y –log z D. log y + log (x + z) Problem 122 (ME April 1997) What is the value of log base 10 of 10003.3? A. 9.9 B. 99.9 C. 10.9 D. 9.5 Problem 123 If logx 2 + log2 x = 2, then the value of x is: A. 1 B. 2 C. 3 D. 4 Problem 124 (CE November 1997) Log6 845 =? A. 4.348 B. 6.348 C. 5.912 D. 3.761 Problem 125 (CE May 1998, similar with November 1998) The logarithms of the quotient and the product of two numbers are 0.352182518 and 1.556302501, respectively. Find the first number? A. 9 B. 10 C. 11 D. 12 Problem 126 The sum of two logarithms of two numbers is 1.748188 and the difference of their logarithms is 0.0579919. One of the numbers is: A. 9 B. 6 C. 8 D. 5 Problem 127 (CE November 199) 𝑒𝑥 Solve for y: y = ln 𝑥−2 . 𝑒 A. 2 B. x C. -2 D. x-2 Problem 128 (ECE April 1998) What is the value of (log 5 to the base 2) + (log 5 to the base 3)? A. 3.97 B. 7.39 C. 9.37 D. 3.79 Problem 129 (ME October 1997) The logarithm of negative number is: A. irrational number B. real number C. imaginary number D. complex number Problem 130(ME April 1998) 38.5 to the x power = 6.5 to the x-2 power, solve for x using logarithms. A. 2.70 B. 2.10 C. -2.10 D. -2.02 Problem 131 (CE November 1996) Find the 6th term of the expansion of (1/2a – 3)16. 22113 A. - 256𝑎11 66339 B. - 128𝑎11 22113 C. - 128𝑎11 66339 D. - 256𝑎11 Problem 132 (ECE April 1998) In the expansion of (x+4y) 12, the numerical coefficient of the 5th term is. A. 253440 B. 126720 C. 63360 D. 506880 Problem 133 The middle term in the expansion of (x2 – 3)8 is: A. -70x8 B. 70x8 C. -5760x8 D. 5760x8 Problem 134 The term involving x9 in the expansion of (x2 + 2/x)12 is: A. 25434x9 B. 52344x9 C. 25344x9 D. 23544x9 Problem 135 1 The constant term in the expansion of ( x + 𝑥3/2 )15 is: A. 3003 B. 5005 C. 6435 D. 7365 Problem 136 Find the sum of the coefficients in the expansion of (x + 2y –z) 8. A. 256 B. 1024 C. 1 D. 6 Problem 137 Find the sum of the coefficients in the expansion of (x + 2y + z) 4 (x + 3y) 5 is: A. 524288 B. 65536 C. 131072 D. 262 144 Problem 138 (ECE April 1995) What is the sum of the coefficients in the expansion of (x + y -z) 8 is: A. less than 2 B. above 10 C. from 2 to 5 D. from 5 to 10 Problem 139 (ECE November 1995) What is the sum of the coefficients of the expansion of (2x -1)20? A. 1 B. 0 C. 215 D. 225 Problem 140 In the quadratic equation Ax2 + Bx + C = 0, the product of the roots is: A. C/A B. –B/A C. –C/A D. B/A Problem 141 If ¼ and -7/2 are the roots of the quadratic equation Ax2 + Bx + C = 0, what is the value of B? A. -28 B. 4 C. -7 D. 26 Problem 142 In the equation 3x2 + 4x + (2h – 5) = 0, find h if the product of the roots is 4. A. -7/2 B. -10/2 C. 17/2 D. 7/2 Problem 143 If the roots of ax2 + bx + c = 0, are u and v, then the roots of cx2 + bx + a = 0 are: A. u and v B. –u and v C. 1/u and 1/v D. -1/u and -1/v Problem 144 If the roots of the quadratic equation ax2 + bx + c = 0 are 3 and 2 and a, b, and c are all whole numbers, find a + b + c. A. 12 B. -2 C. 2 D. 6 Problem 145 (ECE March 1996) The equation whose roots are the reciprocals of the roots of 2x2 – 3x – 5 = 0 is: A. 5x2 + 3x – 2 = 0 B. 3x2 – 5x – 3 = 0 C. 5x2 – 2x – 3 = 0 D. 2x2 – 5x -3 = 0 Problem 146 (ECE November 1997) The roots of a quadratic equation are 1/3 and ¼. What is the equation? A. 12x2 + 7x + 1 = 0 B. 12x2 + 7x – 1 = 0 C. 12x2 – 7x + 1 = 0 D. 12x2 – 7x – 1 = 0 Problem 147 Find k so that the expression kx2 – 3kx + 9 is a perfect square. A. 3 B. 4 C. 12 D. 6 Problem 148 (EE October 1990) Find k so that 4x2+kx+1=0 will only have one real solution. A. 1 B. 4 C. 3 D. 2 Problem 149 The only root of the equation x2 – 6x + k = 0 is: A. 3 B. 2 C. 6 D. 1 Problem 150 Two engineering students are solving a problem leading to a quadratic equation. One student made a mistake in the coefficient of the first-degree term, got roots of 2 and -3. The other student made a mistake in the coefficient of the constant term got roots of -1 and 4. What is the correct equation? A. x2 – 6x – 3 = 0 B. x2 + 6x + 3 = 0 C. x2 + 3x + 6 = 0 D. x2 – 3x – 6 = 0 Age, Mixture, Work, Clock, Number Problem Problem 151 Two times the father’s age is 8 more than six times his son’s age. Ten years ago, the sum of their ages was 44. The age of the son is: A. 49 B. 15 C. 20 D. 18 Problem 152 Peter’s age 13 years ago was 1/3 of his age 7 years hence. How old is Peter? A. 15 B. 21 C. 23 D. 27 Problem 153 A man is 41 years old and in seven years he will be four times as old as his son is at that time. How old is his son now? A. 9 B. 4 C. 5 D. 8 Problem 154 A father is three times as old as his son. Four years ago, he was four times as old as his son was at that time. How old is his son? A. 36 years B. 24 years C. 32 years D. 12 years Problem 155 The ages of the mother and her daughter are 45 and 5 years, respectively. How many years will the mother be three times as old as her daughter? A. 5 B. 10 C. 15 D. 20 Problem 156 Mary is 24 years old. Mary is twice as old as Ana was when Mary was as old as Ana is now. How old is Ana? (ECE November 1995) A. 16 B. 18 C. 19 D. 20 Problem 157 The sum of the parent’s ages is twice the sum of their children’s ages. Five years ago, the sum of the parent’s ages is four times the sum of their children’s ages. In fifteen years the sum of the parent’s ages will be equal to the sum of their children’s ages. How many children were in the family? A. 2 B. 3 C. 4 D. 5 Problem 158 Two thousand kilogram of steel containing 8% of nickel is to be made by mixing stell containing 14% nickel with another steel containing 6% nickel. How much of the steel containing 14% nickel is needed? A. 1500 kg B. 800 kg C. 750 kg D. 500kg Problem 159 A 40-gram alloy containing 35% gold is to be melted with a 20-gram alloy containing 50% gold. How much percentage of gold is the resulting alloy? A. B. C. D. 40% 30% 45% 35% Problem 160 In what radio must a peanut costing P240.00 per kg. be mixed with a peanut costing P340.00 per kg so that the profit of 20% is made by selling the mixture at 360.00 per kg? A. 1:2 B. 3:2 C. 2:3 D. 3:5 Problem 161 A 100-kilogram salt solution originally 4% by weight. Salt in water is boiled to reduce water content until the concentration is 5% by weight salt. How much water is evaporated? A. 10 B. 15 C. 20 D. 25 Problem 162 A pound of alloy of lead and nickel weights 14.4 ounces in water, where lead losses 1/11 of its weight and nickel losses 1/9 of its weight. How much of each metal is in alloy? A. Lead = 7.2 ounces; Nickel = 8.8 ounces B. Lead = 8.8 ounces; Nickel = 7.2 ounces C. Lead = 6.5 ounces; Nickel = 5.4 ounces D. Lead = 7.8 ounces; Nickel = 4.2 ounces Problem 163 An alloy of silver and gold weighs 15 oz. in air and 14 oz. in water. Assuming that silver losses 1/10 of its weight in water and gold losses 1/18 of its weight, how many oz. at each metal are in the alloy? A. Silver = 4.5 oz.; Gold = 10.5 oz. B. Silver = 3.75 oz.; Gold = 11.25 oz. C. Silver = 5 oz.; Gold = 10 oz. D. Silver = 7.8 oz.; Gold = 4.2 oz. Problem 164(ME April 1998) A pump can pump out a tank in 11 hours. Another pump can pump out the same tank in 20 hours. How long it will take both pumps together to pump out the tank? A. ½ hour B. ½ hour C. 6 hours D. 7 hours Problem 165 Mr. Brown can wash his car in 15 minutes, while his son John takes twice as long as the same job. If they work together, how many minutes can they do the washing? A. 6 B. 8 C. 10 D. 12 Problem 166 One pipe can fill a tank in 5 hours and another pipe can fill the same tank in 4 hours. A drainpipe can empty the full content of the tank in 20 hours. With all the three pipes open, how long will it take to fill the tank? A. 2 hours B. 2.5 hours C. 1.92 hours D. 1.8 hours Problem 167 A swimming pool is filled through its inlet pipe and then emptied through its outlet pipe in a total of 8 hours. If water enters through its inlet and simultaneously allowed to leave through its outlet, the pool is filled in 7 ½ hours. Find how long will it take to fill the pool with the outlet closed. A. 6 B. 2 C. 3 D. 5 Problem 168 Three persons can do a piece of work alone in 3 hours, 4 hours and 6 hours respectively. What fraction of the job can they finish in one hour working together? A. ¾ B. 4/3 C. ½ D. 2/3 Problem 169 A father and his son can dig a well if the father works 6 hours and his son works 12 hours or they can do it if the father works 9 hours and son works 8 hours. How long will it take for the son to dig the well alone? A. 5 hours B. 10 hours C. 15 hours D. 20 hours Problem 170 Peter and Paul can do a certain job in 3 hours. On a given day, they work together for 1 hour then Paul left and Peter finishes the rest work in 8 more hours. How long will it take for Peter to do the job alone? A. 10 hours B. 11 hours C. 12 hours D. 13 hours Problem 171 (ECE November 1995) Pedro can paint a fence 50% faster than Juan and 20% faster than Pilar and together they can paint a given fence in 4 hours. How long will it take Peter to paint the same fence if he had to work alone? A. 10 hrs. B. 11hrs. C. 13hrs. D. 15hrs. Problem 172 Nonoy can finish a certain job in 10 days if Imelda will help for 6 days. The same work can be done by Imelda in 12 days if Nonoy helps for 6 days. If they work together, how long will it take for them to do the job? A. 8.9 B. 8.4 C. 9.2 D. 8 Problem 173 A pipe can fill up a tank with the drain open in three hours. If the pipe runs with the drain open for one hour and then the drain is closed it will take 45 more minutes for the pipe to fill the tank. If the drain will be closed right at the start of filling, how long will it take for the pipe to fill the tank? A. 1.15hrs. B. 1.125hrs C. 1.325hrs. D. 1.525hrs. Problem 174 Delia can finish a job in 8 hours. Daisy can do it in 5 hours. If Delia worked for 3 hours and then Daisy was asked to help her finish it, how long will Daisy have to work with Delia to finish the job? A. 2/5 hours B. 25/14 hours C. 28 hours D. 1.923 hours Problem 175 (CE November 1998) A job could be done by eleven workers in 15 days. Five workers started the job. They were reinforced with four more workers at the beginning of the 6th day. Find the total number of days it took them to finish the job. A. 22.36 B. 21.42 C. 23.22 D. 20.56 Problem 176 On one job, two power shovels excavate 20000m3 of earth, the larger the shovel working for 40 hours and the smaller shovel for 35 hours. Another job, they removed 40000m3 with the larger shovel working for 70 hours and the smaller working 90 hours. How much earth can the larger shovel move in one hour? A. B. C. D. 173.91 347.83 368.12 162.22 Problem 177 (EE April 1996) A and B can do a piece of work in 42 days, B and C in 31 days, and A and C in 20 days. Working together, how many days can all of them finish the work? A. 18.9 B. 19.4 C. 17.8 D. 20.9 Problem 178 Eight men can dig 150 ft of trench in 7hrs. Three men can backfill 100ft of the trench in 4hrs. The time it will take 10 men to dig and fill 200 ft of trench is: A. 9.867hrs. B. 9.687hrs. C. 8.967hrs. D. 8.687hrs. Problem 179 In two hours, the minute hand of the clock rotates through an angle of : A. 45° B. 90° C. 360° D. 720° Problem 180 In one day (24 hours), how many times will the hour hand and minute hand of a continuously driven clock be together A. 21 B. 22 C. 23 D. 24 Problem 181 How many minutes after 3:00 will the minute hand of the clock overtakes the hour hand? A. 14/12 minutes B. 16-11/12 minutes C. 16-4/11 minutes D. 14/11 minutes Problem 182 How many minutes after 10:00 o’clock will the hands of the clock be opposite of the other for the first time? A. 21.41 B. 22.31 C. 21.81 D. 22.61 Problem 183 What time between the hours of 12:00 noon and 1:00 pm would the hour hand and the minute hand of a continuously driven clock be in straight line? A. 12:33 pm B. 12:30 pm C. 12:37 pm D. 12:287 pm Problem 184 (GE February 1997) At what time after 12:00 noon will the hour hand and the minute hand of a clock first form a n angle of 120°? A. 21.818 B. 12:21.818 C. 21.181 D. 12:21.181 Problem 185 (GE February 1994) From the time 6:15 PM to the time 7:45 PM of the same day, the minute hand of a standard clock describes an arc of: A. 360° B. 120° C. 540° D. 720° Problem 186 It is now between 3 and 4 o’clock and in twenty minutes the minute hand will be as much as the hour-hand as it is now behind it. What is the time now? A. 3:06.06 B. 3:07.36 C. 3:09.36 D. 3:08.36 Problem 187 (EE October 1990) A man left his home at past 3:00 o’clock PM as indicated in his wall clock. Between two to three hours after, he returned home and noticed that the hands of the clock interchanged. At what time did he left his home? A. 3:27.27 B. 3:31.47 C. 3:22.22 D. 3:44.44 Problem 188 The sum of the reciprocals of two numbers is 11. Three times the reciprocal of one of the numbers is three more than twice the reciprocal of the other number. Find the numbers. A. 5 and 6 B. 7 and 4 C. 1/5 and 1/6 D. 1/7 and ¼ Problem 189 If a two digit number has x for its unit’s digit and y for its ten’s digit, represent the number. A. yx B. 10y + x C. 10x + y D. x + y Problem 190 One number if five less than the other number. If their sum is 135, what are the numbers? A. 70&75 B. 60&65 C. 65&70 D. 75&80 Problem 191 In a two-digit number, the unit’s digit is 3 greater than the ten’s digit. Find the number if it is 4 times as large as the sum of its digits. A. 47 B. 58 C. 63 D. 25 Problem 192 Find two consecutive even integers such that the square of the larger is 44 greater than the square of the smaller integer. A. B. C. D. 10&12 12&14 8&10 14&16 Problem 193 Twice the middle digit of a three-digit number is the sum of the other two. If the number is divided by the sum of its digit, the answer is 56 and the remainder is 12. If the digits are reversed, the number becomes smaller by 594. Find the number. A. 258 B. 567 C. 852 D. 741 Problem 194 The product f three consecutive integers is 9240. Find the third integer. A. 20 B. 21 C. 22 D. 23 Problem 195 The product if two numbers is 1400. If three (3) is subtracted from each number, their product becomes 1175. Find the bigger number. A. 28 B. 50 C. 32 D. 40 Problem 196 The sum of the digits of the three-digit number is 14. The hundreds digit being 4 times the units digit. If 594 is subtracted from the number, the order of the digits will be reversed. Find the number. A. 743 B. 563 C. 653 D. 842 Problem 197 (ECE March 1996) The sum of two numbers is 21, and one number is twice the other. Find the numbers. A. 7 and 14 B. 6 and 15 C. 8 and 13 D. 9 and 12 Problem 198 (ECE March 1996) Ten less than four times a certain number is 14. Determine the number. A. 4 B. 5 C. 6 D. 7 Problem 199 (ECE November 1997) The denominator of a certain fraction is three more than twice the numerator. If 7 is added to both terms of the fraction, the resulting fraction is 3/5. Find the original fraction. A. 8/5 B. 5/13 C. 13/5 D. 3/5 Problem 200 Three times the first of the three consecutive odd integers is three more than twice the third. Find the third integer. A. 9 B. 11 C. 13 D. 15 Motion Variation, Percent, Miscellaneous Problems Problem 201 Nonoy left Pikit to drive to Davao at 6:15 PM and arrived at 11:45 PM averaged 30 mph and stopped 1 hour for dinner, how far is Davao from Pikit. A. 128 B. 135 C. 160 D. 256 Problem 202 A man fires a target 420 m away hears the bullet strikes to 2 second after he pulled the trigger. An observer 525 m away from the target and 455 m from the man heard the bullet strike the target one second after he heard the report of the rifle. Find the velocity of the bullet. A. 525 m/s B. 360 m/s C. 350 m/s D. 336 m/s Problem 203 A man travels in a motorized banca at rate of 12 kph from his barrio to the poblacion and come back to his barrio at the rate of 10 kph. If his total time of travel back and forth is 3 hours and 10 minutes, the distance from the barrio to the poblacion is : A. 17.27 km B. 17.72 km C. 12.77 km D. 17.32 km Problem 204 It takes Michael 60 seconds to run around a 440yard track. How long does it take Jordan to run around the track if they meet in 32 second after they start together in a race around the track in opposite direction? A. 58.76 seconds B. 68.57 seconds C. 65.87 seconds D. 86.57 seconds Problem 205 Juan can walk from his home to his office at the rate of 5 mph and back at the rate 2 mph. What is his average speed in mph? A. 2.86 B.3.56 C.4.12 D.5.89 Problem 206 Kim and Ken traveled at the same time at the rate of20m/min,from the same pointon a circular track of radius 600 m. If Kim walks along a circumference and Kim towards the center,find their distance after 10 minutes. A.193 m B.202 m C.241 m D.258 m Problem 207 Two ferryboats ply back and forth across a river with constant but different speeds, turning at the river banks without loss of time. They leave the opposite shores at the same instant, meet for the first time 900 meters from one shore, and meet for the second time 500 meters from the opposite shore. What is the width of the river? A. 1500 m B. 1700 m C. 2000 m D. 2200 m Problem 208 (CE May 1998) A boat takes 2/3 as much time to travel downstream from C to D, as to return, If the rate of the river’s current is 8 kph, what is the speed of the boat in still water? A. B. C. D. 38 39 40 41 Problem 209 (ECE November 1998) A man rows downstream at the rate of 5mph and upstream at the rate of 2mph. How far downstream should he go if he is to return in 7/4 hours after leaving? A. 2 mi B. 3.5 mi C. 3 mi D. 2.5 mi Problem 210 (EE April 1997) A jogger starts a course at a steady rate of 8kph. Five minutes later, a second jogger the same course at 10 kph. How long will it take for the second jogger to catch the first? A. 20 min B. 25 min C. 30 min D. 35 min Problem 211 (CE May 1999) At 2:00 pm, an airplane takes off at 340mph on an aircraft carrier. The aircraft carrier moves due south at 25kph in the same direction as the plane. At 4:05 pm, the communication between the plane and aircraft carrier was lost. Determine the communication range in miles between the plane and the carrier. A. 656 miles B. 785 miles C. 557 miles D. 412 miles Problem 212 A boat going across a lake 8km wide proceed 2 km at a certain speed and then completes the trip at a speed 1/2kph faster. By doing this, the boat arrives 10 minutes earlier than if the original speed had been maintained. Find the original speed of the boat. A. 2 kph B. 4 kph C. 9 kph D. 5 kph If x varies directly as y and inversely as z, and x=14 when y=7 and z=2, find x, when z=4 and y=16. A. 14 B. 4 C. 16 D. 8 Problem 215 The electrical resistance of a cable varies directly as its length and inversely as the square of its diameter. If a cable 600 meters long and 25 mm in diameter has a resistance of 0.1 ohm, find the length of the cable 75 mm in diameter with resistance of 1/6 ohm. A. 6000 m B. 7000 m C. 8000 m D. 9000 m Problem 216 The electrical resistance offered by an electric wire varies directly as the length and inversely as the square of the diameter of the wire. Compare the electrical resistance offered by two pieces of wire of the same material, one being 100 m long and 5 mm diameter, and the other is 50 m long and 3 mm in diameter. A. R1 = 0.57 R2 B. R1 = 0.72 R2 C. R1 = 0.84 R2 D. R1 = 0.95 R2 Problem 217 The time required for an elevator to lift a weight varies directly with the weight and the distance through which it is to be lifted and inversely as the power of the motors. If it takes 20 seconds for a 5hp motor to lift 50 lbs. through 40 feet, what weight can an 80-hp motor lift through a distance of 40 feet within 30 seconds? A. 1000 lbs. B. 1150 lbs. C. 1175 lbs. D. 1200 lbs. Problem 213 (CE May 1993) Given that w varies directly as the product of x and y and inversely as the square of z and that w=4 when x=2, y=6, and z=3. Find w when x=1, y=4 and z=2. A. 4 B. 2 C. 1 D. 3 Problem 218 (ECE November 1995) The time required by an elevator to lift a weight, vary directly with the weight and the distance through which it is to be lifted and inversely as the power of the motor. If it takes 30 seconds for a 10hp motor to lift 100lbs through 50 feet, what size of motor is requires to lift 800 lbs. in 40 seconds through a distance of 40 feet? A. 48 hp B. 50 hp C. 56 hp D. 58 hp Problem 214 (ECE November 1993) Problem 219 In a certain department store, the salary of saleslady is partly constant and varies as the value of her sales for the month, when the value of her sales for the month is P10000.00, her salary for that month is P900.00. When her sales goes up to P 2000.00 her monthly salary goes up to P1000.00. What must be the value of her sales for the month so that her salary for that month will be P2000.00? A. P25000.00 B. P28000.00 C. P32000.00 D. P36000.00 Problem 220 A man sold 100 eggs, eighty of them were sold at gain of 30% while the twenty eggs were sold at a loss of 40%. What is the percentage gain or loss of the whole eggs? A. 14% B. 15% C. 16% D. 17% Problem 221 The population of the country increases 5% each year. Find the percentage it will increase in three years. A. 5% B. 15% C. 15.15% D. 15.76% By selling balut at P5.00 each, a vendor gains 20%. The cost price of egg rises by 12.5%. If he sells the balut at the same price as before, find his new gain in percent. A. 7.5% B. 5% C. 8% D. 6.25% Problem 226 The enrollment at college A and college B both grew up by 8% from 1980 to 1985. If the enrollment in college A grew up by 800 and the enrollment in college B grew up by 840, the enrollment at college B was how much greater than the enrollment in college A in 1985? A. 650 B. 504 C. 483 D. 540 Problem 227 A group consists of n boys and n girls. If two of the boys are replaced by two other girls, then 49% of the group members will be boys. Find the value of n. A. 100 B. 49 C. 50 D. 51 Problem 222 Pedro bought two cars, one for P600000.00 and the other for P400000.00. He sold the first at a gain of 10% and the second at a loss of 12%. What was his total percentage gain or loss? A. 6% gain B. 0% gain C. 1.20% gain D. 6% loss Problem 228 On his Christmas Sale, a merchant marked a pair of slipper P180.00, which is 20% off the normal retail price. If the retail price is 50% higher than the whole sale price, what is the wholesale price of the slipper? A. P18.00 B. P17.00 C. P15.00 D. P22.50 Problem 223 A grocery owner raises the prices of his goods by 10%. Then he starts his Christmas sale by offering the customers a 10% discount. How many percent of discount does the customers actually get? A. nothing B. 1% discount C. 9% discount D. they pay 1% more Problem 229 A certain XEROX copier produces 13 copies every 10 seconds. If the machine operates without interruption, how many copies will it produce in an hour? A. 780 B. 46800 C. 1835 D. 4680 Problem 224 Kim sold a watch for P3500.00 at a loss of 30% on the cost price. Find the corresponding loss or gain if he sold it for P5050.00. A. 1% loss B. 10% loss C. 1% gain D. 10% gain Problem 230 At a certain printing plant, each of the machines prints 6 newspapers every second. If all machines work together but independently without interruption, how many minutes will it take to print the entire 18000 newspapers? ( Hint: let x = number of machines) A. 50x B. 3000/x C. 50/x Problem 225 D. 3000x Problem 231 (ME April 1996) A manufacturing firm maintains one product assembly line to produce signal generators. Weekly demand for the generators is 35 units. The line operates for 7 hours per day, 5 days per week. What is the maximum production time per unit in hours required for the line to meet the demand? A. 1 hour B. 0.75 hour C. 3 hours D. 2.25 hours Problem 232 Of the 316 people watching a movie, there a re 78 more children than women and 56 more women than men. The number of men in the movie house is: A. 176 B. 98 C. 42 D. 210 Problem 233 A certain department store has an inventory of Q units of a certain product at time t=0. The store sells the product at a steady rate of Q/A units per week, and exhausts the inventory in A weeks. The amount of product in inventory at any time t is: A. Q – (Q/A) t B. Q + (Q/A) t C. Qt – Q/A D. Qt – (Q/A) t Problem 234 (ECE March 1996) A merchant has three items on sale: namely, a radio for P50, a clock for P30, and a flashlight for P1. At the end of the day, she has sold a total of 100 of three items and has taken exactly 1000 on the total sales. How many radios did he sale? A. 80 B. 4 C. 16 D. 20 Problem 235 The price of 8 calculators ranges from P200 to P1000.If their average price is P950,what is the lowest possible price of any one of the calculators? A. 500 B. 550 C. 600 D. 650 Problem 236 A deck of 52 playing cards is cut into two piles. The first pile contains 7 times as many black cards as red cards. The second pile contains the number of red cards that is an exact multiple as the number of black cards. How many cards are there in the first pile. A. B. C. D. 14 15 16 17 Problem 237 (ECE November 1997) The population of the Philippines doubled in the last 30 years from 1967 to 1997.Assuming that the rate of population rate increase will remain the same in what year wills the population triple? A. 2030 B. 2027 C. 2021 D. 2025 Problem 238 Determine the unit digit in the expansion of 3 855. A. 3 B. 9 C. 7 D. 1 Problem 239 (ECE April 1998) Find the 1987th digit in the decimal equivalent of 1785/9999 starting from the decimal point. A. 1 B. 7 C. 8 D. 5 Problem 240 Find the sum of all positive integral factors of 2048. A. 4095 B. 3065 C. 4560 D. 1254 Problem 241 In how many ways can two integers be selected from the numbers 1,2,3,…50 so that their difference is exactly 5? A. 50 B. 5 C. 45 D. 41 Problem 242 A box contains 8 balls, 6 black balls.a8 red balls, and 13 yellow balls. How many balls must be drawn to ensure that there will be three balls of the same color? A. 8 B. 9 C. 10 D. 11 Problem 243 A shore sells 10 different sizes of shoes, each in both high-cut and low-cut variety, each either rubber or leather, and each with white or black color. How many different kinds of shoes does he sell? A. 64 B. 80 C. 72 D. 92 Problem 244(ME October 1999) An engineer was told that a survey had been made on a certain rectangular field but the dimension had been lost .An assistant remembered that if the field had been 100 ft longer and 25 ft narrower, the area would have been increased by 2500 sq. ft, and that if it had been 100 ft shorter 50 ft wider, the area would have been decreased 5000 sq.ft. What was the area of the field? A. 25.000 ft2 B. 15,000 ft2 C. 20,000 ft2 D. 22,000 ft2 Problem 245 (EE April 1994) A 10-meter tape is 5 mm short. What is the correct length in meters? A. 9.995 m B. 10.05 m C. 9.95 m D. 10.005 m Problem 246 (ME OCTOBER 1997) The distance between two points measured with a steel tape was recorded as 916.58 ft. later. The tape was checked and to be only 99.9 ft long. What is the true distance between the points? A. 035.66 ft B. 966.15 ft C. 955.66 ft D. 915.66 ft Problem 247 (ME April 1996) A certain steel tape is known to be 100000 feet long when the temperature of 70 ℉ . When the tape is at a temperature of 10℉, what reading corresponds to a distance of 90 ft? Coefficient of linear expansion of the tape is 5.833 × 10-6 per℉. A. 85.935 B. 88.031 C. 90.031 D. 93.031 Problem 248 (ME April 1996) A line was measured with a steel tape when the temperature was 30℃. The measured length of the line was found to be 1,256.271 feet. The tape was afterwards tasted when the temperature was 10 ℃ and it was found to be 100.042 feet long. What was the true length of the line if the coefficient of expansion of the tape was 0.000011 per℃? A. 1,275.075 feet B. 1,375.575 feet C. 1,256.547 feet D. 1,249.385 feet Problem 249 (ME April 1997) The standard deviation of the numbers 1, 4, &7 is: A. 2.3567 B. 2.4495 C. 3.2256 D. 3.8876 Problem 250 Three cities are connected by roads forming a triangle, all of different lengths. It is 30 km around the circuit. One of the roads is 10 km long and the longest is 10 km longer than the shortest. What is the length of the longest road? A. 5 km B. 10 km C. 15 km D. 20 km Progression, Matrix, Determinant, Venn diagram Problem 251 (ECE November 1996) How many terms of the sequence -9, -6, -3 … must be taken so that the sum is 66? A. 13 B. 12 C. 4 D. 11 Problem 252 (CE November 1997) The sum of the progression 5, 8, 11, 14 …. is 1025. How many terms are there? A. 22 B. 23 C. 24 D. 25 Problem 253 (CE May 1998) There are seven arithmetic means between 3 and 35. Find the sum of all terms. A. 169 B. 171 C. 167 D. 173 Problem 254 (CE May 1999) There are line (9) arithmetic means between 11 and 51. The sum of the progression is: A. 279 B. 341 C. 376 D. 254 Problem 255 The sum of all even numbers from 0 to 420 is: A. 43410 B. 44300 C. 44310 D. 44130 Problem 256 (CE May 1997) Which of the following numbers should be changed to make all the numbers form an arithmetic progression when properly arranged? A. 27/14 B. 33/28 C. 45/28 D. 20/14 Problem 257 The first term of an arithmetic progression (A.P.) is 6 and the 10th term is 3 times the second number. What is the common difference? A. 1 B. 2 C. 3 D. 4 Problem 258 The sum of five arithmetic means between 34 and 42 is: A. 150 B. 160 C. 190 D. 210 Problem 259 The positive values of a so that 4x, 5x + 4, 3x2 – 1will be in arithmetic progression is: A. 2 B. 3 C. 4 D. 5 Problem 260 Solve for x if x + 3x + 5x + 7x + … + 49x = 625 A. ¼ B. ½ C. 1 D. 1 ¼ Problem 261 The 10th term of the series a, a-b, a-2b, … is: A. a-6b B. a-9b C. 2a-b D. a+9b Problem 262 If the sum of the first 13 terms of two arithmetic progressions are in the ratio 7:3, find the ratio of their corresponding 7th term. A. 3:7 B. 1:3 C. 7:3 D. 6:7 Problem 263 If 1/x, 1/y, 1/z are in arithmetic progression, then y is equal to: A. B. C. D. X-z ½(x+2z) (x+z)/2xz 2xz/(x+z) Problem 264 (ECE November 1997) Find the 30th term of the A.P. 4, 7, 10 … A. 88 B. 91 C. 75 D. 90 Problem 265 (ECE November 1997) Find the 100th term of the sequence 1.01, 1.00, 0.99…. A. 0.05 B. 0.04 C. 0.03 D. 0.02 Problem 266 The sum of all numbers between 0 and 10000 which is exactly divisible by 77 is: A. 546546 B. 645568 C. 645645 D. 645722 Problem 267 (ME April 1998) What is the sum of the following finite sequence of terms? 18, 25, 32, 39, .., 67. A. 234 B. 181 C. 213 D. 340 Problem 268 Find x in the series: 1, 1/3, 0.2, x. A. 1/6 B. 1/8 C. 1/7 D. 1/9 Problem269 (ECE November 1995) Find the fourth term of the progression ½, 0.2, 0.125, … A. 0.102 B. 1/10 C. 1/11 D. 0.099 Problem 270 The 10th term of the progression 6/4, 4/3, 3/2, … is: A. 12 B. 10/3 C. 12/3 D. 13/3 Problem 271 (ME October 1997) The geometric mean of 4 and 64 is: A. 48 B. 16 C. 34 D. 24 Problem 272 (ME October 1997) The geometric mean of a nd b is: A. √𝑎𝑏 B. (a+b)/2 C. 1/b D. ab/2 Problem 273 (CE May 1998) Determine the sum of the infinite geometric series of 1, -1/5,+1/25, …? A. 4/5 B. 5/7 C. 4/6 D. 5/6 Problem 274 There are 6 geometric means between 4 and 8748. Find the sum of all terms. A. 13120 B. 15480 C. 10250 D. 9840 Problem 275 (ECE April 1998) Find the sum of the infinite progression 6, -2, -2/3 … A. 5/2 B. 9/2 C. 7/2 D. 11/2 Problem 276 (ECE April 1998) Find the sum of the first 10 terms of the Geometric Progression 2, 4, 8, 16 … A. 1023 B. 2046 C. 1596 D. 225 Problem 277 The 1st, 4th, 8th terms of an A.P. are themselves geometric progression (G.P.). What is the common ratio of the G.P.? A. 4/3 B. 5/3 C. 2 D. 7/3 Problem 278 Determine x so that x, 2x+7, 10x-7 will form a geometric progression. A. -7 B. 6 C. 7 D. -6 Problem 279 The fourth term of a geometric progression is 189 and the sixth term is 1701, the 8th term is: A. 5103 B. 1240029 C. 45927 D. 15309 Problem 280 The sum of the numbers in arithmetical progression is 45. If 2 is added to the first number, 3 to the second and 7 to the third, the new numbers will be in geometrical progression. Find the common difference in A.P. A. -5 B. 10 C. 6 D. 5 Problem 281 The geometric mean and the harmonic mean of two numbers are 12 and 36/5 respectively. What are the numbers? A. 36 & 4 B. 72 & 8 C. 36 & 8 D. 72 & 4 Problem 282 If x, 4x+8, 30x +24 are in geometrical progression, find the common ratio. A. 2 B. 4 C. 6 D. 8 Problem 283 (ECE April 1995) A besiege fortress is held by 5700 men who have provision for 66 days. If the garrison loses 20 men each day, for how many days can the provision hold out? A. 60 B. 72 C. 76 D. 82 Problem 284 (ECE April 1999) If one third of the air in the tank is removed by each stroke of an air pump, what fractional part of the total air is removed in 6 strokes? A. 0.9122 B. 0.0877 C. 0.8211 D. 0.7145 Problem 285 A rubber ball is dropped from a height of 15m. On each rebound, it rises 2/3 of the height from which it last fell. Find the distance traveled by the ball before it becomes to rest. A. 75m B. 96m C. 100m D. 85m Problem 286 (CE May 1991) In the recent Bosnia conflict, The NATO forces captured 6400 soldiers. The provisions on hand will last for 216 meals while feeding 3 meals a day. The provisions lasted 9 more days because of daily deaths. At an average, how many died per day? A. 15.2 B. 17.8 C. 18.3 D. 19.4 Problem 287 To build a dam, 60 men must work 72 days. If all 60 men are employed at the start but the number is decreased by 5 men at the end of each 12-day period, how long will it take to complete the dam? A. 108 days B. 9 days C. 94 days D. 60 days D. 46 Problem 292 (CE November 1996) Compute the value of x from the following: 4 −1 2 3 2 0 2 1 | x =| 10 3 0 1 14 2 4 5 A. B. C. D. 27 -28 26 -29 Problem 293 Evaluate the following determinant: 1 4 2 −1 D=| −2 3 0 2 A. 5 B. -4 C. 4 D. -5 2 −1 0 −3 | 1 2 1 4 Problem 288 (CE November 1994) In a benefit show, a number of wealthy men agreed that the first one to arrive would pay 10 centavos to enter and each later arrival would pay twice as much as the preceding man. The total amount collected from all of them was P104857.50. How may wealthy men had paid? A. 18 B. 19 C. 20 D. 21 Problem 294 Problem 289 Problem 295 (CE May 1996) 1 2 |; Elements of Elements of matrix B = | 0 −5 3 6 | matrix C = | 4 1 Find the elements of the product of the two matrices, matrix BC. 11 8 | A. | −20 −5 15 9| B. | −22 4 12 10 | C. | 20 −4 15 15 | D. | −17 −6 7 8 | Evaluate the following determinant: | 9 4 A. 64 B. 44 C. 54 D. -44 Problem 290 The following equation involves two determinants: 3 𝑥 2 −1 | |=| | 2 2 𝑥 −3 The value of x is: A. 1 B. 3 C. 4 D. 3 Problem 291 (CE November 1997) Evaluate the following determinant: 1 5 −2 |2 1 −3| 3 −2 1 A. -24 B. 24 C. -46 1 3 | and matrix B = Given matrix A = | −2 1 −1 −2 | |. Find A + 2B. −1 1 1 0 | A. | 2 1 1 0 | B. | 1 3 −1 3 | C. | 0 1 −1 −1 | D. | 0 3 Problem 296 (CE Board) Solve for x and y from the given relationship: 1 1 𝑥 2 | || | = | | 3 2 𝑦 0 A. x = -2; y = 6 B. x = 2; y = 6 C. x = -2; y = -6 D. x =2; y = -6 Problem 297 (EE October 1993) In a class of 40 students, 27 students like Calculus and 25 like Geometry. How many students liked both Calculus and Geometry? A. 10 B. 14 C. 11 D. 12 Problem 298 A class of 40 took examination in Algebra and Trigonometry. If 30 passed Algebra, 36 passed Trigonometry, and 2 failed in both subjects, the number of students who passed the two subjects is: A. 2 B. 8 C. 28 D. 25 Problem 299 (ECE November 1992) The probability for the ECE board examinees from a certain school to pass the Mathematics subject is 3/7 and that for the Communications subject is 5/7. If none of the examinees failed in both subjects and there are 4 examinees who pass both subjects, how many examinees from the school took the examination? A. 28 B. 27 C. 26 D. 32 Problem 300 (EE March 1998) In a commercial survey involving 1000 persons on brand preferences, 120 were found to prefer brand x only, 200 persons prefer brand y only, 150 persons prefer brand z only, 370 prefer either brand x or y not z, 450 prefer brand y or z but not x, and 370 prefer wither brand z or x but not y, and none prefer all the three brands at a time. How many persons have no brand preference with any of the three brands? A. 120 B. 280 C. 70 D. 320 How many 4 digit number can be formed without repeating any digit, from the following digit 1,2,3,4 and 6. A. 150 B. 120 C. 140 D. 130 Problem 303 How many permutations can made out of the letters of the word ENGINEERING? A. 39,916,800 B. 277,200 C. 55,440 D. 3,326,400 Problem 304 How many ways can 3 men and 4 women be seated on a bench if the women to be together? A. 720 B. 576 C. 5040 D. 1024 Problem 305 In how many ways can 5 people line up to pay their electric bills? A. 120 B. 1 C. 72 D. 24 Problem 306 In how many ways can 5 people line up to pay their electric bills, if two particular persons refuse to follow each other? A. 120 B. 72 C. 90 D. 140 Permutation, Combination, Probability Problem 307 How many ways can 7 people be seated at a round table? A. 5040 B. 120 C. 720 D. 840 Problem 301 How permutation can be made out of the letters in the world island taking four letters at a time? A. 360 B. 720 C. 120 D. 24 Problem 308 In how many relative orders can we seat 7 people at a round table with a certain people side by side. A. 144 B. 5040 C. 720 D. 1008 Problem 302 (CE November 1996) Problem 309 In how many ways can we seat 7 people in a round table with a certain 3 people not in consecutive order? A. 576 B. 3960 C. 5320 D. 689 Problem 310 The captain of a baseball team assigns himself to the 4th place in the batting order. In how many ways can he assign the remaining places to his eight teammates if just three men are eligible for the first position? A. 2160 B. 40320 C. 5040 D. 15120 Problem 311 In how many ways can PICE chapter with 15 directors choose a president, a vice-president, a secretary, a treasurer, and an auditor, if no member can hold more than one position? A. 630630 B. 3300 C. 5040 D. 15120 Problem 312 How many ways can a committee of five be selected from an organization with 35 members? A. 324632 B. 425632 C. 125487 D. 326597 Problem 313 How many line segments can be formed by 13 distinct point? A. 156 B. 36 C. 98 D. 78 Problem 314 In how many ways can a hostess select six luncheon guests from 10 women if she is to avoid having particular two of them together at the luncheon? A. 210 B. 84 C. 140 D. 168 Problem 315 (ECE April 1998) A semiconductor company will hire 7 men and 4 women. In how many ways can the company choose from 9 men and 6 women who qualified for the position? A. 680 B. 840 C. 480 D. 540 Problem 316 How many ways can you invite one or more of five friends to a party? A. 25 B. 15 C. 31 D. 62 Problem 317 A bag contains 4 red balls, 3 green balls, and 5 blue balls. The probability of not getting a red ball in the first draw is: A. 2 B. 2/3 C. 1 D. 1/3 Problem 318 Which of the following cannot be a probability? A. 1 B. 0 C. 1/e D. 0.434343 Problem 319 (CE May 1996) A bag contains 3 white and 5 black balls. If two balls are drawn in succession without replacement, what is the probability that both balls are black? A. 5/28 B. 5/16 C. 5/32 D. 5/14 Problem 320 A bag contains 3 white and 5 red balls. If two balls are drawn at random, find the probability that both are white. A. 3/28 B. 3/8 C. 2/7 D. 5/15 Problem 321 In problem 320, find the probability that one ball is white and the other is red. A. 15/56 B. 15/28 C. ¼ D. 225/784 Problem 322 In the problem 320, find the probability that all are of the same color. A. 13/30 B. 14/29 C. 13/28 D. 15/28 Problem 323 The probability that both stages of a two-stage rocket to function correctly is 0.92. The reliability of the first stage is 0.97. The reliability of the second stage is: A. 0.948 B. 0.958 C. 0.968 D. 0.8924 Problem 324 Ricky and George each throw dice. If Ricky gets a sum of four what is the probability that George will get less than of four? A. ½ B. 5/6 C. 9/11 D. 1/12 Problem 325 Two fair dice are thrown. What is the probability that the sum of the dice is divisible by 5? A. 7/36 B. 1/9 C. 1/12 D. ¼ Problem 326 (ME April 1996) An um contains 4 black balls and 6 white balls. What is the probability of getting one black ball and white ball in two consecutive draws from the urn? A. 0.24 B. 0.27 C. 0.53 D. 0.04 Problem 327 If three balls in drawn in succession from 5 white and a second bag, find the probability that all are of one color, if the first ball is replaced immediately while the second is not replaced before the third draw. A. 10/121 B. 18/121 C. 28/121 D. 180/14641 Problem 328 A first bag contains 5 white balls and 10 black balls. The experiment consists of selecting a bag and then drawing a ball from the selected bag. Find the probability of drawing a white ball. A. 1/3 B. 1/6 C. 1/2 D. 1/8 Problem 329 In problem 328, find the probability of drawing a white ball from the first bag. A. 5/6 B. 1/6 C. 2/3 D. 1/3 Problem 330 If seven coins are tossed simultaneously, find the probability that will just have three heads. A. 33/128 B. 35/128 C. 30/129 D. 37/129 Problem 331 If seven coins are tossed simultaneously, find the probability that there will be at least six tails. A. 2/128 B. 3/128 C. 1/16 D. 2/16 Problem 332 (CE November 1998) A face of a coin is either head or tail. If three coins are tossed, what is are the probability of getting three tails? A. 1/8 B. ½ C. ¼ D. 1/6 Problem 333 The face of a coin is either head or tail. If three coins are tossed, what is the probability of getting three tails or three heads? A. 1/8 B. ½ C. ¼ D. 1/6 Problem 334 Five fair coins were tossed simultaneously. What is the probability of getting three heads and two tails? A. 1/32 B. 1/16 C. 1/8 D. ¼ Problem 335 Throw a fair coin five times. What is the probability of getting three heads and two tails? A. 5/32 B. 5/16 C. 1/32 D. 7/16 Problem 336 (ECE March 1996) The probability of getting credit in an examination is 1/3. If three students are selected at random, what is the probability that at least one of them got a credit? A. 19/27 B. 8/27 C. 2/3 D. 1/3 Problem 337 There are three short questions in mathematics test. For each question, one (1) mark will be awarded for a correct answer and no mark for a wrong answer. If the probability that Mary correctly answers a question in a test is 2/3, determine the probability that Mary gets two marks. A. 4/27 B. 8/27 C. 4/9 D. 2/9 Problem 338 A marksman hits 75% of all his targets. What is the probability that he will hit exactly 4 of his next ten shot? A. 0.01622 B. 0.4055 C. 0.004055 D. 0.001622 Problem 339 A two-digit number is chosen randomly. What is the probability that it is divisible by 7? A. 7/50 B. 13/90 C. 1/7 D. 7/45 Problem 340 One box contains four cards numbered 1, 3,5,and 6. Another box contains three cards numbered 2, 4, and 7. One card is drawn from each bag. Find the probability that the sum is even. A. 5/12 B. 3/7 C. 7/12 D. 5/7 Problem 341 Two people are chosen randomly from 4 married couples. What is probability that they are husband and wife? A. 1/28 B. 1/14 C. 3/28 D. 1/7 Problem 342 One letter is taken from each of the words PARALLEL and LEVEL at random. What is the probability of getting the same letter? A. 1/5 B. 1/20 C. 3/20 D. ¾ Problem 343 In a shooting game, the probability that Botoy and Toto will hit a target is 2/3 and ¾ respectively. What is the probability that the target is hit when both shoot at it once? A. B. C. D. 13/5 5/13 7/12 11/12 Problem 344 A standard deck of 52 playing cards is well shuffled. The probability that the first four cards dealt from the deck will be four aces is closes to: A. 4×10-6 B. 2×10-6 C. 3×10-6 D. 8×10-6 Problem 345 A card is chosen from pack of playing cards. What is the probability that it is either red or a picture card? A. 8/13 B. 10/13 C. 19/26 D. 8/15 Problem 346 In a poker game consisting of 5 cards, what is the probability of holding 2 aces and 2 Queens? A. 5! /52! B. 5/52 C. 33/54145 D. 1264/45685 Problem 347 Dennis Rodman sinks 50% of all his attempts. What is the probability that he will make exactly 3 of his next 10 attempts? A. 1/256 B. 3/8 C. 30/128 D. 15/128 Problem 348 There are 10 defectives per 1000 items of a product in long run. What is the probability that there is one and only one defective in random lot of 100? A. 0.3697 B. 0.3967 C. 0.3796 D. 0.3679 Problem 349 The UN forces for Bosnia uses a type of missile that hits the target with a probability of 0.3. How many missiles should be fired so that there is at least an 80% probability of hitting the target? A. 2 B. 4 C. 5 D. 3 Problem 350 (ME April 1997) In a dice game, one fair is used. The player wins P20.00 if he rolls either 1 or 6. He losses P10.00 if he turns up any other face. What is the expected winning for one roll of the die? A. P40.00 B. P0.00 C. P20.00 D. P10.00 Complex Numbers, Elements Problem 351 (CE May 1994) In the complex number 3 + 4i, the absolute value is: A. 10 B. 7.211 C. 5 D. 5.689 Problem 352 In the complex number 8-21, the amplitude is: A. 104.04° B. 14.04° C. 345.96° D. 165.96° Problem 353 (6 cis 120°)(4 cis 30°) is eual to: A. 10 cis150° B. 24 cis150° C. 10 cis90° D. 24 cis90° Problem 354 20 𝑐𝑖𝑠 80° is equal to: 10 𝑐𝑖𝑠 50° A. 20cis30° B. 3cis130° C. 3cis30° D. 20 cis130° Problem 355 The value of x + y in the complex equation 3 + xi = y + 2i is: A. 5 B. 1 C. 2 D. 3 Problem 356 Multiply (3-2i)(4+3i). A. 12+i B. 18+i C. 6+i D. 20+i Problem 357 (EE October 1997) 4+3𝑖 Divide 2−𝑖 . 11+10𝑖 A. 5 B. 1+2i 5+2𝑖 C. 5 D. 2+2i Problem 358 Find the value of i9. A. i B. –i C. 1 D. -1 Problem 359 (ECE April 1999) Simplify i1997+i1999, where I is an imaginary number. A. 1+i B. I C. 1-i D. 0 Problem 360 Expand (2+√−9)3 A. 46+9i B. 46-9i C. -46-9i D. -46+9i Problem 361 Write -4+3i in polar form. A. 5∟36.87° B. 5∟216.87° C. 5∟323.13° D. 5∟143.13° Problem 362 Simplify: i30-2i25+3i17. A. I+1 B. -1-2i C. -1+i D. -1+5i Problem 363 (ME April 1997) Evaluate the value of √−10 × √−7. A. Imaginary B. -√70 C. √17 D. √70 Problem 364 (EE April 1994) 3 Perform the indicated operation: √−9 × √−343. A. 21 B. 21i C. -21i D. -21 Problem 365 (ECE April 1999) What is the quotient when 4+8i is divided by i 3? A. 8+4i B. -8+4i C. 8-4i D. -8-4i Problem366 What is the exponential form of the complex number 4+3i? A. 5𝑒 𝑖53.13° B. 5𝑒 𝑖36.87° C. 7𝑒 𝑖53.13° D. 7𝑒 𝑖36.87° Problem 367 What is the algebraic form of the complex number 13𝑒 𝑖67.38°? A. 12+5i B. 5-12i C. 12-5i D. 5+12i Problem 368 (ME April 1998) Solve for x that satisfy the equation x2+36 = 9-2x2. A. ±6i B. ±3i C. 9i D. -9i Problem 369 Evaluate ln (5+12i). A. 2.565+1.176i B. 2.365-0.256i C. 5.625+2.112i D. 3.214-1.254i Problem 370 (EE April 1994) Add the given vectors: (-4, 7) + (5, -9) A. (1, 16) B. (1, -2) C. (9, 2) D. (1, 2) Problem 371 (EE April 1994) Find the length of vector (2, 1,1). A. √17 B. √21 C. √20 D. √19 Problem 372 (ECE November 1997) Find the length of the vector (2, 4, 4). A. 8.75 B. 6.00 C. 7.00 D. 5.18 Problem 373 If a=b and b=c, then a=c. This property of real numbers is known as: A. Reflexive Property B. Symmetric Property C. Transitive Property D. Addition Property Problem 374 If a=b, then b=a. This property of real numbers is known as: A. Reflexive Property B. Symmetric Property C. Transitive Property D. Multiplication Property Problem 375 A statement the truth of which is admitted without proof is called: A. An axiom B. A postulate C. A theorem D. A corollary Problem 376 In a proportion of four quantities, the first and the fourth terms are referred to: A. means B. denominators C. extremes D. numerators Problem 377 (ECE November 1997) Convergent series is a sequence of decreasing numbers or when the succeeding term is ____ than the preceding term. A. ten times more B. greater C. equal D. lesser Problem 378 (ECE November 1997) It is the characteristics of a population which is measurable. A. Frequency B. Distribution C. Sample D. Parameter Problem 379 (ECE November 1997) The quartile deviation is a measure of: A. Division B. Central tendency C. Certainty D. Dispersion Problem 380 (ECE November 1995, 1997) In complex algebra, we use a diagram to represent a complex plane commonly called: A. De Moivre’s Diagram B. Funicular Diagram C. Argand Diagram D. Venn Diagram Problem 381 A series of numbers which are perfect square numbers (i.e. 1, 4, 9, 16, …) is called: A. Fourier series B. Fermat’s series C. Euler’s series D. Fibonacci numbers Problem 382 A sequence of numbers where every term is obtained by adding all the preceding terms such as 1, 5, 14, 30… Is called: A. B. C. D. Triangular number Pyramidal number Tetrahedral number Euler’s number Problem 383 (ECE November 1995) The graphical representation of the commulative frequency distribution in a set of statistical data is called: A. Ogive B. Histogram C. Frequency polyhedron D. Mass diagram Problem 384 (ECE March 1996) A sequence of numbers where the succeeding term is greater than the preceding term is called: A. Dissonant series B. Convergent series C. Isometric series D. Divergent series Problem 385 (ECE March 1996) The number 0.123123123…. is A. Irrational B. Surd C. Rational D. Transcendental Problem 386 (ECE November 1996) An array of m × n quantities which represent a single number system composed of elements in rows and columns is know as: A. Transpose of a matrix B. Determinant C. Co-factor of a matrix D. Matrix Problem 387 If equals are added to equals, the sum is equal. A. theorem B. postulate C. axiom D. corollary Problem 388 (ECE November 1996) Terms that differ only in numeric coefficients are nown as: A. unequal terms B. unlike terms C. like terms D. equal terms Problem 389 (ECE November 1996) ______ is a sequence of terms whose reciprocals are in arithmetic progression. A. Geometric progression B. Harmonic progression C. Algebraic Progression D. Ratio and proportion Problem 390 (ECE November 1996) The logarithm of a number to the base e (2.718281828…) is called: A. Naperian logarithm B. Characteristic C. Mantissa D. Briggsian logarithm Problem 391 (ECE November 1996) The ratio or product of two expressions in direct or inverse relation of the other is called: A. Ratio and proportion B. Constant variation C. Means D. Extremes Problem 392 (ECE November 1996) In any square matrix, when the elements of any two rows are the same the determinant is: A. Zero B. Positive integer C. Negative integer D. Unity Problem 393 (ECE November 1996) Two or more equations are equal if and only if they have the same A. Solution set B. Degree C. Order D. Variable set Problem 394 What is the possible outcome of an experiment called? A. a sample space B. a random point C. an event D. a finite set Problem 395 If the roots of an equation are zero, then they are classified as: A. Trivial solutions B. Extraneous roots C. Conditional solutions D. Hypergolic Solutions Problem 396 A complex number associated with a phase-shifted sine wave in polar form whose magnitude is in RMS and angle is equal to the angle of the phaseshifted sine wave is known as: A. Argand’s number B. Imaginary number C. Phasor D. Real number Problem 397 In raw data, the term, which occurs most frequently, is known as: A. B. C. D. Mean Median Mode Quartile Problem 398 Infinity minus infinity is: A. Infinity B. Zero C. Indeterminate D. None of these Problem 399 Any number divided by infinity is equal to: A. I B. Infinity C. Zero D. Indeterminate Problem 400 The term in between any to terms of an arithmetic progression is called: A. Arithmetic mean B. Median C. Middle terms D. Mean Problem 401 Any equation which, because of some mathematical process, has acquired an extra root is sometimes called a: A. Redundant equation B. Literal equation C. Linear equation D. Defective equation Problem 402 A statement that one mathematical expression is greater than or less than another is called: A. inequality B. non-absolute condition C. absolute condition D. conditional expression Problem 403 A relation, in which every ordered pair (x, y) has one and only one value of y that corresponds to the values of x, is called: A. Function B. Range C. Domain D. Coordinates Problem 404 An equation in which a variable appears under the radical sign is called: A. Literal equation B. Radical equation C. Irradical equation D. Irrational equation Problem 405 The number of favorable outcomes divided by the number of possible outcomes: A. Permutations B. Probability C. Combination D. Chance Problem 406 Two factors are considered essentially the same if: A. One is merely the negative of the other B. One is exactly the same of the other C. Both of them are negative D. Both of them are positive Problem 407 An integer is said to be prime if: A. It is factorable by any value B. It is an odd integer C. It has no other integer as factor excepts itself or 1 D. It is an even integer Problem 408 Equations in which the members are equal for all permissible values of integer are called: A. a conditional equation B. an identity C. a parametric equation D. a quadratic equation Problem 409 Equations which satisfy only for some values of unknown are called: A. a conditional equation B. an identity C. a parametric equation D. a quadratic equation Problem 410 (ME April 1996) The logarithm of 1 to any base is: A. indeterminate B. zero C. infinity D. one Part 2 Plane and Spherical Trigonometry Problems- Angles, Trigonometric Identities and Equations Set 10 1. Find the supplement of an angle whose compliment is 62°. A. 28° B. 118° C. 152° D. None of these 2. A certain angle has a supplement 5 times its compliment. Find the angle. A. 67.5° B. 157.5° C. 168.5° D. 186° 3. The sum of the two interior angles of the triangle is equal to the third angle and the difference of the two angles is equal to 2/3 of the third angle. Find the third angle. A. 15° B. 75° C. 90° D. 120° 4. The measure 0f 1 ½ revolutions counterclockwise is: A. 540° B. 520° C. +90° D. -90° 5. The measure of 2.25 revolutions counterclockwise is: A. -835° B. -810° C. 805° D. 810° 6. Solve for Ѳ: sin Ѳ − 𝑠𝑒𝑐 Ѳ + csc Ѳ − tan 20 = −0.0866 A. 40° B. 41° C. 47° D. 43° 7. What are the exact values of the cosine and tangent trigonometric functions of acute angle A, given that sin A = 3/7? 7 A. cos 𝐴 = 2 √10 ; tan 𝐴 = 2√10/3 B. cos 𝐴 = 2√10/7 ; tan 𝐴 = 3√10 /20 7 C. cos 𝐴 = 2√10/3 ; tan 𝐴 = 2 √10 7 D. cos 𝐴 = 2√10/3 ; tan 𝐴 = 2 √10 /20 8. Given three angles A, B, and C whose sum is 180°. If the tan A + tan B + tan C = x, find the value of tan A x tan B x tan C. A. 1 – x B. √𝑥 C. x/2 D. x 9. What is the sine of 820°? A. 0.984 B. -0.866 C. 0.866 D. -0.500 10. csc 270° = ? A. −√3 B. −1 C. √3 D. 1 11. If coversine Ѳ is 0.134, find the value of Ѳ. A. 60° B. 45° C. 30° D. 20° 12. Solve for cos 72° if the given relationship is cos 2A = 2 𝑐𝑜𝑠 2 A – 1. A. 0.309 B. 0.258 C. 0.268 D. 0.315 13. If sin 3A = cos 6B then: A. A + B = 180° B. A + 2B = 30° C. A – 2B = 30° D. A + B = 30° 14. Find the value of sin (arcos 15/17). A. 8/17 B. 17/9 C. 8/21 D. 8/9 15. Find the value of cos [arcsin (1/3) + arctan (2/√5 )] 2 A. (9) (1 + √10) 2 B. (9) (√10 − 11) 2 C. (9) (√10 + 1) 2 D. (9) (√10 − 1) 16. If sin 40° + sin 20° = sin Ѳ, find the value of Ѳ. A. 20° B. 80° C. 120° D. 60° 17. How many different value of x from 0° to 180° for the equation (2sin x – 1)(cos x + 1) = 0? A. 3 B. 0 C. 1 D. 2 18. For what value of Ѳ (less than 2∏) will the following equation be satisfied? 𝑠𝑖𝑛2 Ѳ + 4𝑠𝑖𝑛Ѳ + 3 = 0 A. ∏ B. ∏/4 C. 3∏/2 D. ∏/2 19. Find the value of x in the equation csc x + cot x = 3. A. ∏/4 B. ∏/3 C. ∏/2 D. ∏/5 20. If 𝑠𝑒𝑐 2 𝐴 is 5/2, the quantity 1 − 𝑠𝑖𝑛2 𝐴 is equivalent to: A. 2.5 B. 0.6 C. 1.5 D. 0.4 21. Find sin x if 2 sin x + 3 cos x – 2 = 0. A. 1 & -5/13 B. -1 & 5/13 C. 1 & 5/13 D. -1 & -5/13 22. If sin A = 4/5, A in quadrant II, sin B = 7/25, B in quadrant I, find sin (A + B). A. 3/5 B. 2/5 C. 3/4 D. 4/5 23. If sin A =2.571x, cos A = 3.06, and sin 2A = 3.939x, find the value of x. A. 0.350 B. 0.250 C. 0.100 D. 0.150 24. If cos Ѳ = √3/2, what is the value of x if x = 1 – 𝑡𝑎𝑛2 Ѳ. A. -2 B. -1/3 C. 4/3 D. 2/3 25. If sin Ѳ – cos Ѳ = -1/3, what is the value of sin 2 Ѳ? A. 1/3 B. 1/9 C. 8/9 D. 4/9 26. If x cos Ѳ + y sin Ѳ = 1 and x sin Ѳ – y cos Ѳ = 3, what is the relationship between x and y? A. 𝑥 2 + 𝑦 2 = 20 B. 𝑥 2 − 𝑦 2 = 5 C. 𝑥 2 + 𝑦 2 = 16 D. 𝑥 2 + 𝑦 2 = 10 27. If sin 𝑥 + 1 / sin 𝑥 = √2 , then 𝑠𝑖𝑛2 x + 1 / 𝑠𝑖𝑛2 𝑥 is equal to: A. √2 B. 1 C. 2 D. 0 28. The equation 2 sin Ѳ + 2 cos Ѳ – 1 = √3 is: A. An identity B. A parametric equation C. A conditional equation D. A quadratic equation sin 𝑥 tan 𝑦 29. If x + y = 90°, then sin 𝑦 tan 𝑥 is equal to: A. tan x B. cos x C. cot x D. sin x 30. if cos Ѳ = x / 2 then 1 – 𝑡𝑎𝑛2 Ѳ is equal to: A. (2𝑥 2 + 4) / 𝑥 2 B. (4 − 2𝑥 2 ) / 𝑥 2 C. (2𝑥 2 − 4) / 𝑥 D. (2𝑥 2 − 4) / 𝑥 2 31. Find the value in degrees of arcos (tan 24°). A. 61.48 B. 62.35 C. 63.56 D. 60.84 √3 32. arctan[2 cos( )] 𝑖𝑠 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜: 2 A. ∏/3 B. ∏/4 C. ∏/6 D. ∏/2 33. Solve for x in the equation: arctan (2x) + arctan (x) = ∏/4 A. 0.821 B. 0.218 C. 0.281 D. 0.182 34. Solve for x from the given trigonometric equation: 𝑎𝑟𝑐𝑟𝑡𝑎𝑛(1 − 𝑥 ) + ( arctan 1 + 𝑥 ) = arctan 1/8 A. 4 B. 6 C. 8 D. 2 35. Solve for y if y = (1/sin x – 1/tan x)(1 + cos x) A. sin x B. cos x C. tan x D. sec2 x 36. Solve for x: x = (𝑡𝑎𝑛 Ѳ + 𝑐𝑜𝑡 Ѳ)2 𝑠𝑖𝑛2 Ѳ − 𝑡𝑎𝑛2 Ѳ. A. sin Ѳ B. cos Ѳ C. 1 D. 2 37. Solve for x: 𝑥 = 1 − (𝑠𝑖𝑛Ѳ − 𝑐𝑜𝑠Ѳ)2 . A. sin Ѳ 𝑐𝑜𝑠Ѳ B. -2 cos Ѳ C. cos 2 Ѳ D. sin 2 Ѳ 38. Simplify 𝑐𝑜𝑠 4 Ѳ – 𝑠𝑖𝑛4 Ѳ. A. 2 B. 1 C. 2 𝑠𝑖𝑛2 Ѳ + 1 D. 2 𝑐𝑜𝑠 2 Ѳ − 1 1− 𝑡𝑎𝑛2 𝑎 39. Solve for x: 𝑥 = 1+𝑡𝑎𝑛2 𝑎 A. cos a B. sin 2a C. cos 2a D. sin a 40. which of the following is different from the others? A. 2 cos 2x – 1 B. cos 4x – sin 4x C. cos 3x – sin 3x D. 1 – 2 sin 2x 41. Find the value of y: y = (1 + cos 2 Ѳ) tan Ѳ. A. cos Ѳ B. sin Ѳ C. sin 2 Ѳ D. cos 2 Ѳ 42. The equation 2 sinh x cosh x is equal to: A. 𝑒 𝑥 B. 𝑒 −𝑥 C. sinh 2𝑥 D. Cosh 2x 43. Simplifying the equation 𝑠𝑖𝑛2 Ѳ(1 + 𝑐𝑜𝑡 2 Ѳ) A. 1 B. 𝑠𝑖𝑛2 Ѳ C. 𝑠𝑖𝑛2 Ѳ 𝑠𝑒𝑐 2Ѳ D. 𝑐𝑜𝑠 2 Ѳ 44. If tan Ѳ = 𝑥 2 , which of the following is incorrect? A. 𝑠𝑖𝑛Ѳ = 1 / √1 + 𝑥 4 B. 𝑠𝑒𝑐Ѳ = √1 + 𝑥 4 C. 𝑐𝑜𝑠Ѳ = 1 / √1 + 𝑥 4 D. 𝑐𝑠𝑐Ѳ = √1 + 𝑥 4 / 𝑥 2 45. In an isosceles right triangle, the hypotenuse is how much longer than its sides? A. 2 times B. √2 times C. 1.5 times D. None of these 46. Find the angle in mils subtended by a line 10 yards long at a distance of 5000 yards. A. 2.5 mils B. 2 mils C. 4 mils D. 1 mil 47. The angle or inclination of ascend of a road having 8.25% grade is _____degrees. A. 5.12 degrees B. 4.72 degrees C. 1.86 degrees D. 4.27 degrees 48. The sides of a right triangle is in arithmetic progression whose common difference if 6 cm. its area is: A. 216 𝑐𝑚2 B. 270 𝑐𝑚2 C. 360 𝑐𝑚2 D. 144 𝑐𝑚2 Problems – Triangles, Angles of Elevation & Depression Set 11 49. The hypotenuse of a right triangle is 34 cm. Find the length of the shortest leg if it is 14 cm shorter than the other leg. A. 15 cm B. 16 cm C. 17 cm D. 18 cm 50. A truck travels from point M northward for 30 min. then eastward for one hour, then shifted N 30° W. if the constant speed is 40 Kph, how far directly from M, in km. will be it after 2 hours? A. 43.5 B. 45.2 C. 47.9 D. 41.6 51. Two sides of a triangle measures 6 cm. and 8 cm. and their included angle is 40°. Find the third side. A. 5.144 cm B. 5.263 cm C. 4.256 cm D. 5.645 cm 52. Given a triangle: C = 100°, a = 15, b = 20. Find c: A. 34 B. 27 C. 43 D. 35 53. Given angle A = 32°, angle B = 70°, and side c = 27 units. Solve for side a of the triangle. A. 24 units B. 10 units C. 14.63 units D. 12 units 54. In a triangle, find the side c if the angle C = 100°, side b = 20, and side a = 15. A. 28 B. 27 C. 29 D. 26 55. Two sides of a triangle are 50 m. and 60 m. long. The angle included between these sides is 30 degrees. What is the interior angle (in degrees) opposite the longest side? A. 92.74 B. 93.74 C. 94.74 D. 91.74 56. The sides of a triangle ABC are AB = 15 cm, BC = 18 cm, and CA = 24 cm. Determine the distance from the point of intersection of the angular bisectors to side AB. A. 5.21 cm B. 3.78 cm C. 4.73 cm D. 6.25 cm 57. If AB = 15 m, BC = 18 m and CA = 24 m, find the point of intersection of the angular bisector from the vertex C. A. 11.3 B. 12.1 C. 13.4 D. 14.3 58. In triangle ABC, angle C = 70 degrees; angle A = 45 degrees; AB = 40 m. what is the length of the median drawn from vertex A to side BC? A. 36.8 meters B. 37.1 meters C. 36.3 meters D. 37.4 meters 59. The area of the triangle whose angles are 61°9’32”, 34°14’46”, and 84°35’42” is 680.60. the length of the longest side is: A. 35.53 B. 54.32 C. 52.43 D. 62.54 60. Given a triangle ABC whose angles are A = 40°, B = 95° and side b = 30 cm. find the length of the bisector of angle C. A. 21.74 cm B. 22.35 cm C. 20.45 cm D. 20.98 cm 61. The sides of a triangular lot are 130 m, 180 m, and 190 m. the lot is to be divided by a line bisecting the longest side and drawn from the opposite vertex. The length of this dividing line is: A. 100 meters B. 130 meters C. 125 meters D. 115 meters 62. From a point outside of an equilateral triangle, the distances to the vertices are 10m, 10m, and 18m. Find the dimension of the triangle. A. 25.63 B. 45.68 C. 19.94 D. 12.25 63. Points A and B 1000m apart are plotted on a straight highway running East and West. From A, the bearing of a tower C is 32 degrees N of W and from B the bearing of C is 26 degrees N of E. Approximate the shortest distance of tower C to the highway. A. 264 meters B. 274 meters C. 284 meters D. 294 meters 64. An airplane leaves an aircraft carrier and flies South at 350 mph. The carrier travels S 30° E at 25 mph. If the wireless communication range of the airplane is 700 miles, when will it lose contact with the carrier? A. after 4.36 hours B. after 5.57 hours C. after 2.13 hours D. after 4.54 hours 65. A statue 2 meters high stands on a column that is 3 meters high. An observer in level with the top of the statue observed that the column and the statue subtend the same angle. How far is the observer from the statue? A. 5√2 𝑚𝑒𝑡𝑒𝑟𝑠 B. 2√5 𝑚𝑒𝑡𝑒𝑟𝑠 C. 20 meters D. √10 𝑚𝑒𝑡𝑒𝑟𝑠 66. From the top of a building 100 m high, the angle of depression of a point A due East of it is 30°. From a poit B due South of the building, the angle of elevation of the top is 60°. Find the distance AB. A. 100 + 3√30 B. 200 - √30 C. 100√30 / 3 D. 100√3 / 30 67. An observer found the angle of elevation of the top of the tree to be 27°. After moving 10m closer (on the same vertical and horizontal plane as the tree), the angle of elevation becomes 54°. Find the height of the tree. A. 8.65 meters B. 7.53 meters C. 7.02 meters D. 8.09 meters 68. From a point A at the foot of the mountain, the angle of elevation of the top B is 60°. After ascending the mountain one (1) mile to an inclination of 30° to the horizon, and reaching a point C, an observer finds that the angle ACB is 135°. A. 14386 B. 12493 C. 11672 D. 11223 69. A vertical pole is 10 m from a building. When the angle of elevation of the sum is 45°, te pole cast a shadow on the building 1 m high. Find the height of the pole. A. 0 meter B. 11 meters C. 12 meters D. 13 meters 70. A pole cast a shadow of 15 meters long when the angle of elevation of the sun is 61°. If the pole has leaned 15° from the vertical directly toward the sun, what is the length of the pole? A. 52.43 meters B. 54.23 meters C. 53.25 meters D. 53.24 meters 71. An observer wishes to determine the height of a tower. He takes sights at the top of the tower from A and B, which are 50 ft. apart, at the same elevation on a direct line with the tower. The vertical angle at point A is 30° and at point B is 40°. What is the height of the tower? A. 85.6 feet B. 143.97 feet C. 110.29 feet D. 92.54 feet 72. From the top of tower A, the angle of elevation of the top of the tower B is 46°. From the foot of tower B the angle of elevation of the top of tower A is 28°. Both towers are on a level ground. If the height of tower B is 120m, how high is tower A in m? A. 38.6 B. 42.3 C. 44.1 D. 40.7 73. Points A and B are 100 m apart and are on the same elevation as the foot of a building. The angles of elevation of the top of the building from points A and B are 21° and 32°, respectively. How far is A from the building in m? A. 271.6 B. 265.4 C. 259.2 D. 277.9 74. A man finds the angle of elevation of the top of a tower to be 30 degrees. He walks 85 m. nearer the tower and finds its angle of elevation to be 60 degrees. What is the height of the tower? A. 76.31 meters B. 73.61 meters C. 73.31 meters D. 73.16 meters 75. The angle of elevation of a point C from a pint B is 29°42’; the angle of elevation of C from another point A 31.2 m directly below B is 59°23’. How high is C from the horizontal line through A? A. 47.1 meters B. 52.3 meters C. 35.1 meters D. 66.9 meters 76. A rectangular piece of land 40m x 30m is to be crossed diagonally by a 10-m wide roadway. If the land cost P1,500.00 per square meter, the cost of the roadway is: A. P401.10 B. P60,165.00 C. P601,650.00 D. 651,500.00 77. A man improvises a temporary shield from the sun using a triangular piece of wood with dimensions of 1.4m, 1.5 m, and 1.3 m. with the longer side lying horizontally on the ground, he props up the other corner of the triangle with a vertical pole 0.9m long. What would be the area of the shadow on the ground when the sun is vertically overhead? A. 0.5 𝑚2 B. 0.75 𝑚2 C. 0.84 𝑚2 D. 0.95 𝑚2 78. A rectangular piece of wood 4cm x 12cm tall is titled at an angle of 45°. Find the vertical distance between the lower corner and the upper corner. A. 4√2 𝑐𝑚 B. 2√2 𝑐𝑚 C. 8√2 𝑐𝑚 D. 6√2 𝑐𝑚 79. A clock has a dial face 12 inches in radius. The minute hand is 9 inches long while the hour hand is 6 inches long. The plane of rotation of the minute hand is 2 inches above the plane of rotation of the hour hand. Find the distance between the tips of the hands at 5:40 AM. A. 9.17 inches B. 8.23 inches C. 10.65 inches D. 11.25 inches 80. If the bearing of A from B is 40° W, then the bearing of B from A is: A. N 40° E B. N 40° W C. N 50° E D. N 50° W 81. A plane hillside is inclined at an angle of 28° with the horizontal. A man wearing skis can climb this hillside by following a straight path inclined at an angle of 12° to the horizontal, but one without skis must follow a path inclined at an angle of only 5° with the horizontal. Find the angle between the directions of the two paths. A. 13.21° B. 18.74° C. 15.56° D. 17.22° 82. Calculate the area of a spherical triangle whose radius is 5 m and whose angles are 40°, 65°, and 110°. A. 12.34 sq. m. B. 14.89 sq. m. C. 16.45 sq. m. D. 15.27 sq. m. 83. A right spherical triangle has an angle C = 90°, a = 50°, and c = 80°. Find the side b. A. 45.33° B. 78.66° C. 74.33° D. 75.89° 84. If the time is 8:00 a.m. GMT, what is the time in the Philippines, which is located at 120° East longitude? A. 6 p.m. B. 4 am C. 4 p.m. D. 6 am 85. An airplane flew from Manila (14° 36’N, 121° 05’E) at a course of S 30° E maintaining a certain altitude and following a great circle path. If its groundspeed is 350 knots, after how many hours will it cross the equator? A. 2.87 hours B. 2.27 hours C. 3.17 hours D. 3.97 hours 86. Find the distance in nautical miles between Manila and San Francisco. Manila is located at 14° 36’N latitude and 121° 05’ E longitude. San Francisco is situated at 37° 48’ N latitude and 122° 24’ W longitude. A. 7856.2 nautical miles B. 5896.2 nautical miles C. 6326.2 nautical miles D. 6046.2 nautical miles Part 3 Plane Geometry Problems – Triangles, Quadrilaterals, Polygons Set 12 87. The sides of a right triangle have lengths (a – b), a, and (a + b). What is the ratio of a to b if a is greater than b and b could not be equal to zero? A. 1 : 4 B. 3 : 1 C. 1 : 4 D. 4 : 1 88. Two sides of a triangle measure 8 cm and 12 cm. find its area if its perimeter is 26 cm. A. 21.33 sq. m. B. 32.56 sq. cm. C. 3.306 sq. in. D. 32.56 sq. in. 89. If three sides of a triangle of an acute triangle is 3 cm, 4 cm, and “x” cm, what are the possible values of x? A. 1 < x < 5 B. 0 < x > 5 C. 0 < x < 7 D. 1 < x > 7 90. In triangle ABC, AB = 8m and BC = 20m. one possible dimension of CA is: A. 13 B. 7 C. 9 D. 11 91. In a triangle BCD, BC = 25 m. and CD = 10 m. The perimeter of the triangle may be. A. 72 m. B. 70 m. C. 69 m. D. 71 m. 92. The sides of a triangle ABC are AB = 25 cm, BC = 39 cm, and AC = 40 cm. Find its area. A. 486 sq. cm. B. 846 sq. cm. C. 648 sq. cm. D. 468 sq. cm. 93. The corresponding sides of two similar triangles are in the ratio 3:2. What is the ratio of their areas? A. 3 B. 2 C. 9/4 D. 3/2 94. Find the area of the triangle whose sides are 12, 16, and 21 units. A. 95.45 sq. units B. 102.36 sq. units C. 87.45 sq. units D. 82.78 sq. units 95. The sides of a right triangle are 8, 15 and 17 units. If each side is doubled, how many square units will be the area of the new triangle? A. 240 B. 300 C. 320 D. 420 96. Two triangles have equal bases. The altitude of one triangle is 3 units more than its base and the altitude of the other is 3 units less than its base. Find the altitudes, if the areas of the triangle differ by 21 square units. A. 5 & 11 B. 4 & 10 C. 6 & 12 D. 3 & 9 97. A triangular piece of wood having a dimension 130 cm, 180 cm, and 190 cm is to be divided by a line bisecting the longest side drawn from its opposite vertex. The area of the part adjacent to the 180-cm side is: A. 5126 sq. cm. B. 5162 sq. cm. C. 5612 sq. cm. D. 5216 sq. cm. 98. Find EB if the area of the inner triangle is ¼ of the outer triangle. A. 32.5 B. 55.7 C. 56.2 D. 57.5 99. A piece of wire is shaped to enclose a square whose area is 169 cm2. It is then reshaped to enclose a rectangle whose length is 15 cm. The area of the rectangle is: A. 165 cm2 B. 175 cm2 C. 170 cm2 D. 156 cm2 100. The diagonal of the floor of a rectangular room is 7.50 m. The shorter side of the room is 4.5 m. What is the area of the room? A. 36 sq. m. B. 27 sq. m. C. 58 sq. m. D. 24 sq. m. 101. A man measuring a rectangle “x” meters by “y” meters, makes each side 15% too small. By how many percent will his estimate for the area be too small? A. 23.55% B. 25.67% C. 27.75% D. 72.25% 102. The length of the side of a square is increased by 100%. Its perimeter is increased by: A. 25% B. 100% C. 200% D. 300% 103. A piece of wire of length 52 cm is cut into two parts. Each part is then bent to form a square. It is found that total area of the two squares is 97 sq. cm. the dimension of the bigger square is: A. 4 B. 9 C. 3 D. 6 104. In the figure shown, ABCD is a square and PDC is an equilateral triangle. Find Ѳ. A. 5° B. 15° C. 10° D. 25° 105. One side of a parallelogram is 10 m and its diagonals are 16 m and 24 m, respectively. Its area is: A. 156.8 sq. m. B. 185.6 sq. m. C. 158.7 sq. m. D. 142.3 sq. m. 106. If the sides of the parallelogram and an included angle are 6, 10 and 100 degrees respectively, find the length of the shorter diagonal. A. 10.63 B. 10.37 C. 10.73 D. 10.23 107. The area of a rhombus is 132 square cm. if its shorter diagonal is 12 cm, the length of the longer diagonal is: A. 20 centimeter B. 21 centimeter C. 22 centimeter D. 23 centimeter 108. The diagonals of a rhombus are 10 cm. and 8 cm., respectively. Its area is: A. 10 sq. cm. B. 50 sq. cm. C. 60 sq. cm. D. 40 sq. cm. 109. Given a cyclic quadrilateral whose sides are 4 cm, 5 cm, 8 cm, and 11 cm. Its area is: A. 40.25 sq. cm. B. 48.65 sq. cm. C. 50.25 sq. cm. D. 60.25 sq. cm 110. A rectangle ABCD which measure 18 by 24 cm is folded once, perpendicular to diagonal AC, so that the opposite vertices A and C coincide. Find the length of the fold. A. 2 B. 7/2 C. 54/2 D. 45/2 111. The sides of a quadrilateral are 10m, 8m, 16m and 20m, respectively. Two opposite interior angles have a sum of 225°. Find the area of the quadrilateral in sq. m. A. 140.33 sq. cm. B. 145.33 sq. cm. C. 150.33 sq. cm. D. 155.33 sq. cm. 112. A trapezoid has an area of 36 m2 and altitude of 2 m. Its two bases in meters have ratio of 4:5, the bases are: A. 12, 15 B. 7, 11 C. 16, 20 D. 8, 10 113. Determine the area of the quadrilateral ABCD shown if OB = 80 cm, OA = 120 cm, OD = 150 cm and Ѳ = 25°. A. B. C. D. 2272 sq. cm 7222 sq. cm 2572 sq. cm 2722 sq. cm 114. A corner lot of land is 35 m on one street and 25 m on the other street. The angle between the two lines of the street being 82°. The other to two lines of the lot are respectively perpendicular to the lines of the streets. What is the worth of the lot if its unit price is P2500 per square meter? A. P1,978,456 B. P1,588,045 C. P2,234,023 D. P1,884,050 115. Determine the area of the quadrilateral having (8, -2), (5, 6), (4, 1), and (-7, 4) as consecutive vertices. A. 22 sq. units B. 44 sq. units C. 32 sq. units D. 48 sq. units 116. Find the area of the shaded portion shown if AB is parallel to CD. A. 16 sq. m. B. 18 sq. m. C. 20 sq. m. D. 22 sq. m. 117. The deflection angles of any polygon has a sum of: A. 360° B. 720° C. 180°(n – 3) D. 180° n 118. The sum of the interior angles of a dodecagon is: A. 2160° B. 1980° C. 1800° D. 2520° 119. Each interior angle of a regular polygon is 165°. How many sides? A. 23 B. 24 C. 25 D. 26 120. The sum of the interior angles of a polygon is 540°. Find the number of sides. A. 4 B. 6 C. 7 D. 5 121. The sum of the interior angles of a polygon of n sides is 1080°. Find the value of n. A. 5 B. 6 C. 7 D. 8 122. How many diagonals does a pentedecagon have: A. 60 B. 70 C. 80 D. 90 123. A polygon has 170 diagonals. How many sides does it have? A. 20 B. 18 C. 25 D. 26 124. A regular hexagon with an area of 93.53 square centimeters is inscribed in a circle. The area in the circle not covered by hexagon is: A. 18.38 cm2 B. 16.72 cm2 C. 19.57 cm2 D. 15.68 cm2 125. The area of a regular decagon inscribed in a circle of 15 cm diameter is: A. 156 sq. cm. B. 158 sq. cm. C. 165 sq. cm. D. 185 sq. cm. 126. The sum of the interior angle of polygon is 2,520°. How many are the sides? A. 14 B. 15 C. 16 D. 17 127. The area of a regular hexagon inscribed in a circle of radius 1 is: A. 2.698 sq. units B. 2.598 sq. units C. 3.698 sq. units D. 3.598 sq. units 128. The corners of a 2-meter square are cut off to form a regular octagon. What is the length of the sides of the resulting octagon? A. 0.525 B. 0.626 C. 0.727 D. 0.828 129. If a regular polygon has 27 diagonals, then it is a: A. Hexagon B. Nonagon C. Pentagon D. Heptagon 130. One side of a regular octagon is 2. Find the area of the region inside the octagon. A. 19.3 sq. units B. 13.9 sq. units C. 21.4 sq. units D. 31 sq. units 131. A regular octagon is inscribed in a circle of radius 10. Find the area of the octagon. A. 228.2 sq. units B. 288.2 sq. units C. 282.8 sq. units D. 238.2 sq. units Problems – Circles, Miscellaneous Applications Set 13 132. The area of a circle is 89.4 square inches. What is the circumference? A. 35.33 inches B. 32.25 inches C. 33.52 inches D. 35.55 inches 133. A circle whose area is 452 cm square is cut into two segment by a chord whose distance from the center of the circle is 6 cm. Find the area of the larger segment in cm square. A. 372.5 B. 363.6 C. 368.4 D. 377.6 134. A circle is divided into two parts by a chord, 3 cm away from the center. Find the area of the smaller part, in cm square, if the circles has an area of 201 cm square. A. 51.4 B. 57.8 C. 55.2 D. 53.7 135. A quadrilateral ABCD is inscribed in a semi-circle with side AD coinciding with the diameter of the circle. If sides AB, BC, and CD are 8cm, 10cm, and 12cm long, respectively, find the area of the circle. A. 317 sq. cm. B. 356 sq. cm. C. 456 sq. cm. D. 486 sq. cm. 136. A semi-circle of radius 14 cm is formed from a piece of wire. If it is bent into a rectangle whose length is 1cm more than its width, find the area of the rectangle. A. 256.25 sq. cm. B. 323.57 sq. cm. C. 386.54 sq. cm. D. 452.24 sq. cm 137. The angle of a sector is 30 degrees and the radius is 15 cm. What is the area of the sector? A. 89.5 cm2 B. 58.9 cm2 C. 59.8 cm2 D. 85.9 cm2 138. A sector has a radius of 12 cm. if the length of its arc is 12 cm, its area is: A. 66 sq. cm. B. 82 sq. cm. C. 144 sq. cm. D. 72 sq. cm. 139. The perimeter of a sector is 9 cm and its radius is 3 cm. What is the area of the sector? A. 4 cm2 B. 9/2 cm2 C. 11/2 cm2 D. 27/2 cm2 140. A swimming pool is to be constructed in the space of partially overlapping identical circles. Each of the circles has a radius of 9 m, and each passes through the center of the other. Find the area of the swimming pool. A. 302.33 m2 B. 362.55 m2 C. 398.99 m2 D. 409.44 m2 141. Given are two concentric circles with the outer circle having a radius of 10 cm. If the area of the inner circle is half of the outer circle, find the boarder between the two circles. A. 2.930 cm B. 2.856 cm C. 3.265 cm D. 2.444 cm 142. A circle of radius 5 cm has a chord which is 6cm long. Find the area of the circle concentric to this circle and tangent to the given chord. A. 14 𝜋 B. 16 𝜋 C. 9 𝜋 D. 4 𝜋 143. A reversed curve on a railroad track consists of two circular arcs. The central angle of one side is 20° with radius 2500 feet, and the central angle of the other is 25° with radius 3000 feet. Find the total lengths of t he two arcs. A. 2812 ft. B. 2218 ft. C. 2821 ft. D. 2182 ft. 144. Given a triangle whose sides are 24 cm, 30 cm, and 36 cm. find the radius of a circle which is tangent to the shortest and longest side of the triangle, and whose center lies on the third side. A. 9.111 cm B. 11.91 cm C. 12.31 cm D. 18 cm 145. Find the area of the largest circle that can be cut from a triangle whose sides are 10 cm, 18 cm, and 20 m. A. 11 𝜋cm2 B. 12 𝜋cm2 C. 14 𝜋cm2 D. 15 𝜋cm2 146. The diameter of the circle circumscribed about a triangle ABC with sides a, b , c is equal to: A. a/sin A B. b/sin B C. c/sin C D. all of the above 147. The sides of a triangle are 14 cm., 15 cm., and 13 cm. find the area of the circumscribing circle. A. 207.4 sq. cm. B. 209.6 sq. cm. C. 215.4 sq. cm. D. 220.5 sq. cm. 148. What is the radius of the circle circumscribing an isosceles right triangle having an area of 162 sq. cm? A. 13.52 B. 14.18 C. 12.73 D. 1564 149. If the radius of the circle is decreased by 20%, by how much is its area decreased? A. 36% B. 26% C. 46% D. 56% 150. The distance between the center of the three circles which are mutually tangent to each other externally are 10, 12 and 14 units. The area of the of the largest circle is: A. 72 𝜋 B. 23 𝜋 C. 64 𝜋 D. 16 𝜋 151. The sides of a cyclic quadrilateral measures 8 cm, 9 cm, 12 cm, and 7 cm, respectively. Find the area of the circumscribing circle. A. 8.65 cm2 B. 186.23 cm2 C. 6.54 cm2 D. 134.37 cm 152. The wheel of a car revolves n times, while the car travels x km. the radius of the wheel in meter is: A. 10,000 x/ (𝜋 n) B. 500 x/ (𝜋 n) C. 500,00 x/ (𝜋 n) D. 5,000 x/ (𝜋 n) 153. If the inside wheels of a car running a circular track are going half as fast as the outside wheel, determine the length of the track, described by the outer wheels, if the wheels are 1.5 m apart. A. 4 𝜋 B. 5 𝜋 C. 6 𝜋 D. 8 𝜋 154. A goat is tied to a corner of a 30 ft by 35 ft building. If the rope is 40 ft long and the goat can reach 1 ft farther than the rope length, what is the maximum area the goat can cover? A. 5281 ft 2 B. 4084 ft 2 C. 3961 ft 2 D. 3970 ft 2 155. The interior angles of a triangle measures 2x, x + 15, and 2x + 15. What is the value of x? A. 30° B. 66° C. 42° D. 54° 156. Two complementary angles are in the ratio 2:1. Find the larger angle. A. 30° B. 60° C. 75° D. 15° 157. Two transmission towers 40 feet high is 200 feet apart. If the lowest point of the cable is 10 feet above the ground, the vertical distance from the roadway to the cable 50 feet from the center is: A. 17.25 feet B. 17.5 feet C. 17.75 feet D. 18 feet 158. What is the area bounded by the curves 𝑦 2 = 4𝑥 and 𝑥 2 = 4𝑦? A. 6.0 B. 7.333 C. 6.666 D. 5.333 159. What is the area between y = 0, y = 3𝑥 2 , x = 0, and x = 2? A. 8 B. 12 C. 24 D. 6 Part 4 Solid Geometry Problems – Prisms, Pyramids, Cylinders, Cones Set 14 160. If the edge of a cube is doubled, which of the following is incorrect? A. The lateral area will be quadrupled B. The volume is increased 8 times C. The diagonal is doubled D. The weight is doubled 161. The volume of a cube is reduced by how much if all sides are halved? A. 1/8 B. 5/8 C. 6/8 D. 7/8 162. Each side of a cube is increased by 1%. By what percent is the volume of the cube increased? A. 23.4% B. 33.1% C. 3% D. 34.56% 163. If the edge of a cube is increased by 30%, by how much is the surface area increased? A. 67 B. 69 C. 63 D. 65 164. Find the approximate change in the volume of a cube of side x inches caused by increasing its side by 1%. A. 0.3x3 cu. in. B. 0.1x3 cu. in. C. 0.02 cu. in. D. 0.03x3 cu. in. 165. A rectangular bin 4 feet long, 3 feet wide, and 2 feet high is solidly packed with bricks whose dimensions are 8 in. by 4 in. by 2 in. The number of bricks in the bin is: A. 68 B. 386 C. 648 D. 956 166. Find the total surface area of a cube of side 6 cm. A. 214 sq. cm. B. 216 sq. cm. C. 226 sq. cm. D. 236 sq. cm. 167. The space diagonal of a cube is 4√3 m. Find its volume. A. 16 cubic meters B. 48 cubic meters C. 64 cubic meters D. 86 cubic meters 168. A reservoir is shaped like a square prism. If the area of its base is 225 sq. cm, how many liters of water will it hold? A. 3.375 B. 3375 C. 33.75 D. 3375 169. Find the angle formed by the intersection of a face diagonal t the diagonal of a cube drawn from the same vertex. A. 35.26° B. 32.56° C. 33.69° D. 42.23° 170. The space diagonal of a cube (the diagonal joining two non-coplanar vertices) is 6 m. The total surface area of the cube is: A. 60 B. 66 C. 72 D. 78 171. The base edge of a regular hexagonal prism is 6 cm and its bases are 12 cm apart. Find its volume in cu. cm. A. 1563.45 cm3 B. 1058.45 cm3 C. 1896.37 cm3 D. 1122.37 cm3 172. The base edge of a regular pentagonal prism is 6 cm and its bases are 12 cm apart. Find its volume in cu. cm. A. 743.22 cm3 B. 786.89 cm3 C. 567.45 cm3 D. 842.12 cm3 173. The base of a right prism is a hexagon with one side 6 cm long. If the volume of the prism is 450 cc, how far apart are the bases? A. 5.74 cm B. 3.56 cm C. 4.11 cm D. 4.81 cm 174. A trough has an open top 0.30 m by 6 m and closed vertical ends which are equilateral triangles 30 cm on each side. It is filled with water to half its depth. Find the volume of the water in cubic meters. A. 0.058 B. 0.046 C. 0.037 D. 0.065 175. Determine the volume of a right truncated prism with the following dimensions: Let the corner of the triangular base be defined by A, B, and C. the length AB = 10 feet, BC = 9 feet and CA = 12 feet. The sides at A, B and C are perpendicular to the triangular base and have the height of 8.6 feet, 7.1 feet, and 5.5 feet, respectively. A. 413 ft3 B. 311 ft3 C. 313 ft3 D. 391 ft3 176. The volume of a regular tetrahedron of side 5 cm is: A. 13.72 cu. cm B. 14.73 cu.cm C. 15.63 cu. cm D. 17.82 cu. cm 177. A regular hexagonal pyramid whose base perimeter is 60 cm has an altitude of 30 cm, the volume of the pyramid is: A. 2958 cu. cm. B. 2598 cu. cm. C. 2859 cu. cm. D. 2589 cu. cm. 178. A frustum of a pyramid has an upper base 100 m by 10 m and a lower base of 80 m by 8 m. if the altitude of the frustum is 5 m, find its volume. A. 4567.67 cu. m. B. 3873.33 cu. m. C. 4066.67 cu. m. D. 2345.98 cu. m. 179. The altitude of the frustum of a regular rectangular pyramid is 5m the volume is 140 cu. m. and the upper base is 3m by 4m. What are the dimensions of the lower base in m? A. 9 x 10 B. 6 x 8 C. 4.5 x 6 D. 7.50 x 10 180. The frustum of a regular triangular pyramid has equilateral triangles for its bases. The lower and upper base edges are 9m and 3m, respectively. If the volume is 118.2 cu. m.., how far apart are the base? A. 9m B. 8m C. 7m D. 10m 181. A cylindrical gasoline tank, lying horizontally, 0.90 m. in diameter and 3 m long is filled to a depth of 0.60 m. How many gallons of gasoline does it contain? Hint: One cubic meter = 265 gallons A. 250 B. 360 C. 300 D. 270 182. A closed cylindrical tank is 8 feet long and 3 feet in diameter. When lying in a horizontal position, the water is 2 feet deep. If the tank is the vertical position, the depth of water in the tank is: A. 5.67 m B. 5.82 m C. 5.82 ft D. 5.67 ft 183. A circular cylinder is circumscribed about a right prism having a square base one meter on an edge. The volume of the cylinder is 6.283 cu. m. find its altitude in m. meter on an edge. The volume of the cylinder is 6.283 cu. m. Find its altitude in m. A. 5 B. 4.5 C. 69.08 D. 4 184. If 23 cubic meters of water are poured into a conical vessel, it reaches a depth of 12 cm. how much water must be added so that the length reaches 18 cm.? A. 95 cubic meters B. 100 cubic meters C. 54.6 cubic meters D. 76.4 cubic meters 185. The height of a right circular base down is h. If it contains water to depth of 2h/3 the ratio of the volume of water to that of the cone is: A. 1:27 B. 2:3 C. 8:27 D. 26:27 186. A right circular cone with an altitude of 9m is divided into two segments; one is a smaller circular cone having the same vertex with an altitude of 6m. Find the ratio of the volume of the two cones. A. 19:27 B. 2:3 C. 1:3 D. 8:27 187. A conical vessel has a height of 24 cm. and a base diameter of 12 cm. It holds water to a depth of 18 cm. above its vertex. Find the volume of its content in cc. A. 387.4 B. 381.7 C. 383.5 D. 385.2 188. A right circular cone with an altitude of 8 cm is divided into two segments. One is a smaller circular cone having the same vertex with the volume equal to ¼ of the original cone. Find the altitude of the smaller cone. A. 4.52 cm B. 6.74 cm C. 5.04 cm D. 6.12 cm 189. The slant height of a right circular cone is 5m long. The base diameter is 6m. What is the lateral area in sq. m? A. 37.7 B. 47 C. 44 D. 40.8 190. A right circular cone has a volume of 128 𝜋/3 cm3 and an altitude of 8 cm. The lateral area is: A. 16√5 𝜋 𝑐𝑚2 B. 12 √5 𝜋 𝑐𝑚2 C. 16 𝜋 𝑐𝑚2 D. 15 𝜋 𝑐𝑚2 191. The volume of a right circular cone is 36 𝜋. If its altitude is 3, find its radius. A. 3 B. 4 C. 5 D. 6 192. A cone and hemisphere share base that is a semicircle with radius 3 and the cone is inscribed inside the hemisphere. Find the volume of the region outside the cone and inside the hemisphere. A. 24.874 B. 27.284 C. 28.274 D. 28.724 193. A cone was formed by rolling a thin sheet of metal in the form of a sector of a circle 72 cm in diameter with a central angle of 210°. What is the volume of the cone in cc? A. 13,602 B. 13,504 C. 13,716 D. 13,318 194. A cone was formed by rolling a thin sheet of metal in the form of a sector of a circle 72 cm in diameter with a central angle of 150°. Find the volume of the cone in cc. A. 7733 B. 7722 C. 7744 D. 7711 195. A chemist’s measuring glass is conical in shape. If it is 8 cm deep and 3 cm across the mouth, find the distance on the slant edge between the markings for 1 cc and 2 cc. A. 0.82 cm B. 0.79 cm C. 0.74 cm D. 0.92 cm 196. The base areas of a frustum of a cone are 25 sq. cm. and 16 sq. cm, respectively. If its altitude is 6 cm, find its volume. A. 120 cm3 B. 122 cm3 C. 129 cm3 D. 133 cm3 Problems – Spheres, Prismatoid, Solids of Revolutions, Miscellaneous Applications Set 15 197. What is the surface area of a sphere whose volume is 36 cu. m? A. 52.7 m2 B. 48.7 m2 C. 46.6 m2 D. 54.6 m2 198. If the surface area of a sphere is increased by 21%, its volume is increased by: A. 13.31% B. 33.1% C. 21% D. 30% 199. The surface area of the sphere is 4 𝜋𝑟2. Find the percentage increase in its diameter when the surface area increases by 21%. A. 5% B. 10% C. 15% D. 20% 200. Find the percentage increase in volume of a sphere if its surface area is increased by 21% A. 30.2% B. 33.1% C. 34.5% D. 30.9% 201. The volume of a sphere is increased by how much if its surface area is increased by 20%? A. 32.6% B. 33% C. 44% D. 72.8% 202. Given two spheres whose combined volume is known to be 819 cu. m. if their radii are in the ratio 3:4, what is the volume of the smaller sphere? A. 576 cu. m. B. 243 cu. m. C. 343 cu. m. D. 476 cu. m. 203. How much will the surface area of a sphere be increased if its radius is increased by 5%? A. 25% B. 15.5% C. 12.5% D. 10.25% 204. The volume of a sphere is 904.78 cu. m. Find the volume of the spherical segment of height 4m. A. 234.57 cu. m. B. 256.58 cu. m. C. 145.69 cu. m. D. 124.58 cu. m. 205. A sphere of radius r just fits into a cylindrical container of radius r and altitude 2r. Find the empty space in the cylinder. A. (8/9) 𝜋𝑟3 B. (20/27) 𝜋𝑟3 C. (4/5) 𝜋𝑟3 D. (2/3) 𝜋𝑟3 206. If a solid steel ball is immersed in an eight cm. diameter cylinder, it displaces water to a depth of 2.25 cm. the radius of the ball is: A. 3 cm B. 6 cm C. 9 cm D. 12 cm 207. The diameter of two spheres is in the ratio 2:3. If the sum of their volumes is 1,260 cu. m., the volume of the larger sphere is: A. 972 cu. m. B. 927 cu. m. C. 856 cu. m. D. 865 cu. m. 208. A hemispherical bowl of radius 10 cm is filled with water to such a depth that the water surface area is equal to 75 𝜋 𝑠𝑞. 𝑐𝑚. The volume of water is: A. 625/3 𝑐𝑚3 B. 625𝜋/3 𝑐𝑚3 C. 625𝜋/2 𝑐𝑚3 D. 625𝜋 𝑐𝑚3 209. A water tank is in the form of a spherical segment whose base radii are 4m and 3m and whose altitude is 6m. The capacity of the tank in gallon is: A. 91,011 B. 92,011 C. 95,011 D. 348.72 210. Find the volume of a spherical sector of altitude 3 cm. and radius 5 cm. A. 75𝜋 cu. cm. B. 100𝜋 cu. cm. C. 50𝜋 cu. cm. D. 25𝜋 cu. cm. 211. How far from the center of a sphere of a radius 10 cm should a plane be passed so that the ratio of the areas of two zones is 3:7? A. 3 cm B. 4 cm C. 5 cm D. 6 cm 212. A 2-m diameter spherical tank contains1396 liter of water. How many liters of water must be added for the water to reach a depth of 1.75 m? A. 2613 B. 2723 C. 2542 D. 2472 213. Find the volume of a spherical segment of radius 10 m and the altitude 5 m. A. 654.5 cu. m. B. 659.8 cu. m. C. 675.2 cu. m. D. 680.5 cu. m. 214. Find the volume of a spherical wedge of radius 10 cm. and central angle 50°. A. 425.66 sq. m. B. 431.25 sq. m. C. 581.78 sq. m. D. 444.56 sq. m. 215. Determine the area of the zone of a sphere of radius 8 in. and altitude 12 in. A. 192𝜋 𝑠𝑞. 𝑖𝑛. B. 198𝜋 𝑠𝑞. 𝑖𝑛. C. 185𝜋 𝑠𝑞. 𝑖𝑛. D. 195𝜋 𝑠𝑞. 𝑖𝑛. 216. The corners of a cubical block touch the closed spherical shell that encloses it. The volume of the box is 2744 cc. What volume in cc, inside the shell is not occupied by the block? A. 1356 cm3 B. 4721 cm3 C. 3423 cm3 D. 7623 cm3 217. A cubical container that measures 2 inches on each side is tightly packed with 8 marbles and is filled with water. All 8 marbles are in contact with the walls of the container and the adjacent marbles. All of the marbles are of the same size. What is the volume of water in the container? A. 0.38 cu. in. B. 2.5 cu. in. C. 3.8 cu. in. D. 4.2 cu. in. 218. The volume of the water is a spherical tank is 1470.265 cm3. Determine the depth of water if the tank has a diameter of 30 cm. A. 8 B. 6 C. 4 D. 10 219. The volume of water in a spherical tank having a diameter of 4 m. is 5.236 m3. Determine the depth of the water on the tank. A. 1.0 B. 1.4 C. 1.2 D. 1.6 220. A mixture compound from equal parts of two liquids, one white and the other black was placed in a hemispherical bowl. The total depth of the two liquids is 6”. After standing for a short time the mixture separated the white liquid settling below the black. If the thickness of the segment of the black liquid is 2”, find the radius of the bowl in inches. A. 7.53 B. 7.33 C. 7.73 D. 7.93 221. 20.5 cubic meters of water is inside a spherical tank whose radius is 2m. find the height of the water surface above the bottom of the tank, in m. A. 2.7 B. 2.5 C. 2.3 D. 2.1 222. The volume of the sphere is 3𝜋 cu. m. The surface area of this sphere in sq. m. is: A. 36 𝜋 B. 24 𝜋 C. 18 𝜋 D. 12 𝜋 223. Spherical balls 1.5 cm in diameter area packed in a box measuring 6 cm by 3 cm by 3 cm. If as many balls as possible are packed in the box, how much free space remains in the box? A. 28.41 cc B. 20.47 cc C. 29.87 cc D. 25.73 cc 224. A solid has a circular base of radius r. find the volume of the solid if every plane perpendicular to a given diameter is a square. A. 16 r3/3 B. 5 r3 C. 6 r3 D. 19 r3/3 225. A solid has circular base of diameter 20 cm. Find the volume of the solid if every cutting plane perpendicular to the base along a given diameter is an equilateral triangle. A. 2514 cc B. 2107 cc C. 2309 cc D. 2847 cc 226. The base of a certain solid is a triangle of base b and altitude h. if all sections perpendicular to the altitude of the triangle are regular hexagons, find the volume of the solid. 1 A. 2 √3 𝑏2 ℎ B. 2√3 𝑏2 ℎ C. √3 𝑏2 ℎ/3 D. √3 𝑏2 ℎ 227. The volume generated by the circle by the circle 𝑥 2 + 𝑦 2 + 4𝑥 − 6𝑦 − 12 = 0 revolved about the line 2x – 3y – 12 = 0 is: A. 3242 cubic units B. 3342 cubic units C. 3452 cubic units D. 3422 cubic units 228. The volume generated by rotating the curve 9𝑥 2 + 4𝑦 2 = 36 about the line 4x + 3y = 20 is: A. 48 𝜋 B. 58 𝜋2 C. 42 𝜋 D. 48 𝜋2 229. Find the volume generated by revolving 𝑦2 𝑥2 the area bounded by the ellipse + = 1 9 4 about the line x = 3. A. 347.23 cu. units B. 355.31 cu. units C. 378.43 cu. units D. 389.51 cu. units 230. The area in the second quadrant of the circle 𝑥 2 + 𝑦 2 = 36 is revolved about the line y + 10 = 0. What is the volume generated? A. 2218.6 B. 2228.8 C. 2233.4 D. 2208.5 231. A square area of edge “a” revolves about a line through one vertex, making an angle Ѳ with an edge and not crossing the square. Find the volume generated. A. 3 𝜋 a3 (sin Ѳ + cos Ѳ) B. 𝜋 a3 (sin Ѳ + cos Ѳ) / 2 C. 2𝜋 a3 (sin Ѳ + cos Ѳ) D. 𝜋 a3 (sin Ѳ + cos Ѳ) 232. Given an ellipse whose semi-major axis is 6 cm. and semi-minor axis is 3 cm. what is the volume generated if it is revolved about the minor axis? A. 36 𝜋 cu. cm. B. 72 𝜋 cu. cm. C. 96 𝜋 cu. cm D. 144 𝜋 cu. cm 233. A square hole 2” x 2” is cut through a 6inch diameter long along its diameter and perpendicular to its axis. Find the volume of wood that was removed. A. 27.32 cu. in. B. 23.54 cu. in. C. 21.78 cu. in. D. 34.62 cu. in. Part 5 Analytical Geometry Problems – Points, Lines, Circles Set 16 234. State the quadrant in which the coordinate (15, -2) lies. A. I B. IV C. II D. III 235. Of what quadrant is A, if sec A is positive and csc A is negative? A. III B. I C. IV D. II 236. The segment from (-1, 4) to (2, -2) is extended three times its own length. The terminal point is A. (11, -18) B. (11, -24) C. (11, -20) D. (-11, -20) 237. The midpoint of the line segment between P1(x, y) and P2(-2, 4) is Pm(2, -1). Find the coordinate of P1. A. (6, -5) B. (5, -6) C. (6, -6) D. (-6, 6) 238. Find the coordinates of the point P(2,4) with respect to the translated axis with origin at (1,3). A. (1, -1) B. (1, 1) C. (-1, -1) D. (-1, 1) 239. Find the median through (-2, -5) of the triangle whose vertices are (-6, 2), (2, -2), and (-2, -5). A. 3 B. 4 C. 5 D. 6 240. Find the centroid of a triangle whose vertices are (2, 3), (-4, 6) and (2, -6). A. (0, 1) B. (0, -1) C. (1, 0) D. (-1, 0) 241. Find the area of triangle whose vertices are A (-3, -1), B(5, 3) and (2, -8) A. 34 B. 36 C. 38 D. 32 242. Find the distance between the points (4, 2) and (-5, 1) A. 4.897 B. 8.947 C. 7.149 D. 9.487 243. Find the distance between A(4, -3) and B(-2, 5). A. 11 B. 8 C. 9 D. 10 244. If the distance between the points (8, 7) and (3, y) is 13, what is the value of y? A. 5 B. -19 C. 19 or -5 D. 5 or -19 245. The distance between the points (sin x, cos x) and (cos x, -sin x) is: A. 1 B. √2 C. 2 sin x cos x D. 4 sin x cos x 246. Find the distance from the point (2, 3) to the line 3x + 4y + 9 = 0. A. 5 B. 5.4 C. 5.8 D. 6.2 247. Find the distance from the point (5, -3) to the line 7x - 4y - 28 = 0. A. 2.62 B. 2.36 C. 2.48 D. 2.54 248. How far is the line 3x – 4y + 15 = 0 from the origin? A. 1 B. 2 C. 3 D. 4 249. Determine the distance from (5, 10) to the line x – y = 0 A. 3.86 B. 3.54 C. 3.68 D. 3.72 250. The two points on the lines 2x + 3y +4 = 0 which are at distance 2 from the line 3x + 4y – 6 = 0 are: A. (-8, -8) and (-16, -16) B. (-44, 64) and (-5, 2) C. (-5.5, 1) and (-5, 2) D. (64, -44) and (4, -4) 251. The intercept form for algebraic straightline equation is: 𝑎 𝑦 A. 𝑥 + 𝑏 = 1 B. 𝑦 = 𝑚𝑥 + 𝑏 C. 𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0 𝑥 𝑦 D. 𝑎 + 𝑏 = 1 252. Find the slope of the line defined by y – x =5 A. 1 B. -1/2 C. ¼ D. 5 + x 253. The slope of the line 3x + 2y + 5 = 0 is: A. -2/3 B. -3/2 C. 3/2 D. 2/3 254. Find the slope of the line whose parametric equation is y = 5 – 3t and x = 2 + t. A. 3 B. -3 C. 2 D. -2 255. Find the slope of the curve whose parametric equations are x = -1 + t y = 2t A. 2 B. 3 C. 1 D. 4 256. Find the angle that the line 2y – 9x – 18 = 0 makes with the x-axis. A. 74.77° B. 4.5° C. 47.77° D. 77.47° 257. Which of the following is perpendicular to the line x/3 + y/4 = 1? A. x – 4y – 8 = 0 B. 4x – 3y – 6 = 0 C. 3x – 4y – 5 = 0 D. 4x + 3y – 11 = 0 258. Find the equation of the bisector of the obtuse angle between the lines 2x + y = 4 and 4x - 2y = 7 A. 4y = 1 B. 8x = 15 C. 2y = 3 D. 8x + 4y = 6 259. The equation of the line through (1, 2) and parallel to the line 3x – 2y + 4 = 0 is: A. 3x – 2y + 1 = 0 B. 3x – 2y – 1 = 0 C. 3x + 2y + 1 = 0 D. 3x + 2y – 1 = 0 260. If the points (-3, -5), (x, y), and (3, 4) lie on a straight line, which of the following is correct? A. 3x + 2y – 1 = 0 B. 2x + 3y + 1 = 0 C. 2x + 3y – 1 = 0 D. 3x – 2y – 1 = 0 261. One line passes through the points (1, 9) and (2, 6), another line passes through (3, 3) and (-1, 5). The acute angle between the two lines is: A. 30° B. 45° C. 60° D. 135° 262. The two straight lines 4x – y + 3 = 0 and 8x – 2y + 6 = 0 A. Intersects at the origin B. Are coincident C. Are parallel D. Are perpendicular 263. A line which passes through (5, 6) and (3. -4) has an equation of A. 5x + 4y + 1 = 0 B. 5x - 4y - 1 = 0 C. 5x - 4y + 1 = 0 D. 5x + y - 1 = 0 264. Find the equation of the line with slope of 2 and y-intercept of -3. A. y = -3x + 2 B. y = 2x – 3 C. y = 2/3 x + 1 D. y = 3x – 2 265. What is the equation of the line that passes through (4, 0) and is parallel to the line x – y – 2 = 0? A. y + x + 4 = 0 B. y - x + 4 = 0 C. y - x - 4 = 0 D. y + x - 4 = 0 266. Determine B such that 3x + 2y – 7 = 0 is perpendicular to 2x – By + 2 = 0 A. 2 B. 3 C. 4 D. 5 267. The equation of a line that intercepts the x-axis at x = 4 and the y-axis at y = -6 is: A. 2x – 3y = 12 B. 3x + 2y = 12 C. 3x – 2y = 12 D. 2x – 37 = 12 268. How far from the y-axis is the center of the curve 2x2 + 2y2 + 10x – 6y – 55 = 0? A. -3.0 B. 2.75 C. -3.25 D. 2.5 269. Find the area of the circle whose center is at (2,-5) and tangent to the line 4x + 3y – 8 = 0. A. 6𝜋 B. 9𝜋 C. 3𝜋 D. 12𝜋 270. Determine the area enclosed by the curve 𝑥 2 − 10𝑥 + 4𝑦 + 𝑦 2 = 196 A. 15𝜋 B. 225𝜋 C. 12𝜋 D. 144𝜋 271. Find the shortest distance from the point (1, 2) to appoint on the circumference of the circle defined by the equation 𝑥 2 + 𝑦 2 + 10𝑥 + 6𝑦 + 30 = 0. A. 5.61 B. 5.71 C. 5.81 D. 5.91 272. Determine the length of the chord common to the circles 𝑥 2 + 𝑦 2 = 64 and 𝑥 2 + 𝑦 2 − 16𝑥 = 0. A. 13.86 B. 12.82 C. 13.25 D. 12.28 273. If (3, -2) is on a circle with center (-1, 1), then the area of the circle is: A. 5𝜋 B. 25𝜋 C. 4𝜋 D. 3𝜋 274. The radius of the circle 2𝑥 2 + 2𝑦 2 − 3𝑥 + 4𝑦 − 1 = 0 is: A. √33/4 B. 33/16 C. √33/3 D. 17 275. What is the radius of the circle with the following equation? 𝑥 2 − 6𝑥 + 𝑦 2 − 4𝑦 − 12 = 0 A. 3.46 B. 5 C. 7 D. 6 276. The diameter of a circle described by 9𝑥 2 + 9𝑦 2 = 16 is: A. 16/9 B. 4/3 C. 4 D. 8/3 277. Find the center of the circle 𝑥 2 + 𝑦 2 − 6𝑥 + 4𝑦 − 23 = 0. A. (3, -2) B. (3, 2) C. (-3, 2) D. (-3, -2) 278. Determine the equation of the circle whose center is at (4, 5) and tangent to the circle whose equation is 𝑥 2 + 𝑦 2 + 4𝑥 + 6𝑦 − 23 = 0. A. 𝑥 2 + 𝑦 2 − 8𝑥 + 10𝑦 − 25 = 0 B. 𝑥 2 + 𝑦 2 + 8𝑥 − 10𝑦 + 25 = 0 C. 𝑥 2 + 𝑦 2 − 8𝑥 − 10𝑦 + 25 = 0 D. 𝑥 2 + 𝑦 2 − 8𝑥 − 10𝑦 − 25 = 0 279. The equation of the circle with center at (2, 3) and which is tangent to the line 20x – 21y – 42 = 0. A. 𝑥 2 + 𝑦 2 + 4𝑥 − 6𝑦 − 12 = 0 B. 𝑥 2 + 𝑦 2 + 4𝑥 − 6𝑦 + 12 = 0 C. 𝑥 2 + 𝑦 2 + 4𝑥 + 6𝑦 − 12 = 0 D. 𝑥 2 + 𝑦 2 − 4𝑥 − 6𝑦 − 12 = 0 280. A circle has a diameter whose ends are at (-3, 2) and (12, -6). Its Equation is: A. 4𝑥 2 + 4𝑦 2 − 36𝑥 + 16𝑦 + 192 = 0 B. 4𝑥 2 + 4𝑦 2 − 36𝑥 + 16𝑦 − 192 = 0 C. 4𝑥 2 + 4𝑦 2 − 36𝑥 − 16𝑦 − 192 = 0 D. 4𝑥 2 + 4𝑦 2 − 36𝑥 + 16𝑦 − 192 = 0 281. Find the equation of the circle with center on x + y = 4 and 5x + 2y + 1 = 0 and having a radius of 3. A. 𝑥 2 + 𝑦 2 + 6𝑥 − 16𝑦 + 64 = 0 B. 𝑥 2 + 𝑦 2 + 8𝑥 − 14𝑦 + 25 = 0 C. 𝑥 2 + 𝑦 2 + 6𝑥 − 14𝑦 + 49 = 0 D. 𝑥 2 + 𝑦 2 + 6𝑥 − 14𝑦 + 36 = 0 282. If (3, -2) lies on the circle with center (-1, 1) then the equation of the circle is: A. 𝑥 2 + 𝑦 2 + 2𝑥 − 2𝑦 − 23 = 0 B. 𝑥 2 + 𝑦 2 + 4𝑥 − 2𝑦 − 21 = 0 C. 𝑥 2 + 𝑦 2 + 2𝑥 − 𝑦 − 33 = 0 D. 𝑥 2 + 𝑦 2 + 4𝑥 − 2𝑦 − 27 = 0 283. Find the equation of k for which the equation 𝑥 2 + 𝑦 2 + 4𝑥 − 2𝑦 − 𝑘 = 0 represents a point circle. A. 5 B. -5 C. 6 D. -6 Problems – Parabola, Ellipse, Hyperbola, Polar, Space Set 17 284. The vertex of the parabola 𝑦 2 − 2𝑥 + 6𝑦 + 3 = 0 is at: A. (-3, 3) B. (3, 3) C. (-3, 3) D. (-3, -3) 285. The length of the latus rectum of the parabola 𝑦 2 = 4𝑝𝑥 is: A. 4p B. 2p C. P D. -4p 286. Given the equation of the parabola: 𝑦 2 − 8𝑥 − 4𝑦 − 20 = 0. The length of its latus rectum is: A. 2 B. 4 C. 6 D. 8 287. What is the length of the latus rectum of the curve 𝑥 2 = −12𝑦? A. 12 B. -3 C. 3 D. -12 288. Find the equation of the directrix of the parabola 𝑦 2 = 6𝑥. A. x = 8 B. x = 4 C. x = -8 D. x = -4 289. The curve y = -𝑥 2 + 𝑥 + 1 opens: A. Upward B. To the left C. To the right D. Downward 290. The parabola y = -𝑥 2 + 𝑥 + 1 opens: A. To the right B. To the left C. Upward D. Downward 291. Find the equation of the axis of symmetry of the function y = 2𝑥 2 − 7𝑥 + 5. A. 4x + 7 = 0 B. x – 2 = 0 C. 4x – 7 = 0 D. 7x + 4 = 0 292. Find the equation of the locus of the center of the circle which moves so that it is tangent to the y-axis and to the circle of radius one (1) with center at (2,0). A. 𝑥 2 + 𝑦 2 − 6𝑥 + 3 = 0 B. 𝑥 2 − 6𝑥 + 3 = 0 C. 2𝑥 2 + 𝑦 2 − 6𝑥 + 3 = 0 D. 𝑦 2 − 6𝑥 + 3 = 0 293. Find the equation of the parabola with vertex at (4, 3) and focus at (4, -1). A. 𝑦 2 − 8𝑥 + 16𝑦 − 32 = 0 B. 𝑦 2 + 8𝑥 + 16𝑦 − 32 = 0 C. 𝑥 2 + 8𝑥 − 16𝑦 + 32 = 0 D. 𝑥 2 − 8𝑥 + 16𝑦 − 32 = 0 294. Find the area bounded by the curves 𝑥 2 + 8𝑦 + 16 = 0, x – 4 = 0, the x-axis, and the y-axis. A. 10.67 sq. units B. 10.33 sq. units C. 9.67 sq. units D. 8 sq. units 295. Find the area (in sq. units) bounded by the parabolas 𝑥 2 − 2𝑦 = 0 and 𝑥 2 + 2𝑦 − 8 = 0. A. 11.7 B. 10.7 C. 9.7 D. 4.7 296. The length of the latus rectum of the curve (x – 2)2 / 4 = (y + 4)2 / 25 = 1 is: A. 1.6 B. 2.3 C. 0.80 D. 1.52 297. Find the length of the latus rectum of the following ellipse: 25𝑥 2 + 9𝑦 2 − 300𝑥 − 144𝑦 + 1251 = 0 A. 3.4 B. 3.2 C. 3.6 D. 3.0 298. If the length of the major and minor axes of an ellipse is 10 cm and 8 cm, respectively, what is the eccentricity of the ellipse? A. 0.50 B. 0.60 C. 0.70 D. 0.80 299. The eccentricity of the ellipse 𝑥 2 /4 + y2 / 16 = 1 is: A. 0.725 B. 0.256 C. 0.689 D. 0.866 300. An ellipse has the equation 16𝑥 2 + 9y2 + 32x – 128 = 0. Its eccentricity is: A. 0.531 B. 0.66 C. 0.824 D. 0.93 301. The center of the ellipse 4𝑥 2 + 𝑦 2 − 16𝑥 − 6𝑦 − 43 = 0 is at: A. (2, 3) B. (4, -6) C. (1, 9) D. (-2, -5) 302. Find the ratio of the major axis to the minor axis of the ellipse: 9𝑥 2 + 4𝑦 2 − 24𝑦 − 72𝑥 − 144 = 0 A. 0.67 B. 1.8 C. 1.5 D. 0.75 303. The area of the ellipse 9𝑥 2 + 25𝑦 2 − 36𝑥 − 189 = 0 is equal to: A. 15𝜋 sq. units B. 20𝜋 sq. units C. 25𝜋 sq. units D. 30𝜋 sq. units 304. The area of the ellipse is given as A = 3.1416 a b. Find the area of the ellipse 25𝑥 2 + 16𝑦 2 − 100𝑥 + 32𝑦 = 284. A. 86.2 square units B. 62.8 square units C. 68.2 square units D. 82.6 square units 305. The semi-major axis of an ellipse is 4 and its semi-minor axis is 3. The distance from the center to the directrix is: A. 6.532 B. 6.047 C. 0.6614 D. 6.222 306. Given an ellipse x2 / 36 + y2 / 32 = 1. Determine the distance between foci. A. 2 B. 3 C. 4 D. 8 307. How far apart are the directrices of the curve 25𝑥 2 + 9𝑦 2 − 300𝑥 − 144𝑦 + 1251 = 0? A. 12.5 B. 14.2 C. 13.2 D. 15.2 308. The major axis of the elliptical path in which the earth moves around the sun is approximately 186,000,000 miles and the eccentricity of the ellipse is 1/60. Determine the apogee of the earth. A. 94,550,000 miles B. 94,335.100 miles C. 91,450,000 miles D. 93,000,000 miles 309. Find the equation of the ellipse whose center is at (-3, -1), vertex at (2, -1), and focus at (1, -1). A. 9𝑥 2 + 36𝑦 2 − 54𝑥 + 50𝑦 − 116 = 0 B. 4𝑥 2 + 25𝑦 2 + 54𝑥 − 50𝑦 − 122 = 0 C. 9𝑥 2 + 25𝑦 2 + 50𝑥 + 50𝑦 + 109 = 0 D. 9𝑥 2 + 25𝑦 2 + 54𝑥 + 50𝑦 − 119 = 0 310. Point P(x, y) moves with a distance from point (0, 1) one-half of its distance from line y = 4, the equation of its locus is A. 4x2 + 3y2 = 12 B. 2x2 - 4y2 = 5 C. x2 + 2y2 = 4 D. 2x2 + 5y3 = 3 311. The chords of the ellipse 64^2 + 25y^2 = 1600 having equal slopes of 1/5 are bisected by its diameter. Determine the equation of the diameter of the ellipse. A. 5x – 64y = 0 B. 64x – 5y = 0 C. 5x +64y = 0 D. 64x + 5y = 0 312. Find the equation of the upward asymptote of the hyperbola whose equation is (x – 2)2 / 9 – (y + 4)2 / 16 A. 3x + 4y – 20 = 0 B. 4x – 3y – 20 = 0 C. 4x + 3y – 20 = 0 D. 3x – 4y – 20 = 0 313. The semi-conjugate axis of the hyperbola 𝑥 2 /9 − 𝑦 2 / 4 = 1 is: A. 2 B. -2 C. 3 D. -3 314. What is the equation of the asymptote of 𝑥2 𝑦2 the hyperbola 9 − 4 = 1? A. 2x – 3y = 0 B. 3x – 2y = 0 C. 2x – y = 0 D. 2x + y = 0 315. The graph y = (x – 1) / (x + 2) is not defined at: A. 0 B. 2 C. -2 D. 1 316. The equation x2 + Bx + y2 + Cy + D = 0 is: A. Hyperbola B. Parabola C. Ellipse D. Circle 317. The general second degree equation has the form Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 and describes an ellipse if: A. B2 – 4AC = 0 B. B2 – 4AC > 0 C. B2 – 4AC = 1 D. B2 – 4AC < 0 318. Find the equation of the tangent to the circle x2 + y2 – 34 = 0 through point (3, 5). A. 3x + 5y -34 = 0 B. 3x – 5y – 34 = 0 C. 3x + 5y + 34 = 0 D. 3x – 5y + 34 = 0 319. Find the equation of the tangent to the curve x2 + y2 + 4x + 16y – 32 = 0 through (4, 0). A. 3x – 4y + 12 = 0 B. 3x – 4y – 12 = 0 C. 3x + 4y + 12 = 0 D. 3x + 4y - 12 = 0 320. Find the equation of the normal to the curve y2 + 2x + 3y = 0 though point (-5,2) A. 7x + 2y + 39 = 0 B. 7x - 2y + 39 = 0 C. 2x - 7y - 39 = 0 D. 2x + 7y - 39 = 0 321. Determine the equation of the line tangent to the graph y = 2x2 + 1, at the point (1, 3). A. y = 4x + 1 B. y = 4x – 1 C. y = 2x – 1 D. y = 2x + 1 322. Find the equation of the tangent to the curve x2 + y2 = 41 through (5, 4). A. 5x + 4y = 41 B. 4x – 5y = 41 C. 4x + 5y = 41 D. 5x – 4y = 41 323. Find the equation of a line normal to the curve x2 = 16y at (4, 1). A. 2x – y – 9 = 0 B. 2x – y + 9 = C. 2x + y – 9 = 0 D. 2x + y + 9 = 0 324. What is the equation of the tangent to the curve 9x2 + 25y2 – 225 = 0 at (0, 3)? A. y + 3 = 0 B. x + 3 = 0 C. x – 3 = 0 D. y – 3 = 0 325. What is the equation of the normal to the curve x2 + y2 = 25 at (4, 3)? A. 3x – 4y = 0 B. 5x + 3y = 0 C. 5x – 3y = 0 D. 3x + 4y = 0 326. The polar form of the equation 3x + 4y – 2 = 0 is: A. 3r sin Ѳ + 4r cos Ѳ = 2 B. 3r cos Ѳ + 4r sin Ѳ = -2 C. 3r cos Ѳ + 4r sin Ѳ = 2 D. 3r sin Ѳ + 4r tan Ѳ = -2 327. The polar form of the equation 3x + 4y – 2 = 0 is: A. r2 = 8 8 B. 𝑟 = 𝑐𝑜𝑠 2Ѳ+2 C. 𝑟 = 8 8 D. r2= 𝑐𝑜𝑠 2 Ѳ+2 328. the distance between points (5, 30°) and (8, -50°) is: A. 9.84 B. 10.14 C. 6.13 D. 12.14 329. Convert Ѳ = 𝜋/3 to Cartesian equation. A. x = √3𝑥 B. y = x C. 3y = √3𝑥 D. y = √3𝑥 330. The point of intersection of the planes x + 5y – 2z = 9, 3x – 2y + z = 3, and x + y + z = 2 is: A. (2, 1, -1) B. (2, 0, -1) C. (-1, 1, -1) D. (-1, 2, 1) 331. A warehouse roof needs a rectangular skylight with vertices (3, 0, 0), (3, 3, 0), (0, 3, 4), and (0, 0, 4). If the units are in meter, the area of the skylight is: A. 12 sq. m. B. 20 sq. m. C. 15 sq. m. D. 9 sq. m. 332. The distance between points in space coordinates are (3, 4, 5) and (4, 6, 7) is: A. 1 B. 2 C. 3 D. 4 333. What is the radius of the sphere with center at origin and which passes through the point (8, 1, 6)? A. 10 B. 9 C. √101 D. 10.5 Part 6 Differential Calculus Problems – Limits, Differentiation, Rate of Change, Slope Set 18 334. Evaluate lim (1 − 𝑠𝑖𝑛2 Ѳ)1/2 Ѳ⟶0 A. B. C. D. 0 1 2 3 335. Simplify the expression: lim ( 𝑥→4 A. B. C. D. 𝑥 2 − 16 𝑥−4 ) 337. Evaluate the limit ( x – 4 ) / (𝑥 2 − 𝑥 − 12 ) as x approaches 4. A. 0 B. undefined C. 1/7 D. infinity 338. Evaluate the limit (1n x ) / x as x approaches positive infinity. A. 1 B. 0 C. e D. infinity 𝑥+4 339. Evaluate the following limit: lim 𝑥 − 4 𝑥→∞ A. 1 B. Indefinite C. 0 D. 2 1−cos 𝑥 340. Evaluate: lim 𝑥2 𝑥→0 A. 0 B. ½ C. 2 D. -1/2 𝜋𝑥 341. Evaluate the following: lim (2 − 𝑥)tan 2 𝑥→1 A. Infinity B. 𝑒 𝜋 C. 0 D. 𝑒 2/𝜋 342. Find dy/dx if y = 52x-1 A. 52x-1 ln 5 B. 52x-1 ln 25 C. 52x-1 ln 10 D. 52x-1 ln 2 343. Find dy/dx if y = 𝑒 √𝑥 A. 𝑒 √𝑥 / 2√𝑥 B. 𝑒 √𝑥 / √𝑥 C. 𝑒 𝑥 / √𝑥 D. 𝑒 √𝑥−2√𝑥 344. Find dy/dx if y = 𝑥 2 + 3𝑥 + 1 𝑎𝑛𝑑 𝑥 = 𝑡 2 + 2. A. 4t3 + 14t2 B. t3 + 4t C. 4t3 + 14t D. 4t3 + t 345. Evaluate the first derivative of the implicit function: 4x2 + 2xy + y2 = 0 4𝑥+𝑦 A. 𝑥+𝑦 B. − C. 1 8 0 16 𝑥→1 𝑥 2 −1 𝑥 2 + 3𝑥 − 4 4𝑥+𝑦 𝑥+𝑦 4𝑥−𝑦 𝑥+𝑦 4𝑥+𝑦 D. − 336. Evaluate the following limit: lim A. 2/5 B. infinity C. 0 D. 5/2 𝑥−𝑦 346. Find the derivative of (x + 5) / (𝑥 2 − 1) with respect to x. A. DF(x) = (−𝑥 2 − 10𝑥 − 1) / (𝑥 2 − 1)2 B. DF(x) = (𝑥 2 + 10𝑥 − 1) / (𝑥 2 − 1)2 C. DF(x) = (𝑥 2 − 10𝑥 − 1) / (𝑥 2 − 1)2 D. DF(x) = (−𝑥 2 − 10𝑥 + 1) / (𝑥 2 − 1)2 347. If a simple constant, what is the derivative of y = xa? A. a xa-1 B. (a – 1)x C. xa-1 D. ax 348. Find the derivative of the function 2x2 + 8x + 9 with respect to x. A. Df(x) = 4x – 8 B. Df(x) = 2x + 9 C. Df(x) = 2x + 8 D. Df(x) = 4x + 8 349. What is the first derivative dy/dx of the expression (xy)x = e? A. – y(1 + ln xy) / x B. 0 C. – y(1 – ln xy) / x2 D. y/x 350. find the derivative of A. B. C. 3(𝑥+1)2 𝑥 3(𝑥+1)2 𝑥 2(𝑥+1)2 𝑥 3(𝑥+1)2 + − − (𝑥+1)3 (𝑥+1)3 𝑥 𝑥2 (𝑥+1)3 𝑥2 (𝑥+1)3 𝑥2 2(𝑥+1)3 D. 𝑥 + 𝑥2 351. Given the equation: y = (e ln x)2, find y’. A. ln x B. 2 (ln x) / x C. 2x D. 2 e ln x 352. Find the derivatives with respect to x of the function √2 − 3𝑥 2 A. -2𝑥 2 / √2 − 3𝑥 2 B. -3x / √2 − 3𝑥 2 C. -2𝑥 2 / √2 + 3𝑥 2 D. -3x / √2 + 3𝑥 2 353. Differentiate ax2 + b to the ½ power. A. -2ax B. 2ax C. 2ax + b D. ax + 2b 354. Find dy/dx if y = ln √𝑥 A. √𝑥 / ln x B. x / ln x C. 1 / 2x D. 2 / x 355. Evaluate the differential of tan Ѳ. A. ln sec Ѳ dѲ B. ln cos Ѳ dѲ C. sec Ѳ tan Ѳ dѲ D. sec2 Ѳ dѲ 356. If y = cos x, what is dy/dx? A. sec x B. –sec x C. sin x D. –sin x 357. Find dy/dx: y = sin (ln x2). A. 2 cos (ln x2) B. 2 cos (ln x2) / x C. 2x cos (ln x2) D. 2 cos (ln x2) / x2 358. The derivative of ln (cos x) is: A. sec x B. –sec x C. –tan x D. tan x 359. Find the derivative of arcos 4x with respect to x. A. -4 / [1 – (4x)^2]^2 B. -4 / [1 – (4x)]^0.5 C. 4 / [1 – (4x)^2]^0.5 D. -4 / [(4x)^2 - 1]^0.5 360. What is the first derivative of y = arcsin 3x. 3 A. − 1+9𝑥2 3 B. 1+9𝑥2 3 C. − √1−9𝑥2 3 D. √1−9𝑥2 361. If y = x (ln x), find d2y/dx2. A. 1 / x2 B. -1 / x C. 1 / x D. -1 / x2 362. Find the second derivative of y = x-2 at x = 2. A. 96 B. 0.375 C. -0.25 D. -0.875 363. Given the function f(x) = x3 – 5x + 2, find the value of the first derivative at x = 2, f’ (2). A. 7 B. 3x2 – 5 C. 2 D. 8 364. Given the function f(x) = x to the 3rd power – 6x + 2, find the value of the first derivative at x = 2, f’(2) A. 6 B. 3x2 – 5 C. 7 D. 8 365. Find the partial derivatives with respect to x of the function: xy2 – 5y + 6. A. y2 – 5 B. xy – 5y C. y2 D. 2xy 366. Find the point in the parabola y2 = 4x at which the rate of change of the ordinate and abscissa are equal. A. (1, 2) B. (2, 1) C. (4, 4) D. (-1, 4) 367. Find the slope of the line tangent to the curve y = x3 – 2x + 1 at x = 1. A. 1 B. ½ C. 1/3 D. ¼ 368. Determine the slope of the curve x^2 + y^2 – 6x – 4y – 21 = 0 at (0, 7). A. 3/5 B. -2/5 C. -3/5 D. 2/5 369. Find the slope of the tangent to a parabola y = x2 at a point on the curve where x = ½. A. 0 B. 1 C. ¼ D. -1/2 370. Find the slope of the ellipse x2 + 4y2 – 10x + 16y + 5 = 0 at the point where y = -2 + 80.5 and x = 7. A. -0.1654 B. -0.1538 C. -0.1768 D. -0.1463 371. Find the slope of the tangent to the curve y = x4 – 2x2 + 8 through point (2, 16). A. 20 B. 1/24 C. 24 D. 1/20 372. Find the slope of the tangent to the curve y2 = 3x2 + 4 through point (-2, 4) A. -3/2 B. 3/2 C. 2/3 D. -2/3 373. Find the slope of the line whose parametric equations are x = 4t + 6 and y = t – 1. A. -4 B. ¼ C. 4 D. -1/4 374. What is the slope of the curve x2 + y2 – 6x + 10y + 5 = 0 at (1, 0). A. 2/5 B. 5/2 C. -2/5 D. -5/2 375. Find the slope of the curve y = 6(4 + x) ½ at (0, 12). A. 0.67 B. 1.5 C. 1.33 D. 0.75 376. Find the acute angle that the curve y = 1 – 3x2 cut the x-axis. A. 77° B. 75° C. 79° D. 120° 377. Find the angle that the line 2y – 9x – 18 = 0 makes with the x-axis. A. 74.77° B. 4.5° C. 47.77° D. 77.47° 378. Find the equation of the tangent to the curve y = x + 2x1/3 through point (8, 12) A. 7x – 6y + 14 = 0 B. 8x + 5y + 21 = 0 C. 5x – 6y – 15 = 0 D. 3x – 2y – 1 = 0 379. What is the radius of curvature at point (1, 2) of the curve 4x – y2 = 0? A. 6.21 B. 5.21 C. 5.66 D. 6.66 380. Find the radius of curvature at any point of the curve y + ln (cos x) = 0. A. cos x B. 1.5707 C. sec x D. 1 381. Determine the radius of curvature at (4, 4) of the curve y2 – 4x = 0. A. 24.4 B. 25.4 C. 23.4 D. 22.4 382. Find the radius of curvature of the curve x = y3 at (1, 1) A. 4.72 B. 3.28 C. 4.67 D. 5.27 383. The chords of the ellipse 64x^2 + 25y^2 = 1600 having equal slopes of 1/5 are bisected by its diameter. Determine the equation of the diameter of the ellipse. A. 5x – 64y = 0 B. 64x – 5y = 0 C. 5x + 64y = 0 D. 64x + 5y = 0 Problems – Maxima & Minima, Time Rates Set 19 384. A function is given below, what x value maximizes y? y2 + y + x2 – 2x = 5 A. 2.23 B. -1 C. 5 D. 1 385. The number of newspaper copies distributed is given by C = 50 t2 – 200 t + 10000, where t is in years. Find the minimum number of copies distributed from 1995 to 2002. A. 9850 B. 9800 C. 10200 D. 7500 386. Given the following profit-versusproduction function for a certain commodity: P = 200000 – x – 1.1 8 (1+𝑥) Where P is the profit and x is unit of production. Determine the maximum profit. A. 190000 B. 200000 C. 250000 D. 550000 387. The cost C of a product is a function of the quantity x of the product is given by the relation: C(x) = x2 – 4000x + 50. Find the quantity for which the cost is a minimum. A. 3000 B. 2000 C. 1000 D. 1500 388. If y = x to the 3rd power – 3x. find the maximum value of y. A. 0 B. -1 C. 1 D. 2 389. Divide 120 into two parts so that product of one and the square of the other is maximum. Find the numbers. A. 60 & 60 B. 100 & 20 C. 70 & 50 D. 80 & 40 390. If the sum of two numbers is C, find the minimum value of the sum of their squares. A. C2 / 2 B. C2 / 4 C. C2 / 6 D. C2 / 8 391. A certain travel agency offered a tour that will cost each person P 1500.00 if not more than 150 persons will join, however the cost per person will be reduced by P 5.00 per person in excess of 150. How many persons will make the profit a maximum? A. 75 B. 150 C. 225 D. 250 392. Two cities A and B are 8 km and 12 km, respectively, north of a river which runs due east. City B being 15 km east of A. a pumping station is to be constructed (along the river) to supply water for the two cities. Where should the station be located so that the amount of pipe is a minimum? A. 3 km east of A B. 4 km east of A C. 9 km east of A D. 6 km east of A 393. A boatman is at A, which is 4.5 km from the nearest point B on a straight shore BM. He wishes to reach, in minimum time, a point C situated on the shore 9 km from B. How far from C should he land if he can row at the rate of 6 Kph and walk at the rate of 7.5 Kph? A. 1 km B. 3 km C. 5 km D. 8 km 394. The shortest distance from the point (5, 10) to the curve x2 = 12y is: A. 4.331 B. 3.474 C. 5.127 D. 6.445 395. A statue 3 m high is standing on a base of 4 m high. If an observer’s eye is 1.5 m above the ground, how far should he stand from the base in order that the angle subtended by the statue is a maximum? A. 3.41 m B. 3.51 m C. 3.71 m D. 4.41 m 396. An iron bar 20 m long is bent to form a closed plane area. What is the largest area possible? A. 21.56 square meter B. 25.68 square meter C. 28.56 square meter D. 31.83 square meter 397. A Norman window is in the shape of a rectangle surmounted by a semi-circle. What is the ratio of the width of the rectangle to the total height so that it will yield a window admitting the most light for a given perimeter? A. 1 B. 2/3 C. 1/3 D. ½ 398. A rectangular field is to be fenced into four equal parts. What is the size of the largest field that can be fenced this way with a fencing length of 1500 feet if the division is to be parallel to one side? A. 65,200 B. 62,500 C. 64,500 D. 63,500 399. Three sides of a trapezoid are each 8 cm long. How long is the 4th side, when the area of the trapezoid has the greatest value? A. 16 cm B. 15 cm C. 12 cm D. 10 cm 400. An open top rectangular tank with square bases is to have a volume of 10 cubic meters. The material for its bottom cost P 150.00 per square meter, and that for the sides is P 60.00 per square meter. The most economical height is: A. 2 meters B. 2.5 meters C. 3 meters D. 3.5 meters 401. A rectangular box having a square base and open at the top is to have a capacity of 16823 cc. Find the height of the box to use the least amount of material. A. 16.14 cm B. 32.28 cm C. 18.41 cm D. 28.74 cm 402. The altitude of a cylinder of maximum volume that can be inscribed in a right circular cone of radius r and height h is: A. h/3 B. 2h/3 C. 3h/2 D. h/4 Problems – Integration Set 20 7𝑥 3 3 7𝑥 4 4 B. 7𝑥 4 4 4𝑥 2 + 2 4𝑥 3 + + + 𝐶 4𝑥 2 7𝑥 4 − C. + 𝐶 3 + 𝐶 5 4𝑥 2 2 D. +𝐶 Problem 2 (CE May 1999) 4𝑑𝑥 Evaluate: ∫ 3𝑥+2 A. 4 ln(3𝑥 + 2) + 𝐶 1 ln(3𝑥 + 2) + 𝐶 3 B. 4 ln(3𝑥 + 2) + 𝐶 2 ln(3𝑥 + 2) + 𝐶 3 2 𝑒 𝑥 +1 +𝐶 ln 2 𝑥 2 +1 𝑒 C. Problem 6 (ME April 1998) Integrate 𝑥 cos(2 𝑥 2 + 7)𝑑𝑥. A. ¼ sin(2𝑥 2 + 7) + 𝐶 C. ¼ cos(2𝑥 2 + 7) + C B. sin(2𝑥 2 + 7) + 𝐶 D. ¼ (𝑠𝑖𝑛 − 𝜃 )(𝑥 2 + 7) + 𝐶 Problem 7 tan(ln 𝑥) Evaluate ∫ 𝑥 𝑑𝑥 . C. ½ D. Problem 8 Evaluate ∫(cos 𝑥)(ln sin 𝑥)𝑑𝑥 . A. sin 𝑥 (1 − ln sin 𝑥) + 𝐶 sin 𝑥 (ln sin 𝑥 − 1) + 𝐶 B. sin 𝑥 (1 + ln sin 𝑥) + 𝐶 ln √sin 𝑥 + 𝐶 C. D. A. 2(ln sec 𝜃 )2 + 𝐶 (ln sec 𝜃 ) + 𝐶 B. (ln sec 𝜃 )2 + 𝐶 (ln sec 𝜃 )2 + 𝐶 C. ½ D. ½ Problem 10 𝑒 𝑥 𝑑𝑥 Evaluate ∫ 1+𝑒2𝑥 . C. D. A. ½ ln(1 + 𝑒 2𝑥 ) + 𝐶 (1 + 𝑒 2𝑥 )2 + 𝐶 B. ln(1 + 𝑒 2𝑥 ) + 𝐶 arctan(𝑒 𝑥 ) + 𝐶 Problem 11 Evaluate ∫ 𝑑𝑥 ln 𝑥 2 √(ln 𝑥)2 −1 Problem 4 (CE May 1995) What is the integral of 𝑐𝑜𝑠2𝑥𝑒 sin 2𝑥 𝑑𝑥? A. −𝑒 sin 2𝑥 + 𝐶 𝑒 sin 2𝑥 + 𝐶 B. 𝑒 sin 2𝑥 / 2 + 𝐶 – 𝑒 sin 2𝑥 / 2 + 𝐶 D. D. +𝐶 B. 𝑒 2𝑥 + 𝐶 2x𝑒 𝑥 C. Problem 9 Evaluate ∫ tan 𝜃 ln sec 𝜃 𝑑𝜃 . Problem 3 (CE May 1994) 2 Evaluate the integral of 𝑒 𝑥 +1 2𝑥𝑑𝑥. A. A. sec 𝑥 + 𝑐 cos 𝑥 + 𝑐 B. sin 𝑥 + 𝑐 – sin 𝑥 + 𝑐 A. ln cos(ln 𝑥) + 𝐶 tan2 (ln 𝑥) + 𝐶 B. ln sec(ln 𝑥) + 𝐶 tan (ln 𝑥)2 + 𝐶 Problem 1 (ME April 1997) Integrate: (7𝑥 3 + 4𝑥 2 ) 𝑑𝑥 A. Problem 5 (Me October 1997) The integral of cos 𝑥 with respect to 𝑥; ∫ cos 𝑥 𝑑𝑥 = _______. C. D. D. . A. arcsec(ln 𝑥 ) + 𝐶 ln √(ln 𝑥 )2 − 1 + 𝐶 B. (2/3) [(ln 𝑥 )2 − 1]3/2 + 𝐶 arctan(ln 𝑥 ) + 𝐶 Problem 12 C. ½ C. D. Evaluate ∫ 𝑥 3 +1 𝑥+2 𝑑𝑥 . A. 𝑥 3 + 𝑥 2 + 4𝑥 + 7 ln (𝑥 + 2) + 𝐶 𝑥3 C. B. 𝑥 3 − 3 𝑥2 + 4𝑥 − ln(𝑥 + 2) + 𝐶 2 𝑥3 D. − 𝑥 2 + 4𝑥 − 7 ln(𝑥 + 2) + 𝐶 3 + 𝑥 2 + 4𝑥 − ln(𝑥 + 2) + 𝐶 Problem 20 (CE May 1997) Evaluate the integral of 𝑥 (𝑥 − 5)12 𝑑𝑥 with limits from 5 to 6. A. 81/182 C. 83/182 B. 82/182 D. 84/182 Problem 13 Evaluate ∫ 𝑒 2𝑥 𝑒 𝑥 +1 𝑑𝑥 . A. ½ 𝑒 𝑥 + ln(𝑒 𝑥 + 1) + 𝐶 C. ln(𝑒 𝑥 + 1) + 𝐶 𝑥 B. 𝑒 − ln(𝑒 𝑥 + 1) + 𝐶 D. 𝑒 𝑥 + ln(𝑒 𝑥 + 1) + 𝐶 Problem 14 3𝑑𝑥 Evaluate ∫ 𝑥(𝑥+3) . A. 3 ln 𝑥 (𝑥 + 3) + 𝐶 𝑥 ln 𝑥+3 + 𝐶 B. 𝑥 1 ln 𝑥+3 + 𝐶 3 ln 𝑥 (𝑥 + 3) + 𝐶 C. 𝑥 1 A. – 4 cos 2𝑥 + 2 sin 2𝑥 + 𝐶 B. 4 𝑥 2 D. C. 1 cos 2𝑥 − 2 sin 2𝑥 + 𝐶 1 cos 2𝑥 − 4 sin 2𝑥 + 𝐶 𝑥 D. A. 0.022 0.043 B. 0.056 0.031 C. D. Problem 22 (CE May 1996) Find the integral of 12 sin5 𝑥 cos 5 𝑥 𝑑𝑥 if lower limit = 0 and upper limit = 𝜋/2. A. 0.2 C. 0.6 B. 0.8 D. 0.4 1 – 2 cos 2𝑥 + 4 sin 2𝑥 + 𝐶 Problem 16 Evaluate ∫ 𝑒 𝑥 sin 𝑥 𝑑𝑥 . 𝑒𝑥 (sin 𝑥 − cos 𝑥) + 𝐶 – 𝑒 𝑥 (sin 𝑥 − cos 𝑥) + 𝐶 B. – 𝑒 𝑥 (cos 𝑥 + sin 𝑥) + 𝐶 𝑒 𝑥 (cos 𝑥 + sin 𝑥) + 𝐶 A. Problem 21 (CE November 1996) 𝑥𝑑𝑥 Evaluate the integral of (𝑥+1)8 if it has an upper limit of 1 and a lower limit of 0. Problem 15 Evaluate the integral of 𝑥 sin 2𝑥 𝑑𝑥. 𝑥 Problem 19 (CE November 1999) Evaluate the integral of 𝑥 cos 2𝑥 𝑑𝑥 with limits from 0 to 𝜋/4. A. 0.143 C. 0.114 B. 0.258 D. 0.186 2 C. D. Problem 17 Evaluate ∫ arctan 𝑥 𝑑𝑥 . A. arctan 𝑥 − ln √1 + 𝑥 2 𝑥 arctan 𝑥 − √1 + 𝑥^2 + 𝐶 B. arctan 𝑥 + 2ln(1 + 𝑥 2 ) 𝑥 arctan 𝑥 + ln(1 + 𝑥 2 ) + 𝐶 C. D. Problem 18 Integrate the square root of (1 − sin2 𝑥)𝑑𝑥 . 𝑥 A. 2 cos 2 + 𝐶 C. 𝑥 √2 cos 2 + 𝐶 𝑥 B. −2√2 cos + 𝐶 2 𝑥 −2 cos 2 + 𝐶 D. Problem 23 (CE November 1997, Similar to CE November 1994) Using lower limit = 0 and upper limit = 𝜋/2, what is the integral of 15 sin2 𝑥 𝑑𝑥? A. 6.783 C. 6.648 B. 6.857 D. 6.539 Problem 24 (CE November 1998) Evaluate the integral of 3 (sin 𝑥 )3 𝑑𝑥 using lower limit of 0 and upper limit = pi/2. A. 2.0 C. 1.4 B. 1.7 D. 2.3 Problem 25 (CE May 1998, Similar to CE November 95 & May 96) Evaluate the integral of 5 cos 6 𝑥 sin2 𝑥 𝑑𝑥 using lower limit = 0 and upper limit = 𝜋/2. A. 0.5046 C. 0.6107 B. 0.3068 D. 0.4105 Problem 26 (ECE April 1998) Evaluate the integral of cos 8 3𝐴 𝑑𝐴 from 0 to 𝜋/6. A. 35𝜋/768 27𝜋/363 B. 23𝜋/765 12𝜋/81 C. D. Problem 27 (CE November 1996) Evaluate the integral of (3𝑥 2 + 9𝑦 2 )𝑑𝑥 𝑑𝑦 if the interior limit has an upper limit of y and a lower limit of 0, and whose outer limit has an upper limit of 2 and lower limit of 0. A. 10 C. 30 B. 40 D. 20 Problems – Plane Areas, Volumes, Surfaces, Centroid, Etc. Set 21 Problem 28 (CE May 1999) 2 2𝑦 Evaluate ∫1 ∫0 (𝑥 2 + 𝑦 2 )𝑑𝑥𝑑𝑦 . Problem 1 Find the area under the curve y = 𝑥 3 + 3𝑥 2 and the x-axis between x = 1 and x = 3. A. 28 sq. units C. 36 sq. units B. 46 sq. units D. 54 sq. units A. 35/2 B. 19/2 C. 17/2 D. 37/2 Problem 29 (EE April 1997) Evaluate the double integral of 𝑟 sin 𝑢 𝑑𝑟 𝑑𝑢, the limit of r is from 0 to cos 𝑢 and the limit of u is from 0 to pi. A. -1/6 C. 1/3 B. 1/6 D. 1/2 Problem 30 1 2 𝑦 Evaluate ∫0 ∫0 ∫0 𝑑𝑥 𝑑𝑦 𝑑𝑧 . A. 1/3 B. 1/4 C. 1/2 D. 1/6 Problem 2 (ECE April 2005) Find the area bounded by y = (11 − 𝑥 )1⁄2, the lines 3x = 2 and x = 10, and the X-axis. A. 19.456 sq. units C. 22.567 sq. units B. 20.567 sq. units D. 21.478 sq. units Problem 3 Find the area of the region bounded by the curves = 12𝑥 , the x-axis, x = 1, and x = 4. 𝑥 2 +4 A. 4 ln 6 B. ln 24 C. 6 ln 15 D. 6 ln 4 Problem 4 (ECE November 1996) Find the area bounded by the y-axis and x = 4 − 𝑦 2⁄ 3 . A. 25.6 C. 12.8 B. 28.1 D. 56.2 Problem 5 Find the area of the region bounded by one loop of the curve 𝑥 2 = 𝑦 4 (1 − 𝑦 2 ). A. 𝜋 sq. units C. (𝜋/4) sq. units B. (𝜋/2) sq. units D. (𝜋/8) sq. units Problem 6 (CE November 1996, November 1998) Find the area bounded by the curve 𝑟 2 = 𝑎2 cos 2𝜃 A. 3𝑎2 C. 4𝑎2 B. 𝑎2 D. 2𝑎2 Problem 7 (CE November 1997) What is the area within the curve 𝑟 2 = 16 cos 𝜃 ? A. 26 C. 30 B. 28 D. 32 Problem 8 (CE May 1999) Find the area enclosed by 𝑟 2 = 2𝑎2 cos 𝜃 . A. 2𝑎2 C. 4𝑎2 B. 𝑎2 D. 3𝑎2 Problem 9 Find the curved surface (area) of the solid generated by revolving the part of the curve y = x2 from (0, 0) to (√6, 6) about the y-axis. A. 62𝜋 sq. units C. 62𝜋/5 sq. units B. 62𝜋/3 sq. units D. 5/62𝜋 sq. units Problem 10 Find the volume generated by rotating the region bounded by y = x, x = 1, and y2 = 4x, about the xaxis. A. 𝜋 C. 3𝜋 B. 2𝜋 D. 9𝜋 Problem 11 (CE November 1995) The area bounded by the curve y2 = 12x and the line x = 3 is revolved about the line x = 3. What is the volume generated? A. 186 C. 181 B. 179 D. 184 Problem 12 (CE May 1995) Given is the area in the first quadrant bounded by x2 = 8y, the line x = 4 and the x-axis. What is the volume generated by revolving this area about the y-axis? A. 50.26 C. 53.26 B. 52.26 D. 51.26 Problem 13 (CE November 1994) Given is the area in the first quadrant bounded by x2 = 8y, the line y – 2 = 0 and the y-axis. What is the volume generated when this area is revolved about the line y – 2 = 0? A. 28.41 C. 27.32 B. 26.81 D. 25.83 Problem 14 (CE May 1998) Find the length of the arc of x2 + y2 = 64 from x = 1 to x = -3, in the second quadrant. A. 2.24 C. 2.75 B. 2.61 D. 2.07 Problem 15 How far from the y-axis is the centroid of the area bounded by the curve y = x3, the line x = 2, and the y-axis. A. 1.2 C. 1.6 B. 1.4 D. 1.8 Problem 16 (CE May 1998) The area in the first quadrant, bounded by the curve y = 2x1/2, the y-axis and the line y – 6 = 0 is revolved about the line y = 6. Find the centroid of the solid formed. A. (2.2,6) C. (1.8,6) B. (1.6,6) D. (2.0,6) Problem 17 A solid formed by revolving about the y-axis, the area bounded by the curve y = x4, the y-axis, and the line y = 16. Find its centroid. A. (0, 9.6) C. (0, 8.3) B. (0, 12.4) D. (0, 12.8) Problem 18 (CE November 1998) A solid is formed by revolving about the y-axis, the area bounded by the curve x3 = y, the y-axis and the line y = 8. Find its centroid. A. (0, 4.75) C. (0, 5.25) B. (0, 4.5) D. (0, 5) Problem 19 (CE November 1995) Find the moment of inertia of the area bounded by the parabola y2 = 4x, x-axis and the line x = 1, with respect to the x-axis. A. 1.067 C. 0.968 B. 1.244 D. 0.878 Problem 20 Find the work done in stretching a spring of natural length 8 cm, from 10 cm to 13 cm. Assume a force of 6 N is needed to hold it at a length of 11 cm. A. 21 N-m C. 0.21 N-m B. 2.1 N-m D. 0.021 N-m Problem 21 A conical tank that is 5 meters high has a radius of 2 meters, and is filled with a liquid that weighs 800 kg per cubic meter. How much work is done in discharging all the liquid at a point 3 meters above the top of the tank? A. 21,256 𝜋 kg-m C. 23,457 𝜋 kg-m B. 21,896 𝜋 kg-m D. 22,667 𝜋 kg-m Problem 22 How much work is required to pump all the water from a right circular cylindrical tank that is 8 feet in diameter and 9 feet tall, if it is emptied at a point 1 foot above the top of the tank? A. 49,421 𝜋 kg-m C. 54,448 𝜋 kg-m B. 52,316 𝜋 kg-m D. 56,305 𝜋 kg-m Problem 23 A 60-m cable that weighs 4 kg/m has a 500-kg weight attached at the end. How much work is done in winding-up the last 20 m of the cable? A. 9,866 kg-m C. 12,500 kgm B. 10,800 kg-m D. 15,456 kgm Problem 24 A uniform chain the weighs 0.50 kg per meter has a leaky 15-liter bucket attached to it. If the bucket is full of liquid when 30 meters of chain is out and half-full when no chain is out, how much work is done in winding the chain? Assume that the liquid leaks out at a uniform rate and weighs 1 kg per liter. A. 356.2 kg-m C. 562.5 kg-m B. 458.2 kg-m D. 689.3 kg-m Obtain the differential equation of all straight lines with algebraic sum of the intercepts fixed as k. A. (1 + 𝑦 ′)(𝑥𝑦 ′ − 𝑦) = 𝑘𝑦 C. ′ ′ (1 − 𝑦 )(𝑥𝑦 − 𝑦) = 𝑘𝑦 B. (1 − 𝑦 ′)(𝑥𝑦 ′ + 𝑦) = 𝑘𝑦 D. (1 + 𝑦 ′)(𝑥𝑦 ′ + 𝑦) = 𝑘𝑦 Problem 25 (ECE Board November 1995) The velocity of a body is given by v(t) = sin(𝜋𝑡), where the velocity is given in meters per second and t is given in seconds. The distance covered in meters between t = ¼ and ½ second is close to. A. 0.5221 m C. 0.2251 m B. -0.2251 m D. -0.5221 m Problem 3 Obtain the differential equation of all straight lines at a fixed distance p from the origin. Problem 26 (ECE November 1995) The rate of change of a function of y with respect to x equals 2 – y, and y = 8 when x = 0. Find y when x = ln(2). A. 2 C. 5 B. -5 D. -2 A. (𝑥𝑦 ′ − 𝑦)2 = 𝑝[1 + (𝑦 ′)2 ] (𝑥𝑦 ′ − 𝑦)2 = 𝑝[1 + (𝑦 ′)2 ] B. (𝑥𝑦 ′ + 𝑦)2 = [1 + (𝑦 ′)2 ] 𝑥𝑦 ′ − 𝑦) = [1 + 𝑦 ′] C. D. Problem 4 (CE May 1997) Determine the differential equation of the family of lines passing through the origin. A. 𝑥 𝑑𝑦 − 𝑦 𝑑𝑥 = 0 C. 𝑥 𝑑𝑦 + 𝑦 𝑑𝑥 = 0 B. 𝑥 𝑑𝑥 − 𝑦 𝑑𝑦 = 0 D. 𝑥 𝑑𝑥 + 𝑦 𝑑𝑦 = 0 Problem 5 Obtain the differential equation of all circles with center on line y = -x and passing through the origin. A. (𝑦 2 + 2𝑥𝑦 − 𝑥 2 )𝑑𝑥 − (𝑦 2 − 2𝑥𝑦 + 𝑥 2 )𝑑𝑦 = 0 B. (𝑦 2 − 2𝑥𝑦 − 𝑥 2 )𝑑𝑥 − (𝑦 2 − 2𝑥𝑦 − 𝑥 2 )𝑑𝑦 = 0 C. (𝑦 2 − 2𝑥𝑦 − 𝑥 2 )𝑑𝑥 − (𝑦 2 − 2𝑥𝑦 − 𝑥 2 )𝑑𝑦 = 0 D. (𝑦 2 + 2𝑥𝑦 − 𝑥 2 )𝑑𝑥 + (𝑦 2 − 2𝑥𝑦 − 𝑥 2 )𝑑𝑦 = 0 Problem 6 Obtain the differential equation of a parabola with axis parallel to the x-axis. 2 A. 3(𝑦 ′′) − 𝑦 ′ 𝑦 ′′′ = 0 C. 2 2(𝑦 ′′) − 𝑦 ′ 𝑦 ′′′ = 0 B. 3(𝑦 ′′ )2 + 𝑦 ′ 𝑦 ′′′ = 0 2(𝑦 ′′ )2 − 𝑦 ′′ 𝑦 ′′′ = 0 Problems – Differential Equations & Application Set 22 Problem 1 Obtain the differential equation of the family of straight lines with slope and y-intercept equal. A. 𝑦 𝑑𝑦 + (𝑥 + 1)𝑑𝑥 = 0 C. 𝑦 𝑑𝑥 + (𝑥 + 1)𝑑𝑦 = 0 B. 𝑦 𝑑𝑦 − (𝑥 + 1)𝑑𝑥 = 0 D. 𝑦 𝑑𝑥 − (𝑥 + 1)𝑑𝑦 = 0 Problem 2 D. Problem 7 𝑑𝑟 Obtain the particular solution of 𝑑𝑡 = −4𝑟𝑡 when 𝑡 = 0, 𝑟 = 𝑟o. 2 A. 𝑟 = 2𝑟o 𝑒 −2𝑡 C. 𝑟 = 2𝑡 2 𝑟o 𝑒 2 B. 𝑟 = 𝑟o 𝑒 −𝑡 D. 𝑟 = −2𝑡 2 𝑟o 𝑒 Problem 8 Obtain the general solution of the differential 𝑥𝑦𝑑𝑥 − (𝑥 + 2)𝑑𝑦 = 0 A. 𝑒 𝑥 = 𝑦(𝑥 + 2)2 C. 𝑥 𝑒 = 𝑐𝑦(𝑥 + 2) B. 𝑒 𝑥 = 𝑐𝑦(𝑥 + 2)2 𝑒 𝑥 = 2𝑐𝑦(𝑥 + 2) Problem 9 Obtain the general solution of 𝑦 ′ = 𝑥𝑦 2 . A. 𝑦(𝑥 2 + 𝑐 ) + 2 = 0 (𝑥 2 + 𝑐 ) + 2 = 0 B. 𝑦(𝑥 2 + 𝑐 ) − 2 = 0 𝑦 (𝑥 + 𝑐 ) − 2 = 0 D. Problem 16 Solve the equation 𝑥 𝑑𝑦 − 𝑦 𝑑𝑥 = 2𝑥 3 𝑑𝑥. A. 𝑦 = 𝑥 2 + 𝑐𝑥 C. 𝑦 = 3 𝑥 + 𝑐𝑥 B. 𝑦 = 𝑥 3 + 𝑐 D. 𝑦 = 2 3 𝑥 + 𝑐𝑥 C. D. Problem 17 Solve the equation 𝑥 𝑑𝑦 + 𝑦 𝑑𝑥 = 2𝑥 2 𝑦 𝑑𝑥. A. ln|𝑥𝑦| = 𝑥 3 + 𝑐 C. ln|𝑥𝑦| = 𝑥 2 + 𝑐 B. ln|𝑥 + 𝑦| = 𝑥 2 + 𝑐 D. 2 | | Ln 𝑥 − 𝑦 = 2𝑥 + 𝑐 Problem 10 Solve the equation 𝑥𝑦𝑑𝑥 − (𝑥 + 2𝑦)2𝑑𝑦 = 0. A. 𝑒 𝑥⁄𝑦 = 𝑐 𝑦 3 (𝑥𝑦) C. 𝑥 ⁄𝑦 3 𝑒 = 𝑐 𝑦 (𝑥 + 𝑦) B. 𝑒 𝑥⁄𝑦 = 𝑐 𝑦 3 (𝑥 − 𝑦) D. 𝑥 ⁄𝑦 4( 𝑒 = 𝑐 𝑦 𝑥 − 1) Problem 18 𝑑𝑦 2 Solve the equation 𝑑𝑥 + 𝑥 𝑦 = 6𝑥 3 . Problem 11 Obtain the particular solution of (3𝑥 2 − 2𝑦 2 )𝑦 ′ = 2𝑥𝑦; when 𝑥 = 0, 𝑦 = −1. A. 𝑥 2 = 2𝑦 2 (𝑦 + 1) C. 𝑥 2 = 2𝑦 2 (𝑦 − 1) B. 𝑥 2 = 𝑦 2 (𝑦 + 1) D. 𝑥 2 = 𝑦 2 (𝑦 + 1) A. 𝑥 2𝑦 = 6𝑥 5 + 𝑐 𝑥 2 𝑦 2 = 6𝑥 5 + 𝑐 B. 𝑥 2 = 5𝑥 6 + 𝑐 𝑥 2𝑦 = 𝑥 6 + 𝑐 Problem 12 Solve the equation (2𝑥𝑦 + 𝑦)𝑑𝑥 + (𝑥 2 − 𝑥 )𝑑𝑦 = 0. A. 𝑦 = 𝑐𝑥(𝑥 + 1)3 + 2𝑥 C. 𝑦 = 3𝑥 + 𝑐𝑥 (𝑥 − 1)3 B. 𝑦 = 𝑐𝑥(𝑥 + 1)−3 − 2 D. 𝑦 = −3 ( ) 𝑐𝑥 𝑥 − 1 Problem 15 Solve (𝑥 2 + 𝑦) 𝑑𝑥 + (𝑦 3 + 𝑥 ) 𝑑𝑦 = 0. 𝑥3 3 +𝑥+ 𝑥𝑦 − B. 𝑥3 3 𝑥3 3 𝑦4 4 𝑦4 4 =𝑐 C. =𝑐 +𝑥+𝑦+ − 𝑥𝑦 2 + 𝑦4 𝑦4 4 4 =𝑐 =𝑐 D. Problem 20 𝑑𝑦 𝑑𝑥 = 3𝑥 2 +𝑦 2 2𝑥𝑦 exact by using the integrating factor: A. 𝑥 2 B. 1/𝑥 2 1/𝑦 2 Problem 14 Solve the equation (𝑟 + sin 𝜃 − cos 𝜃) 𝑑𝑟 + 𝑟(sin 𝜃 + cos 𝜃) 𝑑𝜃 = 0. A. 𝑟 2 + 𝑟(sin 𝜃 + cos 𝜃) = 𝑐 C. 𝑟 − 2𝑟(sin 𝜃 − cos 𝜃) = 𝑐 B. 𝑟 2 + 2𝑟(sin 𝜃 − cos 𝜃) = 𝑐 D. 3 𝑟 + 2𝑟(sin 𝜃 + cos 𝜃) = 𝑐 A. D. Problem 19 Solve the equation (𝑥 5 + 3𝑦)𝑑𝑥 − 𝑥 𝑑𝑦 = 0. A. 2𝑦 = 𝑥 5 + 𝑐𝑥 3 C. 3 2 2𝑦 = 𝑥 + 𝑐𝑥 B. 𝑦 = 𝑥 5 + 𝑐𝑥 3 D. 𝑦 2 = 5𝑥 4 + 𝑐𝑥 3 The differential equation Problem 13 Solve the equation (6𝑥 + 𝑦 2 )𝑑𝑥 + 𝑦(2𝑥 − 3𝑦)𝑑𝑦 = 0. A. 𝑥𝑦 2 + 3𝑥 2 − 𝑦 3 = 𝑐 C. 2 2 3 𝑥𝑦 + 2𝑥 − 3𝑦 = 𝑐 B. 3𝑥𝑦 2 − 3𝑥 2 − 𝑦 3 = 𝑐 D. 2 2 2 𝑥𝑦 + 3𝑥 − 𝑦 = 𝑐 C. 𝑥3 3 + can be made C. 𝑦 2 D. Problem 21 What is not true for the differential equation 𝑦 ′ + 𝑦/𝑥 2 = 1/𝑥 2 ? A. it is linear B. it is homogeneous C. it is separable D. it can be solved using the integrating factor 𝑒 −1/𝑥 Problem 22 A tank contains 400 liters of brine holding 100 kg of salt in solution. Water containing 125 g of salt per liter flows into the tank at the rate of 12 liters per minute, and the mixture, kept uniform by stirring, flows out at the same rate. Find the amount of salt at the end of 90 minutes. A. 53.36 kg C. 53.63 kg B. 0 D. 65.33 kg Problem 23 Under certain conditions, cane sugar in water is converted into dextrose at a rate proportional to the amount that is unconverted at any time. If, of 75 kg at time t = 0, 8kg are converted during the first 30 minutes, find the amount converted in 2 hours. A. 72.73 kg C. 27.23 kg B. 23.27 kg D. 32.72 kg Problem 24 A thermometer reading 18 oC is brought into a room where the temperature is 70 oC; 1 minute later the thermometer reading is 31 oC. Determine the thermometer reading 5 minutes after it is brought into the room. A. 62.33 oC C. 56.55 oC B. 58.99 oC D. 57.66 oC Problem 25 Solve the equation (4𝑥𝑦 + 3𝑦 2 − 𝑥 )𝑑𝑥 + 𝑥(𝑥 + 2𝑦)𝑑𝑦 = 0. A. 4𝑥𝑦 + 4𝑦 2 − 𝑥 = 𝑐 C. 𝑥 2 (4𝑥𝑦 − 4𝑦 2 − 𝑥 ) = 𝑐 B. 𝑥 3 (4𝑥𝑦 + 4𝑦 2 − 𝑥 ) = 𝑐 D. 𝑥 3 (4𝑥 2 𝑦 + 4𝑦 2 + 𝑥 2 ) = 𝑐 Problem 26 The equation 𝑦 2 = 𝑐𝑥 is the general equation of A. 𝑦 ′ = 2𝑥/𝑦 C. 𝑦 ′ = 𝑦/2𝑥 B. 𝑦 ′ = 2𝑦/𝑥 D. ′ 𝑦 = 𝑥/2𝑦 Problem 27 Given the following simultaneous differential equations: 2𝑑𝑥/𝑑𝑡 − 3𝑑𝑦/𝑑𝑡 + 𝑥 − 𝑦 = 𝑘 3𝑑𝑥/𝑑𝑡 + 2𝑑𝑦/𝑑𝑡 − 𝑥 + cos 𝑡 = 0. Solve for 𝑑𝑦/𝑑𝑡. A. (2/9)[cos 𝑡 + (3/2)𝑥 − (5/2)𝑦 − (3/2)𝑘] B. (−1/6)[sin 𝑡 + (1/9)𝑥 + 𝑦^2 − (3/2)𝑘] C. (1/13)(5𝑥 − 𝑦 − 3𝑘 − 2 cos 𝑡) D. (2/13)[cos 𝑡 + (5/2)𝑥 − (3/2)𝑦 − (3/2)𝑘] Problems – Statics, Translation, Rotation Set 23 Problem 1 The weight of a mass of 10 kilograms at a location where g = 9.77 m/s2 is: A. 79.7 N C. 97.7 N B. 77.9 N D. 977 N Problem 2 (ME April 1997) What is the resultant velocity of a point of xcomponent 𝑉x = 𝑡 3 − 1, and y-component 𝑉y = 𝑡 2 − 𝑡 at time 𝑡 = 4? A. 63.1327 C. 64.1327 B. 62.1327 D. 74.1327 Problem 3 A boat has a speed of 8 mph in still water attempts to go directly across a river with a current of 3 mph. What is the effective speed of the boat? A. 8.35 mph C. 7.42 mph B. 8.54 mph D. 6.33 mph Problem 4 A ship moving North at 10 mph. A passenger walks Southeast across the deck at 5 mph. In what direction and how fast is the man moving, relative to the earth’s surface. A. N 28o40’W; 7.37 mph C. N o 61 20’W; 7.37 mph B. N 61o20’E; 7.37 mph D. N 28o40’E; 7.37 mph Problem 5 A man wishes to cross due west on a river which is flowing due north at the rate of 3 mph. If he can row 12 mph in still water, what direction should he take to cross the river? A. S 14.47o W C. S 81.36 o W o B. S 75.52 W D. S 84.36 o W Problem 6 A plane is moving due east with air speed of 240 kph. If a wind of 40 kph is blowing from the south, find the ground speed of the plane. A. 243 kph C. 200 kph B. 423 kph D. 240 kph Problem 7 Three forces 20N, 30N, and 40N are in equilibrium. Find the angle between the 30-40N. A. 26.96o C. 40o B. 28.96o D. 25.96o Problem 8 A 10-kg weight is suspended by a rope from a ceiling. If a horizontal force of 5.80 kg is applied to the weight, the rope will make an angle with the vertical equal to: A. 60o C. 45o o B. 30 D. 75o Problem 9 A 100 kN block slides down a place inclined at an angle of 30o with the horizontal. Neglecting friction, find the force that causes the block to slide. A. 86.6 kN C. 20 kN B. 80 kN D. 50 kN Problem 10 What tension must be applied at the ends of a flexible wire cable supporting a load of 0.5 kg per horizontal meter in a span of 100 m if the sag is to be limited to 1.25 m? A. 423.42 kg C. 500.62 kg B. 584.23 kg D. 623.24 kg Problem 11 The allowable spacing of towers to carry an aluminum cable weighing 0.03 kg per horizontal meter if the maximum tension at the lowest point is not to exceed 1150 at sag of 0.50 m is: A. 248 m C. 408 m B. 392 m D. 422 m Problem 12 A wooden plank “x” meters long has one end leaning on top of a vertical wall 1.5 m high and the other end resting on a horizontal ground. Neglecting friction, find x if a force (parallel to the plank) of 100 N is needed to pull a 400 N block up the plank. A. 6 m C. 4 m B. 5 m D. 3 m Problem 13 A block of wood is resting on a level surface. If the coefficient of friction between the block and the surface is 0.30, how much can the plane be inclined without causing the block to slide down? A. 16.7o C. 21.2o o B. 30.2 D. 33.3o Problem 14 A 500 kg block is resting on a 30 o inclined plane with a 𝜇 = 0.3. Find the required force P acting horizontally that will prevent the block from sliding. A. 1020 N C. 4236 N B. 1160 N D. 5205 N Problem 15 A 500 kg block is resting on a 30 o inclined plane with a 𝜇 = 0.3. Find the required force P acting horizontally that will start the block up the plane. A. 4236 N C. 5205 N B. 1160 N D. 2570 N Problem 16 (ME April 1996) What is the acceleration of the body that increases in velocity from 20 m/s to 40 m/s in 3 seconds? Answer in S.I. units. A. 8 m/s2 C. 5 m/s2 2 B. 6.67 m/s D. 7 m/s2 Problem 17 (CE May 1996) From a speed of 75 kph, a car decelerates at the rate of 500 m/min2 along a straight path. How far in meters will it travel in 45 sec? A. 795 C. 797 B. 791 D. 793 Problem 18 (CE November 1997) With a starting speed of 30 kph at point A, a car accelerates uniformly. After 18 minutes, it reaches point B, 21 km from A. Find the acceleration of the car in m/s2. A. 0.126 m/s2 C. 0.0206 m/s2 2 B. 0.0562 m/s D. 3.42 m/s2 Problem 19 (CE November 1996) A train upon passing point A at a speed of 72 kph accelerates at 0.75 m/s2 for one minute along a straight path the decelerates at 1.0 m/s2. How far in kilometers from point A will it be 2 minutes after passing point A? A. 4.95 C. 4.85 B. 4.75 D. 4.65 Problem 20 A car starting from rest moves with a constant acceleration of 10 km/hr2 for 1 hour, then decelerates at a constant -5 km/hr2 until it comes to a stop. How far has it traveled? A. 10 km C. 12 km B. 20 km D. 15 km Problem 21 (ECE November 1997) The velocity of an automobile starting from rest is given by 𝑑𝑠/𝑑𝑡 = 90𝑡 / (𝑡 + 10) ft/sec. Determine its acceleration after an interval of 10 seconds (in ft/sec2). A. 2.10 C. 2.25 B. 1.71 D. 2.75 Problem 22 (CE may 1998) A train running at 60 kph decelerated at 2 m/min2 for 14 minutes. Find the distance traveled, in kilometers within this period. A. 12.2 C. 13.8 B. 13.2 D. 12.8 Problem 23 (ECE November 1997) An automobile accelerates at a constant rate of 15 mi/hr to 45 mi/hr in 15 seconds, while travelling in a straight line. What is the average acceleration? A. 2 ft/s2 C. 2.12 ft/s2 B. 2.39 ft/s2 D. 2.93 ft/s2 Problem 24 (CE November 1998) A car was travelling at a speed of 50 mph. The driver saw a road block 80 m ahead and stepped on the brake causing the car to decelerate uniformly at 10 m/s2. Find the distance from the roadblock to the point where the car stopped. Assume perception reaction time is 2 seconds. A. 12.48 m C. 10.28 m B. 6.25 m D. 8.63 m Problem 25 A man driving his car at 45 mph suddenly sees an object in the road 60 feet ahead. What constant deceleration is required to stop the car in this distance? A. -36.3 ft/s2 C. -33.4 ft/s2 2 B. -45.2 ft/s D. -42.3 ft/s2 Problem 26 (ECE March 1996) A mango falls from a branch 5 meters above the ground. With what speed in meters per second will it strike the ground? Assume g = 10 m/s2. A. 8 C. 10 B. 12 D. 14 Problem 27 A man standing at a window 5 meters tall watches a falling stone pass by the window in 0.3 seconds. From how high above the top of the window was the stone released? A. 12.86 m C. 9.54 m B. 11.77 m D. 15.21 m Problem 28 A bullet is fired at an initial velocity of 350 m/s and an angle of 50o with the horizontal. Neglecting air resistance, what maximum height could the bullet rises? A. 3,646 m C. 3,466 m B. 4,366 m D. 3,664 m Problem 29 A bullet is fired at an initial velocity of 350 m/s and an angle of 50o with the horizontal. Neglecting air resistance, find its range on the horizontal plane through the point it was fired. A. 12,298 m C. 12.298 m B. 12.298 km D. 12,298 cm Problem 30 A bullet is fired at an initial velocity of 350 m/s and an angle of 50o with the horizontal. Neglecting air resistance, how long will the bullet travel before hitting the ground? A. 54.66 min. C. 54.66 sec B. 56.42 sec D. 56.42 min. Problem 31 (ME Board October 1997) The muzzle velocity of a projectile is 1,500 fps and the distance of the target is 10 miles. What must be the angle of elevation of the gun? A. 25o 32’ C. 24o 32’ B. 23o 34’ D. 26o 34’ Problem 32 (ME October 1997) A shot is fired at an angle of 45o with the horizontal and a velocity of 300 ft per second. Find the height and the range of projectile. A. 600 ft and 2500 ft C. 1000 ft and 4800 ft B. 700 ft and 2800 ft D. 750 ft and 3000 ft Problem 33 (ECE April 1998) A baseball is thrown from a horizontal plane following a parabolic path with an initial velocity of 100 m/s at an angle of 30 degrees above the horizontal. How far from the throwing point will the ball attain its original level? A. 890 m C. 883 m B. 875 m D. 880 m Problem 34 What is the range of a projectile if the initial velocity is 30 m/s at an angle of 30o with the horizontal? A. 100 m C. 79.45 m B. 92 m D. 110 m Problem 35 A bullet is fired at an angle of 75o with the horizontal with an initial velocity of 420 m/s. How high can it travel after 2 seconds? A. 840 m C. 750 m B. 792 m D. 732 m Problem 36 A coin is tossed vertically upward from the ground at a velocity of 12 m/s. How long will the coin touch the ground? A. 4.35 sec. C. 2.45 sec. B. 3.45 sec. D. 1.45 sec. Problem 37 (CE May 1997) A stone is projected from the ground with a velocity of 15 m/s at an angle of 30o with the horizontal ground. How high in m will it rise? Use g = 9.817 m/s. A. 2.865 m C. 2.586 m B. 2.685 m D. 8.652 m Problem 38 (CE November 1996) A ball is thrown from a tower 30m. high above the ground with a velocity of 300 m/s directed at 20o from the horizontal. How fast will the ball hit the ground? A. 24.2 C. 21.2 B. 23.2 D. 22.2 Problem 39 A man in a hot air balloon dropped an apple at a height of 150 m. If the balloon is rising at 15 m/s, find the highest point reached by the apple. A. 151.5 m C. 171.5 m B. 161.5 m D. 141.5 m Problem 40 A balloon is ascending at the rate of 10 kph and is being carried horizontally by a wind at 20 kph. If a bomb is dropped from the balloon such that it takes 8 sec. to reach the ground, the balloon’s altitude when the bomb was released is: A. 336.14 m C. 252 m B. 322.13 m D. 292 m Problem 41 A plane is flying horizontally 350 kph at an altitude of 420 m. At this instant, a bomb is released. How far horizontally from this point will the bomb hit the ground? A. 625 m C. 785 m B. 577 m D. 900 m Problem 42 A car whose wheels are 30 cm in radius is traveling with a velocity of 110 kph. If it is decelerated at a constant rate of 2 m/s2, how many complete revolutions does it make before it comes to rest? A. 121 C. 123 B. 122 D. 124 Problem 43 The wheel of an automobile revolves at the rate of 700 rpm. How fast does it move, in km per hr., if the radius of its wheel is 250 mm? A. 73.3 C. 5.09 B. 18.33 D. 34.67 Problem 44 Using a constant angular acceleration, a water turbine is brought to its normal operating speed of 180 rev/min in 6 minutes. How many complete revolutions did the turbine make in coming to normal speed? A. 550 C. 560 B. 530 D. 540 Problem 45 A horizontal platform 6 m in diameter revolves so that a point on its rim moves 6.28 m/s. Find its angular speed in rpm. A. 15 C. 25 B. 20 D. 12 Problem 46 (CE May 1998) A horizontal platform with a diameter of 6 m revolves about its center at 20 rpm. Find the tangential speed, in m/s of a point at the edge of the platform. A. 6.34 C. 6.46 B. 6.28 D. 6.12 Problem 47 A flywheel rotating at 500 rpm decelerates uniformly at 2 rad/sec2. How many seconds will it take for the flywheel to stop? A. 24.5 s C. 25.1 s B. 28.4 s D. 26.2 s Problem 48 A cyclist on a circular track of radius r = 800 feet is traveling at 27 ft/sec. His speed in the tangential direction increases at the rate of 3 ft/s2. What is the cyclist’s total acceleration? A. 2.8 ft/s2 C. -5.1 ft/s2 2 B. -3.12 ft/s D. 3.13 ft/s2 Problem 49 The radius of the earth is 3,960 miles. The gravitational acceleration at the earth’s surface is 32.16 ft/s2. What is the velocity of escape from the earth in miles/s? A. 6.94 C. 9.36 B. 8.62 D. 7.83 Problem 50 The radius of the moon is 1080 mi. The gravitational acceleration at the moon’s surface is 0.165 times the gravitational acceleration at the earth’s surface. What is the velocity of escape from the moon in miles/second? A. 2.38 C. 3.52 B. 1.47 D. 4.26 Problems – Kinetics, Work, Energy, Momentum, Etc. Set 24 Problem 1 (ME October 1997) A 10-lbm object is acted upon by a 40-lb force. What is the acceleration in ft/min2? A. 8.0 x 10 to the 4th power ft/min2 C. 7.8 x 10 to the 4th power ft/min2 B. 9.2 x 10 to the 4th power ft/min2 D. 4.638 x 10 to the 4th power ft/min2 Problem 2 What horizontal force P can be applied to a 100-kg block in a level surface with coefficient of friction of 0.2, that will cause and acceleration of 2.50 m/s2? A. 343.5 N C. 106 N B. 224.5 N D. 446.2 N Problem 3 A skier wishes to build a rope tow to pull her up a ski hill that is inclined at 15o with the horizontal. Calculate the tension needed to give the skier’s 54kg body an acceleration of 1.2 m/s2. Neglect friction. A. 202 N C. 106 N B. 403 N D. 304 N Problem 4 (ME April 1997) A pick-up truck is traveling forward at 25 m/s. The truck bed is loaded with boxes, whose coefficient of friction with the bed is 0.4. What is the shortest time that the truck can be brought to a stop such that the boxes do not shift? A. 4.75 sec C. 5.45 sec B. 2.35 sec D. 6.37 sec Problem 5 (CE November 1996) A 40-kg block is resting on an inclined plane making an angle of 20o from the horizontal. If the coefficient of friction is 0.60, determine the force parallel to the incline that must be applied to cause impending motion down the plane. Use g = 9.81 m/s. A. 77 C. 72 B. 82 D. 87 Problem 6 (ECE November 1997) A 50-kilogram block of wood rest on the top of the plane whose length is 3 m. and whose altitude is 0.8 m. How long will it take for the block to slide to the bottom of the plane when released? A. 1.51 seconds C. 2.51 seconds B. 2.41 seconds D. 2.14 seconds Problem 7 (CE May 1999) A body weighs 40 lbs. starts from rest and inclined on a plane at an angle of 30o from the horizontal for which the coefficient of friction 𝜇 = 0.3. How long will it move during the third second? A. 19.99 ft C. 18.33 ft B. 39.63 ft D. 34.81 ft Problem 8 A car and its load weigh 27 kN and the center of gravity is 600 mm from the ground and midway between the front and rear wheel which are 3 m apart. The car is brought to rest from a speed of 54 kph in 5 seconds by means of the brakes. Compute the normal force on each of the front wheels of the car. A. 7.576 kN C. 5.478 kN B. 9.541 kN D. 6 kN Problem 9 (ME April 1998, CE November 1999 “Structural”) An elevator weighing 2,000 lb attains an upward velocity of 16 fps in 4 sec with uniform acceleration. What is the tension in the supporting cables? A. 1,950 lb C. 2495 lb B. 2,150 lb D. 2,250 lb Problem 10 (CE November 1997 “Structural”) A block weighing 200 N rests on a plane inclined upwards to the right at a slope of 4 vertical to 3 horizontal. The block is connected to a cable initially parallel to the plane, passing through the pulley and connected to another block weighing 100 N moving vertically downward. The coefficient of kinetic friction between the 200 N block and the inclined plane is 0.10. Which of the following most nearly gives the acceleration of the system? A. a = 2.93 m/sec2 C. a = 1.57 2 m/sec B. a = 0.37 m/sec2 D. a = 3.74 2 m/sec Problem 11 (ME October 1997) A car travels on the horizontal unbanked circular track of radius r. Coefficient of friction between the tires and the track is 0.3. If the car’s velocity is 10 m/s, what is the smallest radius it may travel without skidding? A. 50 m C. 15 m B. 60 m D. 34 m Problem 12 If a car travels at 15 m/s and the track is banked 5o, what is the smallest radius it can travel so that friction will not be necessary to resist sliding? A. 262.16 m C. 278.14 m B. 651.23 m D. 214.74 m Problem 13 (CE May 1999) A vertical bar of length L with a mass of 40 kg is rotated vertically about its one end at 40 rpm. Find the length of the bar if it makes an angle of 45o with the vertical? A. 1.58 m C. 3.26 m B. 2.38 m D. 1.86 m Problem 14 The seats of a carousel are attached to a vertical rotating shaft by a flexible cable 8 m long. The seats have a mass of 75 kg. What is the maximum angle of tilt for the seats if the carousel operates at 12 rpm? A. 30o C. 45o B. 35o D. 39o Problem 15 (CE November 1998) A highway curve is super elevated at 7o. Find the radius of the curve if there is no lateral pressure on the wheels of a car at a speed of 40 mph. A. 247.4 m C. 229.6 m B. 265.6 m D. 285.3 m Problem 16 (CE November 1997 “Structural”) A 2-N weight is swung in a vertical circle of 1-m radius at the end of the cable that will break if the tension exceeds 500 N. Find the angular velocity of the weight when the cable breaks: A. 49.4 rad/s C. 24.9 rad/s B. 37.2 rad/s D. 58.3 rad/s Problem 17 (ME April 1998) Traffic travels at 65 mi/hr around a banked highway curve with a radius of 3000 ft. What banking angle is necessary such that friction will not be required to resist the centrifugal force? A. 5.4o C. 3.2o o B. 18 D. 2.5o Problem 18 (ME April 1997) A concrete highway curve with a radius of 500 feet is banked to give a lateral pressure equivalent to f = 0.15. For what coefficient of friction will skidding impend for a speed of 60 mph? A. < 0.360 C. > 0.310 B. < 0.310 D. > 0.360 Problem 19 (ME October 1997) A 3500 lbf car is towing a 500 lbf trailer. The coefficient of friction between all tires and the road is 0.80. How fast can the car and trailer travel around an unbanked curve of radius 0.12 mile without either the car or trailer skidding? A. 87 mph C. 26 mph B. 72 mph D. 55 mph Problem 20 (ME October 1997) A cast-iron governor ball 3 inches in diameter has its center 18 inches from the point of support. Neglecting the weight of the arm itself, find the tension in the arm if the angle with the vertical axes is 60o. A. 7.63 lb C. 7.56 lb B. 6.36 lb D. 7.36 lb Problem 21 An object is placed 3 feet from the center of a horizontally rotating platform. The coefficient of friction is 0.3. The object will begin to slide off when the platform speed is nearest to: A. 17 rpm C. 22 rpm B. 12 rpm D. 27 rpm Problem 22 (ME April 1998) A force of 200 lbf acts on a block at an angle of 28o with respect to horizontal. The block is pushed 2 feet horizontally. What is the work done by this force? A. 320 J C. 480 J B. 540 J D. 215 J Problem 23 (ME April 1998) A 10-kg block is raised vertically 3 meters. What is the change in potential energy? Answer in SI units closest to: A. 350 N-m C. 350 kg2 2 m /s B. 294 J D. 320 J Problem 24 At her highest point, a girl on the swing is 7 feet above the ground, and at her lowest point, she is 3 feet above the ground. What is her maximum velocity? A. 10 fps C. 14 fps B. 12 fps D. 16 fps Problem 25 An automobile has a power output of 1 hp. When it pulls a cart with a force of 300 N, what is the cart’s velocity? A. 249 m/s C. 2.49 m/s B. 24.9 m/s D. 0.249 m/s Problem 26 A hunter fires a 50 gram bullet at a tiger. The bullet left the gun with a speed of 600 m/s. What is the momentum of the bullet? A. 15 kg-m/s C. 300 kg-m/s B. 30 kg-m/s D. 150 kg-m/s Problem 27 An elevator can lift a load of 5000 N from the ground level to a height of 20 meters in 10 seconds. Find the horsepower rating of the elevator. A. 10000 C. 13.4 B. 13400 D. 1340 Problem 28 The average horsepower required to raise a 150-kg box to a height of 20 meters over a period of one minute is: A. 450 hp C. 2960 hp B. 0.657 hp D. 785 hp Problem 29 What is the force of attraction between two 90-kg bodies spaced 40 m apart? Assume gravitational constant, G = 6.67 x 10-11 N-m2/kg2 A. 45.6 × 10−6 N C. 4.26 × −6 10 N B. 3.38 × 10−6 N D. 33.8 × −6 10 N Problem 30 What is the efficiency of the pulley system, which lifts a 1 tonne load, a distance of 2 m by the application of the force 150 kg for a distance of 15 m? A. 11% C. 75% B. 46% D. 89% Problem 31 How much mass is converted to energy per day in a nuclear power plant operated at a level of 100 MW? A. 9.6 × 10−4 kg C. 9.6 × 10−5 kg B. 9.6 × 10−7 kg D. 9.6 × 10−6 kg Problem 32 (ECE April 1998) What is the kinetic energy of a 4000-lb automobile, which is moving at 44 ft/s? A. 2.10 × 105 ft-lb C. 1.12 × 105 ft-lb B. 1.20 × 105 ft-lb D. 1.8 × 105 ft-lb Problem 33 (ME October 1997) A 4000-kg elevator starts from rest accelerates uniformly to a constant speed of 2.0 m/s and decelerates uniformly to stop 20 m above its initial position. Neglecting the friction and other losses, what work was done on the elevator? A. 785 × 103 Joule C. 900 × 103 Joule B. 700 × 103 Joule D. 685 × 103 Joule Problem 34 The brakes of a 1000-kg automobile exert 3000 N. How long will it take for the car to come to a complete stop from a velocity of 30 m/s? A. 15 sec C. 5 sec B. 10 sec D. 2 sec Problem 35 (ME April 1997) A car weighing 40 tons is switched to a 2% upgrade with a velocity of 30 mph. If the train resistance is 10 lbs/ton, how far does the grade will it go? A. 1124 ft C. 1204 ft B. 2104 ft D. 1402 ft Problem 36 (ME October 1997) A body weighing 1000 lbs. fall 6 inches and strikes a 2000 lbs (per inch) spring. What is the deformation of the spring? A. 3 inches C. 100 mm B. 6 inches D. 2 inches Problem 37 A 16-gram mass is moving at 30 cm/s while a 4gram mass is moving in an opposite direction at 50 cm/s. They collide head on and stick together. Their velocity after collision is: A. 14 cm/s C. 13 cm/s B. 15 cm/s D. 18 cm/s Problem 38 A 5-kg rifle fires a 15-g bullet at a muzzle velocity of 600 m/s. What is the recoil velocity of the rifle? A. 1800 m/s C. 18 m/s B. 180 m/s D. 1.80 m/s Problem 39 A 0.50-kg ball with a speed of 20 m/s strikes and sticks to a 70-kg block resting on a frictionless surface. Find the block’s velocity. A. 142 m/s C. 1.42 m/s B. 14.2 m/s D. 0.142 m/s Problems – Stress, Strain, Torsion, Shear & Moment, Etc. Set 25 Problem 1 Determine the outside diameter of a hollow steel tube that will carry a tensile load of 500 kN at a stress of 140 MPa. Assume the wall thickness to be one-tenth of the outside diameter. A. 123 mm C. 103 mm B. 113 mm D. 93 mm Problem 2 (ME April 1998) A force of 10 N is applied to one end of a 10 inches diameter circular rod. Calculate the stress. A. 0.20 kPa C. 0.10 kPa B. 0.05 kPa D. 0.15 kPa Problem 3 What force is required to punch a 20-mm diameter hole through a 10-mm thick plate? The ultimate strength of the plate material is 450 MPa. A. 241 kN C. 386 kN B. 283 kN D. 252 kN Problem 4 A steel pipe 1.5m in diameter is required to carry an internal pressure of 750 kPa. If the allowable tensile stress of steel is 140 MPa, determine the required thickness of the pipe in mm. A. 4.56 C. 4.25 B. 5.12 D. 4.01 Problem 5 A spherical pressure vessel 400-mm in diameter has a uniform thickness of 6 mm. The vessel contains gas under a pressure of 8,000 kPa. If the ultimate stress of the material is 420 MPa, what is the factor of safety with respect to tensile failure? A. 3.15 C. 2.15 B. 3.55 D. 2.55 Problem 6 (CE November 1996) A metal specimen 36-mm in diameter has a length of 360 mm and a force of 300 kN elongates the length bar to 1.20-mm. What is the modulus of elasticity? A. 88.419 GPa C. 92.658 GPa B. 92.564 GPa D. 95.635 GPa Problem 7 During a stress-strain test, the unit deformation at a stress of 35 MPa was observed to be 167 × 10−6 m/m and at a stress of 140 MPa it was 667 × 10−6 m/m. If the proportional limit was 200 MPa, what is the modulus of elasticity? What is the strain corresponding to stress of 80 MPa? A. E = 210,000 MPa; 𝜀 = 381 × 10−4 m/m B. E = 200,000 MPa; 𝜀 = 318 × 10−6 m/m C. E = 211,000 MPa; 𝜀 = 318 × 10−4 m/m D. E = 210,000 MPa; 𝜀 = 381 × 10−6 m/m Problem 8 An axial load of 100 kN is applied to a flat bar 20 mm thick, tapering in width from 120 mm to 40 mm in a length of 10 m. Assuming E = 200 GPa, determine the total elongation of the bar. A. 3.43 mm C. 4.33 mm B. 2.125 mm D. 1.985 mm Problem 9 Steel bar having a rectangular cross-section 15mm × 20mm and 150m long is suspended vertically from one end. The steel has a unit mass of 7850 kg/m3 and a modulus of elasticity E of 200 GPa. If a loaf of 20 kN is suspended at the other end of the rod, determine the total elongation of the rod. A. 43.5 mm C. 35.4 mm B. 54.3 mm D. 45.3 mm Problem 10 A steel bar 50 mm in diameter and 2 m long is surrounded by a shell of cast iron 5 mm thick. Compute the load that will compress the bar a total of 1 mm in the length of 2 m. Use Esteel = 200 GPa and Ecast-iron = 100 GPa. A. 200 kN C. 280 kN B. 240 kN D. 320 kN Problem 11 A 20-mm diameter steel rod, 250 mm long is subjected to a tensile force of 75 kN. If the Poisson’s ratio 𝜇 is 0.30, determine the lateral strain of the rod. Use E = 200 GPa. A. 𝜀y = 3.581 × 10−4 mm/mm C. 𝜀y = −4 −2.467 × 10 mm/mm B. 𝜀y = −3.581 × 10−4 mm/mm D. 𝜀y = −4 2.467 × 10 mm/mm Problem 12 A solid aluminum shaft of 100-mm diameter fits concentrically in a hollow steel tube. Determine the minimum internal diameter of the steel tube so that no contact pressure exists when the aluminum shaft carries an axial compressive load of 600 kN. Assume Poisson’s ratio 𝜇 = 1/3 and the modulus of elasticity of aluminum E be 70 GPa. A. 100.0364 mm C. 100.0303 mm B. 100.0312 mm D. 100.0414 mm Problem 13 (CE May 1996) The maximum allowable torque, in kN-m, for a 50mm diameter steel shaft when the allowable shearing stress is 81.5 MPa is: A. 3.0 C. 4.0 B. 1.0 D. 2.0 Problem 14 (CE May 1997) The rotation or twist in degrees of a shaft, 800 mm long subjected to a torque of 80 N-m, 20 mm in diameter and shear modulus G of 80,000 MPa is: A. 3.03 B. 4.04 C. 2.92 D. 1.81 Problem 15 Compute the value of the shear modulus G of steel whose modulus of elasticity E is 200 GPa and Poisson’s ratio 𝜇 is 0.30. A. 72,456 MPa C. 79,698 MPa B. 76,923 MPa D. 82,400 MPa Problem 16 Determine the length of the shortest 2-mm diameter bronze wire, which can be twisted through two complete turns without exceeding a stress of 70 MPa. Use G = 35 GPa. A. 6.28 m C. 6.89 m B. 5.23 m D. 8.56 m Problem 17 A hollow steel shaft 2540 mm long must transmit torque of 34 kN-m. The total angle of twist must not exceed 3 degrees. The maximum shearing stress must not exceed 110 MPa. Find the inside diameter and the outside diameter of the shaft the meets these conditions. Use G = 83 GPa. A. D = 129 mm; d = 92 mm C. D = 132 mm; d = 100 mm B. D = 125 mm; d = 65 mm D. D = 112 mm; d = 85 mm Problem 18 Determine the maximum shearing stress in a helical steel spring composed of 20 turns of 20-mm diameter wire on a mean radius of 80 mm when the spring is supporting a load of 2 kN. A. 110.6 MPa C. 120.6 MPa B. 101.1 MPa D. 136.5 MPa Problem 19 A load P is supported by two springs arranged in series. The upper spring has 20 turns of 29-mm diameter wire on a mean diameter of 150 mm. The lower spring consist of 15 turns of 10-mm diameter wire on a mean diameter of 130 mm. Determine the value of P that will cause a total deflection of 80 mm. Assume G = 80 GPa for both spring. A. 223.3 N C. 214.8 N B. 228.8 N D. 278.4 N Problem 20 A 10-meter long simply supported beam carries a uniform load of 8 kN/m for 6 meters from the left support and a concentrated load of 15 kN 2 meters from the right support. Determine the maximum shear and moment. A. Vmax = 33.2 kN; Mmax = 85.92 KN-m C. Vmax = 36.6 kN; Mmax = 83.72 KN-m B. Vmax = 31.3 kN; Mmax = 81.74 KN-m D. Vmax = 41.8 kN; Mmax = 92.23 KN-m Problem 21 (ECE November 1996) A simple beam, 10 m long carries a concentrated load of 500 kN at the midspan. What is the maximum moment of the beam? A. 1250 kN-m C. 1520 kN-m B. 1050 kN-m D. 1510 kN-m Problem 22 (CE May 1997) A small square 5cm by 5cm is cut out of one corner of a rectangular cardboard 20cm wide by 30cm long. How far, in cm from the uncut longer side, is the centroid of the remaining area? A. 9.56 C. 9.48 B. 9.35 D. 9.67 Problem 23 (ECE April 1998) What is the inertia of a bowling ball (mass = 0.5 kg) of radius 15 cm rotating at an angular speed of 10 rpm for 6 seconds? A. 0.0045 kg-m2 C. 0.005 kgm2 B. 0.001 kg-m2 D. 0.002 kgm2 Problem 24 (ECE November 1997) What is the moment of inertia of a cylinder of radius 5 m and mass of 5 kg? A. 62.5 kg-m2 C. 72.5 kg-m2 2 B. 80 kg-m D. 120 kg-m2 A. 89.36 kPa B. 56.25 kPa Problems – Pressure, Buoyancy, Fluid Flow, Pipes Set 26 Problem 1 The mass of air in a room which is 3m × 5m × 20m is known to be 350 kg. Find its density. A. 1.167 kg/m3 C. 1.617 kg/m3 B. 1.176 kg/m3 D. 1.716 kg/m3 Problem 2 (ME October 1997) One hundred (100) grams of water are mixed with 150 grams of alcohol (𝜌 = 790 kg/cu m). What is the specific gravity of the resulting mixtures, assuming that the two fluids mix completely? A. 0.96 C. 0.63 B. 0.82 D. 0.86 Problem 3 (ME April 1998) One hundred grams of water are mixed with 150 grams of alcohol (𝜌 = 790 kg/cu m). What is the specific volume of the resulting mixtures, assuming that the fluids mix completely? A. 0.88 cu cm/g C. 0.82 cu cm/g B. 1.20 cu cm/g D. 0.63 cu cm/g Problem 4 The pressure 34 meters below the ocean is nearest to: A. 204 kPa C. 344 kPa B. 222 kPa D. 362 kPa Problem 5 (ME April 1997) What is the atmospheric pressure on a planet where the absolute pressure is 100 kPa and the gage pressure is 10 kPa? A. 90 kPa C. 100 kPa B. 80 kPa D. 10 kPa Problem 6 A pressure gage 6 m above the bottom of the tank containing a liquid reads 90 kPa; another gage height 4 m reads 103 kPa. Determine the specific weight of the liquid. A. 6.5 kN/m3 C. 3.2 kN/m3 B. 5.1 kN/m3 D. 8.5 kN/m3 Problem 7 The weight density of a mud is given by w = 10 + 0.5h, where w is in kN/m3 and h is in meters. Determine the pressure, in kPa, at a depth of 5 m. C. 62.5 kPa D. 78.54 kPa Problem 8 (ME April 1997) What is the resulting pressure when one pound of air at 15 psia and 200oF is heated at constant volume to 800oF? A. 28.6 psia C. 36.4 psia B. 52.1 psia D. 15 psia Problem 9 (ECE November 1997) The volume of a gas under standard atmospheric pressure 76 cm Hg is 200 in3. What is the volume when the pressure is 80 cm Hg, if the temperature is unchanged? A. 190 in3 C. 110 in3 3 B. 90 in D. 30.4 in3 Problem 10 A two-meter square plane surface is immersed vertically below the water surface. The immersion is such that the two edges of the square are horizontal. If the top of the square is 1 meter below the water surface, what is the total water pressure exerted on the plane surface? A. 43.93 kN C. 64.76 kN B. 52.46 kN D. 78.48 kN Problem 11 Find the total water pressure on a vertical circular gate, 2 meters in diameter, with its top 3.5 meters below the water surface. A. 138.7 kN C. 169.5 kN B. 107.9 kN D. 186.5 kN Problem 12 (CE Board) An iceberg having specific gravity of 0.92 is floating on salt water of specific gravity of 1.03. If the volume of ice above the water surface is 1000 cu. m., what is the total volume of the ice? A. 8523 m3 C. 9364 m3 B. 7862 m3 D. 6325 m3 Problem 13 A block of wood requires a force of 40 N to keep it immersed in water and a force of 100 N to keep it immersed in glycerin (sp. gr. = 1.3). Find the weight and specific gravity of the wood. A. 0.7 C. 0.9 B. 0.6 D. 0.8 Problem 14 (ME April 1998) Reynolds number may be calculated from: A. diameter, density, and absolute viscosity B. diameter, velocity, and surface tension C. diameter, velocity, and absolute viscosity D. characteristic length, mass flow rate per unit area, and absolute viscosity Problem 15 (ME April 1998) The sum of the pressure load, elevation head, and the velocity head remains constant, this is known as: A. Bernoulli’s Theorem C. Archimedes’ Principle B. Boyle’s Law D. Torricelli’s Theorem Problem 16 (ME October 1997) What is the expected head loss per mile of closed circular pipe (17-in inside diameter, friction factor of 0.03) when 3300 gal/min of water flows under pressure? A. 38 ft C. 3580 ft B. 0.007 ft D. 64 ft kPa, how far apart can they be placed? (Assume f = 0.031) A. 23.7 m C. 12.6 m B. 32.2 m D. 19.8 m Problem 23 A 20-mm diameter commercial steel pipe, 30 m long is used to drain an oil tank. Determine the discharge when the oil level in the tank is 3 m above the exit of the pipe. Neglect minor losses and assume f = 0.12. A. 0.000256 m3/s C. 0.000113 3 m /s B. 0.000179 m3/s D. 0.000869 3 m /s Problem 17 What is the rate of flow of water passing through a pipe with a diameter of 20 mm and speed of 0.5 m/sec? A. 1.24 × 10−4 m3/s C. 1.57 × 10−4 m3/s B. 2.51 × 10−4 m3/s D. 1.87 × 10−4 m3/s Problem 18 An orifice has a coefficient of discharge of 0.62 and a coefficient of contraction of 0.63. Determine the coefficient of velocity for the orifice. A. 0.98 C. 0.97 B. 0.99 D. 0.96 Problem 19 The theoretical velocity of flow through the orifice 3m above the surface of water in a tall tank is: A. 8.63 m/s C. 6.38 m/s B. 9.85 m/s D. 7.67 m/s Problems – Simple Interest, Compound Interest Set 27 Problem 1 Find the interest on P6800.00 for 3 years at 11% simple interest. A. P1,875.00 C. P2,144.00 B. P1,987.00 D. P2,244.00 Problem 20 Oil having specific gravity of 0.869 and dynamic viscosity of 0.0814 Pa-s flows through a cast iron pipe at a velocity of 1 m/s. The pipe is 50 m long and 150 mm in diameter. Find the head lost due to friction. A. 0.73 m C. 0.68 m B. 0.45 m D. 1.25 m Problem 2 A man borrowed P10,000.00 from his friend and agrees to pay at the end of 90 days under 8% simple interest rate. What is the required amount? A. P10,200.00 C. P9,500.00 B. P11,500.00 D. P10,700.00 Problem 21 What commercial size of new cast iron pipe shall be used to carry 4490 gpm with a lost of head of 10.56 feet per smile? Assume f = 0.019. A. 625 mm C. 479 mm B. 576 mm D. 352 mm Problem 3 (EE Board) Annie buys a television set from a merchant who offers P 25,000.00 at the end of 60 days. Annie wishes to pay immediately and the merchant offers to compute the required amount on the assumption that money is worth 14% simple interest. What is the required amount? A. P20,234.87 C. P24,429.97 B. P19,222.67 D. P28,456.23 Problem 22 Assume that 57 liters per second of oil (𝜌 = 860 kg/m3) is pumped through a 300 mm diameter pipeline of cast iron. If each pump produces 685 Problem 4 What is the principal amount if the amount of interest at the end of 2½ year is P4500 for a simple interest of 6% per annum? A. P35,000.00 C. P40,000.00 B. P30,000.00 D. P45,000.00 Problem 5 How long must a P40,000.00 not bearing 4% simple interest run to amount to P41,350.00? A. 340 days C. 304 days B. 403 days D. 430 days Problem 6 If P16,000 earns P480 in 9 months, what is the annual rate of interest? A. 1% C. 3% B. 2% D. 4% Problem 7 (CE May 1997) A time deposit of P110,000 for 31 days earns P890.39 on maturity date after deducting the 20% withholding tax on interest income. Find the rate of interest per annum. A. 12.5% C. 12.25% B. 11.95% D. 11.75% Problem 8 (ME April 1998) A bank charges 12% simple interest on a P300.00 loan. How much will be repaid if the load is paid back in one lump sum after three years? A. P408.00 C. P415.00 B. P551.00 D. P450.00 Problem 9 (CE May 1999) The tag price of a certain commodity is for 100 days. If paid in 31 days, there is a 3% discount. What is the simple interest paid? A. 12.15% C. 22.32% B. 6.25% D. 16.14% Problem 10 Accumulate P5,000.00 for 10 years at 8% compounded quarterly. A. P12,456.20 C. P10,345.80 B. P13,876.50 D. P11,040.20 Accumulate P5,000.00 for 10 years at 8% compounded annually. A. P10,794.62 C. P10,987.90 B. P8,567.98 D. P7,876.87 Problem 14 How long will it take P1,000 to amount to P1,346 if invested at 6% compounded quarterly? A. 3 years C. 5 years B. 4 years D. 6 years Problem 15 How long will it take for an investment to double its amount if invested at an interest rate of 6% compounded bi-monthly? A. 10 years C. 13 years B. 12 years D. 14 years Problem 16 If the compound interest on P3,000.00 in 2 years is P500.00, then the compound interest on P3,000.00 in 4 years is: A. P956.00 C. P1,125.00 B. P1,083.00 D. P1,526.00 Problem 17 The salary of Mr. Cruz is increased by 30% every 2 years beginning January 1, 1982. Counting from that date, at what year will his salary just exceed twice his original salary? A. 1988 C. 1990 B. 1989 D. 1991 Problem 18 If you borrowed P10,000.00 from a bank with 18% interest per annum, what is the total amount to be repaid at the end of one year? A. P11,800.00 C. P28,000.00 B. P19,000.00 D. P10,180.00 Problem 19 What is the effective rate for an interest rate of 12% compounded continuously? A. 12.01% C. 12.42% B. 12.89% D. 12.75% Problem 11 Accumulate P5,000.00 for 10 years at 8% compounded semi-annually. A. P10,955.62 C. P9,455.67 B. P10,233.67 D. P11,876.34 Problem 20 How long will it take for an investment to fivefold its amount if money is worth 14% compounded semi-annually? A. 11 C. 13 B. 12 D. 14 Problem 12 Accumulate P5,000.00 for 10 years at 8% compounded monthly. A. P15,456.75 C. P14,768.34 B. P11,102.61 D. P12,867.34 Problem 21 An interest rate of 8% compounded semiannually is how many percent if compounded quarterly? A. 7.81% C. 7.92% B. 7.85% D. 8.01% Problem 13 Problem 22 A man is expecting to receive P450,000.00 at the end of 7 years. If money is worth 14% compounded quarterly, how much is it worth at present? A. P125,458.36 C. P162,455.63 B. P147,456.36 D. P171,744.44 Problem 23 A man has a will of P650,000.00 from his father. If his father deposited an account of P450,000.00 in a trust fund earning 8% compounded annually, after how many years will the man receive his will? A. 4.55 years C. 5.11 years B. 4.77 years D. 5.33 years Problem 24 Mr. Adam deposited P120,000.00 in a bank who offers 8% interest compounded quarterly. If the interest is subject to a 14% tax, how much will he receive after 5 years? A. P178,313.69 C. P170,149.77 B. P153,349.77 D. P175,343.77 Problem 25 What interest compounded monthly is equivalent to an interest rate of 14% compounded quarterly? A. 1.15% C. 10.03% B. 13.84% D. 11.52% Problem 26 (ME April 1996) What is the present worth of two P100.00 payments at the end of the third and the fourth year? The annual interest rate is 8%. A. P152.87 C. P187.98 B. P112.34 D. P176.67 Problem 27 (ME April 1996) Consider a deposit of P600.00, to be paid up in one year by P700.00. What are the conditions on the rate of interest, i% per year compounded annually, such that the net present worth of the investment is positive? Assume i ≥ 0. A. 0 ≤ i < 16.7% C. 12.5% ≤ i < 14.3% B. 0 ≤ i < 14.3% D. 16.7% ≤ i ≤ 100% Problem 28 (ME April 1996) A firm borrows P2000.00 for 6 years at 8%. At the end of 6 years, it renews the loan for the amount due plus P2000 more for 2 years at 8%. What is the lump sum due? A. P5,679.67 C. P6,034.66 B. P6,789.98 D. P5,888.77 Problem 29 At an annual rate of return of 8%, what is the future worth of P1000 at the end of 4 years? A. P1388.90 C. P1765.56 B. P1234.56 D. P1360.50 Problem 30 (ME October 1997) A student has money given by his grandfather in the amount of P20,000.00. How much money in the form of interest will he get if the money is put in a bank that offers 8% rate compounded annually, at the end of 7 years? A. P34,276.48 C. P36,279.40 B. P34,270.00 D. P34,266.68 Problem 31 (ME October 1997) If the interest rate on an account is 11.5% compounded yearly, approximately how many years will it take to triple the amount? A. 11 years C. 9 years B. 10 years D. 12 years Problem 32 (ME October 1997) The nominal interest rate is 4%. How much is my P10,000.00 worth in 10 years in a continuously compounded account? A. P13,620.10 C. P14,918.25 B. P13,650.20 C. P13,498.60 Problem 33 (ME October 1997) How much must be invested on January 1, year 1, in order to accumulate P2,000.00 on January 1, year 6 at 6%. A. P1,295.00 C. P1,495.00 B. P1,695.00 D. P1,595.00 Problem 34 (ME April 1998) If P5000.00 shall accumulate for 10 years at 8% compounded quarterly. Find the compounded interest at the end of 10 years. A. P6,005.30 C. P6,000.00 B. P6,040.20 D. P6,010.20 Problem 35 (ME April 1998) A sum of P1,000.00 is invested now and left for eight years, at which time the principal is withdrawn. The interest that has accrued is left for another eight years. If the effective annual interest rate is 5%, what will be the withdrawal amount at the end of the 16th year? A. P706.00 C. P500.00 B. P774.00 D. P799.00 Problem 36 (ME April 1998) It is the practice of almost all banks in the Philippines that when they grant a loan, the interest for one year is automatically deducted from the principal amount upon release of money to a borrower. Let us therefore assume that you applied for a loan with a bank and the P80,000.00 was approved at an interest rate of 14% of which P11,200.00 was deducted and you were given a check of P68,800.00. Since you have to pay the amount of P80,000.00 one year after, what then will be the effective interest rate? A. 15.90% C. 16.28% B. 16.30% D. 16.20% Problem 37 (ME April 1998) The amount of P1,500.00 was deposited in a bank account offers a future worth P3,000.00. Interest is paid semi-annually. Determine the interest rate paid on this account. A. 3.5% C. 2.9 B. 2.5% D. 4% Problem 42 (ME October 1995) A company invests P10,000 today to be repaid in five years in one lump sum at 12% compounded annually. If the rate of inflation is 3% compounded annually, how much profit in present day pesos is realized over five years? A. P5,626.00 C. P3,202.00 B. P7,623.00 D. P5,202.00 Problem 43 Compute the effective rate for an interest rate of 16% compounded annually. A. 16% C. 16.98% B. 16.64% D. 17.03% Problem 44 Compute the effective rate for an interest rate of 16% compounded quarterly. A. 16% C. 16.98% B. 16.64% D. 17.03% Problem 38 (ME April 1998) A merchant puts in his P2,000.00 to a small business for a period of six years. With a given interest rate on the investment of 15% per year, compounded annually, how much will he collect at the end of the sixth year? A. P4,400.00 C. P4,390.00 B. P4,200.00 D. P4,626.00 Problem 39 A person invests P4500 to be collected in 8 years. Given that the interest rate on the investment is 14.5% per year compounded annually, what sum will be collected in 8 years? A. P13,678.04 C. P14,888.05 B. P13,294.02 D. P14,234.03 Problem 40 The following schedule of funds is available to form a sinking fund. current year (n) 50,000.00 n+1 40,000.00 n+2 30,000.00 n+3 20,000.00 At the end of the fourth year, equipment costing P250,000.00 will have to be purchased as a replacement for old equipment. Money is valued at 20% by the company. At the time of purchase, how much money will be needed to supplement the sinking fund? A. P12,000.00 C. P10,000.00 B. P11,000.00 D. P9,000.00 Problem 41 (ME October 1995) In year zero, you invest P10,000.00 in a 15% security for 5 years. During that time, the average annual inflation is 6%. How much, in terms of year zero pesos, will be in the account at the maturity? A. P15,386.00 C. P13,382.00 B. P15,030.00 D. P6,653.00 Problem 45 Convert 12% compounded semi-annually to x% compounded quarterly. A. 11.83% C. 11.23% B. 11.71% D. 11.12% Problem 46 Convert 12% compounded semi-annually to x% compounded monthly. A. 11.83% C. 11.23% B. 11.71% D. 11.12% Problem 47 (ME October 1995) A bank is advertising 9.5% accounts that yield 9.84% annually. How often is the interest compounded? A. daily C. bi-monthly B. monthly D. quarterly Problem 48 (ECE November 1995, November 1998) By the conditions of a will, the sum of P25,000 is left to a girl to be held in a trust fund by her guardian until it amount to P45,000. When will the girl receive the money if the fund is invested at 8% compounded quarterly? A. 7.42 years C. 7 years B. 7.25 years D. 6.8 years Problem 49 (ECE April 1995) A man expects to receive P20,000 in 10 years. How much is that money worth now considering interest at 6% compounded quarterly? A. P11,025.25 C. P15,678.45 B. P17,567.95 D. P12,698.65 Problem 50 (ECE March 1996) What is the effective rate corresponding to 16% compounded daily? Take 1 year = 360 days. A. 16.5% C. 17.35% B. 16.78% D. 17.84% Problem 51 What amount will be accumulated by a present investment of P17,200 in 6 years at 2% compounded quarterly? A. P19,387.15 C. P19,856.40 B. P20,456.30 D. P19,232.30 Problem 58 (CE May 1996) P200,000 was deposited on Jan. 1, 1988 at an interest rate of 24% compounded semi-annually. How much would the sum on Jan. 1, 1993? A. P421,170 C. P401,170 B. P521,170 D. P621,170 Problem 52 What rate of interest compounded annually must be received if an investment of P54,000 made now with result in a receipt of P72,000 5 years hence? A. 5.12% C. 5.92% B. 5.65% D. 5.34% Problem 53 With interest at 6% compounded annually, how much is required 7 years hence to repay an P8 M loan made today? A. P12,456,789 C. P12,029,042 B. P12,345,046 D. P12,567,000 Problem 54 If money is worth 6% compounded annually, what payment 12 years from now is equivalent to a payment of P7000 9 years from now? A. P8765.10 C. P8337.10 B. P8945.20 D. P8234.60 Problem 55 If money is worth 6% compounded annually, how much can be loaned now if P6000 will be repaid at the end of 8 years? A. P3567.30 C. P3456.34 B. P3444.44 D. P3764.50 Problem 56 A person invests P4500 to be collected in 8 years. Given that the interest rate on the investment is 14.5% per year, compounded annually, what sum, in pesos, will be collected eight years hence? A. P4504 C. P13294 B. P9720 D. P10140 Problem 57 (CE November 1996) If P500,000 is deposited at a rate of 11.25% compounded monthly, determine the compounded interest after 7 years and 9 months. A. P660,592 C. P680,686 B. P670,258 D. P690,849 Problems – Compound Interest, Annuity Set 28 Problem 1 (ECE November 1996) Find the nominal rate that if converted quarterly could be used instead for 25% compounded semiannually? A. 14.93% C. 15.56% B. 14.73% D. 15.90% Problem 2 (CE November 1999) Which of the following has the least effective annual interest rate? A. 12% compounded quarterly C. 11.7% compounded semi-annually B. 11.5% compounded monthly D. 12.2% compounded annually Problem 3 (CE November 1998) One hundred thousand pesos was placed in a time deposit that earns 9% compounded quarterly, tax free. After how many years would it be able to earn a total interest of fifty thousand pesos? A. 4.56 C. 4.32 B. 4.47 D. 4.63 Problem 4 (ECE November 1996) The amount of P2,825.00 in 8 years at 5% compounded quarterly is: A. P4,166.77 C. P4,188.56 B. P4,397.86 D. P4,203.97 Problem 5 The amount of P2,825.00 in 8 years at 5% compounded continuously is: A. P4,166.77 C. P4,397.86 B. P4,188.56 D. P4,214.97 A savings association pays 4% interest quarterly. What is the effective annual interest rate? A. 18.045% C. 16.985% B. 17.155% D. 17.230% Problem 6 (CE May 1998) What rate (%) compounded quarterly is equivalent to 6% compounded semi-annually? A. 5.93 C. 5.96 B. 5.99 D. 5.9 Problem 15 (ME October 1997) A bank offers 1.2% effective monthly interest. What is the effective annual rate with monthly compounding? A. 15.4% C. 14.4% B. 8.9% D. 7.9% Problem 7 (ECE April 1998) The amount of P12,800 in 4 years at 5% compounded quarterly is: A. P15,614.59 C. P16,311.26 B. P14,785.34 D. P15,847.33 Problem 8 Fifteen percent (15%) when compounded semiannually is what effective rate? A. 17.34% C. 16.02% B. 18.78% D. 15.56% Problem 9 (ECE November 1997) What rate of interest compounded annually is the same as the rate of interest of 8% compounded quarterly? A. 8.24% C. 6.88% B. 8.42% D. 7.90% Problem 10 (ECE November 1997) How long will it take the money to triple itself if invested at 10% compounded semi-annually? A. 13.3 years C. 11.9 years B. 11.3 years D. 12.5 years Problem 11 (ECE November 1997) What is the accumulated amount after three (3) years of P6,500.00 invested at the rate of 12% per year compounded semi-annually? A. P9,500.00 C. P9,221.00 B. P9,321.00 D. P9,248.00 Problem 12 (ECE November 1997) What interest rate, compounded monthly is equivalent to 10% effective rate? A. 9.75% C. 9.68% B. 9.50% D. 9.57% Problem 13 (ECE November 1997) A man wishes his son to receive P500,000.00 ten years from now. What amount should he invest now if it will earn interest of 12% compounded annually during the first 5 years and 15% compounded quarterly during the next 5 years? A. P135,868.19 C. P123,433.23 B. P134,678.90 D. P145,345.34 Problem 14 (ME October 1997) Problem 16 (ME October 1997) What is the present worth of P27,000.00 due in 6 years if money is worth 13% and is compounded semi-annually? A. P12,681.00 C. P15,250.00 B. P13,500.00 D. P21,931.00 Problem 17 (ME October 1997) A student deposits P1,500.00 in a 9% account today. He intends to deposit another P3,000.00 at the end of two years. He plans to purchase in five years his favorite shoes worth P5,000.00. Calculate the money that will be left in his account one year after the purchase. A. P1,280.00 C. P1,300.00 B. P1,250.00 D. P1,260.00 Problem 18 If money is worth 4% compounded monthly, what payment at the end of each quarter will replace payments of P500.00 monthly? A. P1,500.00 C. P1,505.00 B. P1,525.000 D. P1,565.00 Problem 19 What amount would have to be invested at the end of each year for the next 9 years at 4% compounded semi-annually in order to have P5,000.00 at the end of the time? A. P541.86 C. P542.64 B. P553.82 D. P548.23 Problem 20 A contractor bought a concrete mixer at P120,000.00 if paid in cash. The mixer may also be purchased by installment to be paid within 5 years. If money is worth 8%, the amount of each annual payment, if all payments are made at the beginning of each year, is: A. P27,829.00 C. P31,005.00 B. P29,568.00 D. P32,555.00 Problem 21 A contract calls for semiannual payments of P40,000.00 for the next 10 years and an additional payment of P250,000.00 at the end of that time. Find the equivalent cash value of the contract at 7% compounded semiannually? A. P444,526.25 C. P694,138.00 B. P598,589.00 D. P752,777.00 Problem 22 A man is left with an inheritance from his father. He has an option to receive P2 M at the end of 10 years; however he wishes to receive the money at the end of each year for 5 years. If interest rate is 8%, how much would he receive every year? A. P400,000.00 C. P232,020.00 B. P352,533.00 D. P200,000.00 Problem 23 (CE November 1999) To maintain its newly acquired equipment, the company needs P40,000 per year for the first five years and P60,000 per year for the next five years. In addition, an amount of P140,000 would also be needed at the end of the fifth and the eighth years. At 6%, what is the present worth of these costs? A. P689,214 C. P549,812 B. P512,453 D. P586,425 Problem 24 A man receives P125,000.00 credits for his old car when buying a new model costing P375,000.00. What cash payment will be necessary so that the balance can be liquidated by payments of P12,500.00 at the end of each month for 18 months when interest is charged at the rate of 6% compounded monthly? A. P23,400.00 C. P33,650.00 B. P28,750.00 D. P35,340.00 Problem 25 Determine the present worth of an annual payment of P2500.00 at the end of each year for 12 years at 8% compounded annually. A. P18,840.20 C. P15,000.00 B. P30,000.00 D. P17,546.04 Problem 26 A man borrowed P200,000.00 from a bank at 12% compounded monthly, which is payable monthly for 10 years (120 payments). If the first payment is to be made after 3 months, how much is the monthly payment? A. P2,869.42 C. P3,013.10 B. P2,927.10 D. P3,124.12 Problem 27 What is the present worth of a P1000.00 annuity over a 10-year period, if interest rate is 8%? A. P7896.00 B. P8976.00 C. P6234.80 D. P6710.00 Problem 28 (ME October 1995) How much money must you invest today in order to withdraw P1000 per year for 10 years if interest rate is 12%? A. P5650.00 C. P5560.00 B. P6550.00 D. P7550.00 Problem 29 A machine is under consideration for investment. The cost of the machine is P25,000. Each year it operates, the machine will generate a savings of P15,000. Given an effective annual interest of 18%, what is the discounted payback period, in years, on the investment in the machine? A. 1.566 C. 2.155 B. 2.233 D. 2.677 Problem 30 (ME April 1996) What is the present worth of a P100 annuity starting at the end of the third year and continuing to the end of the fourth year, if the annual interest rate is 8%? A. 153.44 C. 154.99 B. 152.89 D. 156.33 Problem 31 Consider a project which involves the investment of P100,000 now and P100,000 at the end of one year. Revenues of P150,000 will be generated at the end of years 1 and 2. What is the net present value of this project if the effective annual interest rate is 10%? A. P65,421.50 C. P68,421.50 B. P67,421.50 D. P69,421.50 Problem 32 An investment of x pesos is made at the end of each year for three years, at an interest rate of 9% per year compounded annually. What will be the value of the investment upon the deposit of the third payment? A. 3.278x C. 3.728x B. 3.287x D. 3.782x Problem 33 (ME October 1995) If P500 is invested at the end of each year for 6 years, at an effective annual interest rate of 7%, what is the total amount available upon the deposit of the 6th payment? A. P3455.00 C. P3577.00 B. P3544.00 D. P3688.00 Problem 34 How much money must you deposit today to an account earning 12% so that you can withdraw P25,000 yearly indefinitely starting at the end of the 10th year? A. P125,000 C. P73,767 B. P89,456 D. P75,127 Problem 35 (ME April 1996) In five years, P18,000 will be needed to pay for a building renovation. In order to generate this sum, a sinking fund consisting of three annual payments is established now. For tax purposes, no further payments will be made after three years. What payment is necessary if money is worth 15% per annum? A. P3,345.65 C. P3,919.53 B. P3,789.34 D. P3,878.56 Problem 36 An investment of P40,000.00 has revenue of x pesos at the end of the first and second year. Given a discount rate of 15% compounded annually, find x so that the net present worth of the investment is zero. A. P33,789.54 C. P24,604.65 B. P27,789.78 D. P21,879.99 Problem 37 Mr. Jones borrowed P150,000 two years ago. The terms of the loan are 10% interest for 10 years with uniform payments. He just made his second annual payment. How much principal does he still owe? A. P130,235.20 C. P132,456.20 B. P134,567.30 D. P129,456.78 Problem 38 Given that the discount rate is 15%, what is the equivalent uniform annual cash flow of the following stream of cash flows? year 0 P 100,000.00 year 1 200,000.00 year 2 50,000.00 year 3 75,000.00 A. P158,124.60 C. P157,345.98 B. P158,897.50 D. P155,789.34 Problem 39 Mr. Bean borrowed P100,000 at 10% effective annual interest rate. He must pay back the loan over 30 years with uniform monthly payments due on the first day of each month. What does he pay each month? A. P768.67 C. P856.30 B. P987.34 D. P839.20 Problem 40 (ECE November 1995) An employee obtained a loan of P10,000 at the rate of 6% compounded annually in order to repair a house. How much must he pay monthly to amortize the loan within a period of ten years? A. P198.20 C. P110.22 B. P150.55 D. P112.02 Problem 41 What is the accumulated value of a payment of P12,500 at the end of each year for 9 years with interest at 5% compounded annually? A. P138,738.05 C. P178,338.50 B. P137,832.05 D. P187,833.50 Problem 42 What is the accumulated value of a payment of P6,000 every six months for 16 years with interest at 7% compounded semiannually? A. P312,345.00 C. P347,898.00 B. P345,678.00 D. P344,007.00 Problem 43 A mining property is offered for sale for P5.7M. On the basis of estimated production, an annual return of P800,000 is foreseen for a period of 10 years. After 10 years, the property will be worthless. What annual rate of return is in prospect? A. 6.7% C. 5.6% B. 6.1% D. 5.2% Problem 44 If a down payment of P600,000 is made on a house and P80,000 a year for the next 12 years is required, what was the price of the house if money is worth 6% compounded annually? A. P1,270,707 C. P1,345,555 B. P1,130,450 D. P1,678,420 Problem 45 What annuity over a 10-year period at 8% interest is equivalent to a present worth of P100,000? A. P14,903 C. P13,803 B. P15,003 D. P12,003 Problem 46 (CE May 1998) The present value of an annuity of “R” pesos payable annually for 8 years, with the first payment at the end of 10 years, is P187,481.25. Find the value of R if money is worth 5%. A. P45,000 C. P42,000 B. P44,000 D. P43,000 Problem 47 (ECE April 1998) How much money must you invest today in order to withdraw P2,000 annually for 10 years if the interest rate is 9%? A. P12,385.32 C. P12,835.32 B. P12,853.32 D. P12,881.37 Problem 48 (ECE April 1998) Money borrowed today is to be paid in 6 equal payments at the end of each of 6 quarters. If the interest is 12% compounded quarterly, how much was initially borrowed if quarterly payment is P2000.00? A. P10,382.90 C. P10,834.38 B. P10,200.56 D. P10,586.99 Problem 49 (ME October 1997) A car was bought on installment basis with a monthly installment of P10,000.00 for 60 months. If interest is 12% compounded annually, calculate the cash price of the car. A. P455,875.00 C. P678,519.75 B. P567,539.75 D. P345,539.75 Problem 50 (ME October 1997) A steel mill estimates that one of its furnaces will require maintenance P20,000.00 at the end of 2 years, P40,000.00 at the end 4 years and P80,000.00 at the end of 8 years. What uniform semi-annual amounts could it set aside over the next eight years at the end of each period to meet these requirements of maintenance cost if all the funds would earn interest at the rate of 6% compounded semiannually? A. P7,897.35 C. P8,897.35 B. P9,397.35 D. P6,897.35 Problem 51 (ME April 1998) A house and lot can be acquired at a down payment of P500,000.00 and a yearly payment of P100,000.00 at the end of each year for a period of 10 years, starting at the end of 5 years from the date of purchase. If money is worth 14% compounded annually, what is the cash price of the property? A. P810,100.00 C. P808,836.00 B. P801,900.00 D. P805,902.00 Problem 52 (ME April 1998) How much must be deposited at 6% each year beginning on January 1, year 1, in order to accumulate P5,000.00 on the date of the last deposit, January 1, year 6? A. P751.00 C. P717.00 B. P715.00 D. P725.00 Problem 53 (ME April 1998) A piece of machinery can be bought for P10,000.00 cash, or for P2,000.00 down and payments of P750.00 per year for 15 years. What is the annual interest rate for the time payments? A. 4.61% C. 3.81% B. 5.71% D. 11.00% Problem 54 (ME April 1998) An instructor plans to retire in exactly one year and want an account that will pay him P25,000.00 a year for the next 15 years. Assuming a 6% annual effective interest rate, what is the amount he would need to deposit now? (The fund will be depleted after 15 years.) A. P249,000.00 C. P242,806.00 B. 248,500.00 D. P250,400.00 Problem 55 A man invested P1,000.00 per month on a bank that offers 6% interest. How much can he get after 5 years? A. P60,000.00 C. P72,540.00 B. P69,770.00 D. P69,491.00 Problem 56 (CE November 1995) Find the present value in pesos, of perpetuity of P15,000 payable semi-annually if money is worth 8%, compounded quarterly. A. P371,287 C. P392,422 B. P386,227 D. P358,477 Problem 57 (CE May 1999, May 1995) A man paid 10% down payment of P200,000 for a house and lot and agreed to pay the balance on monthly installments for 60 months at an interest rate of 15% compounded monthly. Determine the required monthly payment. A. P4,282.00 C. P58,477.00 B. P42,822.00 D. P5,848.00 Problem 58 (CE November 1998) A debt of x pesos, with interest rate of 7% compounded annually will be retired at the end of 10 years through the accumulation of deposit in the sinking fund invested at 6% compounded semiannually. The deposit in the sinking fund every end of six months is P21,962.68. What is the value of x? A. P300,000 C. P350,000 B. P250,000 D. P400,000 Problems – Depreciation, Capitalized Cost, Bonds, Etc. Set 29 Problem 1 What is the value of an asset after 8 years of use if it depreciation from its original value of P120,000.00 to its salvage value of 3% in 12 years? A. P44,200.00 C. P44,002.00 B. P44,020.00 D. P42,400.00 Problem 2 A man bought an equipment which cost P524,000.00. Freight and installation expenses cost him P31,000.00. If the life of the equipment is 15 years with an estimated salvage value of P120,000.00, find its book value after 8 years. A. P323,000.00 C. P259,000.00 B. P244,000.00 D. P296,000.00 Problem 3 An equipment costing P250,000 has an estimated life of 15 years with a book value of P30,000 at the end of the period. Compute the depreciation charge and its book value after 10 years using straight line method. A. d = P14,666.67; BV = P103,333.30 C. d = P13,333.33; BV = P103,333.30 B. d = P14,666.67; BV = P105,666.67 D. d = P13,333.33; BV = P105,666.67 Problem 4 An equipment costing P250,000 has an estimated life of 15 years with a book value of P30,000 at the end of the period. Compute the depreciation charge and its book value after 10 years using sinking fund method assuming i = 8%. A. d = P8,102.50; BV = P103,333.30 C. d = P7,567.50; BV = P138,567.60 B. d = P6,686.67; BV = P125,666.67 D. d = P8,102.50; BV = P132,622.60 Problem 5 An equipment costing P250,000 has an estimated life of 15 years with a book value of P30,000 at the end of the period. Compute the depreciation charge and its book value after 10 years using declining balance method. A. d = P9,456.78; BV = P67,456.98 C. d = P9,235.93; BV = P60,832.80 B. d = P8,987.45; BV = P60,832.80 D. d = P9,235.93; BV = P59,987.34 Problem 6 An equipment costing P250,000 has an estimated life of 15 years with a book value of P30,000 at the end of the period. Compute the depreciation charge and its book value after 10 years using the sum of year’s digit method. A. d = P11,000; BV = P67,500 C. d = P11,500; BV = P60,000 B. d = P10,500; BV = P58,000 D. d = P11,000; BV = P57,500 Problem 7 An asset costing P50,000 has a life expectancy of 6 years and an estimated salvage value of P8,000. Calculate the depreciation charge at the end of the fourth period using fixed-percentage method. A. P7144.20 C. P3878.40 B. P5264.00 D. P2857.60 Problem 8 (CE May 1996) A machine costing P45,000 is estimated to have a salvage value of P4,350 when retired at the end of 6 years. Depreciation cost is computed using a constant percentage of the declining book value. What is the annual rate of depreciation in %? A. 33.25% C. 35.25% B. 32.25% D. 34.25% Problem 9 (CE November 1997, November 1994) An engineer bought an equipment for P500,000. Other expenses including installation amounted to P30,000. At the end of its estimated useful life of 10 years, the salvage value will be 10% of the first cost. Using straight line method of depreciation, what is the book value after 5 years? A. P281,500.00 C. P301,500.00 B. P291,500.00 D. P271,500.00 Problem 10 (ECE November 1997) A machine costs P8,000.00 and an estimated life of 10 years with a salvage value of P500.00. What is its book value after 8 years using straight line method? A. P2,500.00 C. P3,000.00 B. P4,000.00 D. P2,000.00 Problem 11 (ME October 1997) A factory equipment has an initial cost of P200,000.00. Its salvage value after ten years is P20,000.00. As a percentage of the initial cost, what is the straight-line depreciation rate of the equipment? A. 5% C. 9% B. 6% D. 8% Problem 12 (ME October 1997) An asset is purchased for P120,000.00. Its estimated economic life is 10 years, after which it will be sold for P12,000.00. Find the depreciation for the first year using the sum-of-the-year’s digit, (SOYD). A. P20,000.00 C. P21,080.00 B. P18,400.00 D. P19,636.00 Problem 13 (ME April 1998) An asset is purchased for P9,000.00. Its estimated life is 10 years, after which it will be sold for P1,000.00. Find the book value during the third year if sum-of-the-year’s digit (SOYD) depreciation is used. A. P6,100.00 C. P4,500.00 B. P5,072.00 D. P4,800.00 Problem 14 (ME April 1998) An asset is purchased for P500,000.00. The salvage value in 25 years is P100,000.00. What is the total depreciation in the first three years using straight line method? A. P48,000.00 C. P24,000.00 B. P32,000.00 D. P16,000.00 Problem 15 (ME April 1998) A machine has an initial cost of P50,000.00 and a salvage value of P10,000.00 after 10 years. What is the book value after five years using straight-line depreciation? A. P35,000.00 C. P25,000.00 B. P15,500.00 D. P30,000.00 Problem 16 (ME April 1998) A company purchased an asset for P10,000.00 and plans to keep it for 20 years. If the salvage value is zero at the end of the 20th year, what is the depreciation in the third year? Use sum-of-the-years digits depreciation. A. P1000.00 C. P857.00 B. P937.00 D. P747.00 Problem 17 (ME April 1998) An asset is purchased for P9,000.00. Its estimated life is 10 years, after which it will be sold for P1,000.00. Find the book value during the first year if sum-of-the-year’s digit (SOYD) depreciation is used. A. P8,000.00 C. P6,500.00 B. P7,545.00 D. P6,000.00 Problem 18 (CE November 1998) A machine having a first cost of P60,000.00 will be retired at the end of 8 years. Depreciation cost is computed using a constant percentage of the declining book value. What is the total cost of depreciation, in pesos, up to the time the machine is retired if the annual rate of depreciation is 28.72%? A. 56,000 C. 58,000 B. 57,000 D. 59,000 Problem 19 (ECE November 1998) XYZ Corporation makes it a policy that for any new equipment purchased; the annual depreciation cost should not exceed 20% of the first cost at any time with no salvage value. Determine the length of service life necessary if the depreciation used is the sum-of-the-year’s digit (SOYD) method. A. 7 years C. 9 years B. 8 years D. 6 years Problem 20 Determine the capitalized cost of an equipment costing P 2M with and annual maintenance of P200,000.00 if money is worth 20% per annum. A. P 2.5M C. P 3M B. P 2.75M D. P 3.5M Problem 21 (CE November 1996) At 6%, find the capitalized cost of a bridge whose cost is P250M and life is 20 years, if the bridge must be partially rebuilt at a cost of P100M at the end of each 20 years. A. 245.3 C. 210 B. 215 D. 220 Problem 22 (ME October 1997) An item is purchased for P100,000.00. Annual costs are P18,000.00. Using 8%, what is the capitalized cost of perpetual service? A. P350,000.00 C. P320,000.00 B. P335,000.00 D. P325,000.00 Problem 23 (CE May 1997) A company uses a type of truck which costs P2M, with life of 3 years and a final salvage value of P320,000. How much could the company afford to pay for another type of truck for the same purpose, whose life is 4 years with a final salvage value of P400,000, if money is worth 4%? A. P2,565,964.73 C. P2,585,964.73 B. P2,855,964.73 D. P2,585,864.73 Problem 24 A P100,000, 6% bond, pays dividend semi-annually and will be redeemed at 110% on July 1, 1999. Find its price if bought on July 1, 1996, to yield an investor 4%, compounded semi-annually. A. P100,000.00 C. P113,456.98 B. P112,786.65 D. P114,481.14 Problem 25 A community wishes to purchase an existing utility valued at P500,000 by selling 5% bonds that will mature in 30 years. The money to retire the bond will be raised by paying equal annual amounts into a sinking fund that will earn 4%. What will be the total annual cost of the bonds until they mature? A. P44,667.98 C. P34,515.05 B. P37,345.78 D. P33,915.05 Problem 26 A man paid P110,000 for a P100,000 bond that pays P4000 per year. In 20 years, the bond will be redeemed for P105,000. What net rate of interest will the man obtain on his investment? A. 3.37% C. 3.56% B. 3.47% D. 3.40% Problem 27 (ECE November 1996) A man wants to make 14% nominal interest compounded semi-annually on a bond investment. How much should the man be willing to pay now for a 12%, P10,000 bond that will mature in 10 years and pays interest semi-annually? A. P2,584.19 C. P8,940.50 B. P3,118.05 D. P867.82 Machine cost = $15,000; Life = 8 years; Salvage Value = $3,000. What minimum cash return would the investor demand annually from the operation of this machine if he desires interest annually at the rate of 8% on his investment and accumulates a capital replacement fund by investing annual deposits at 5%? A. $5246.66 C. $2456.66 B. $2546.66 D. $4256.66 Problem 28 It is estimated that a timber tract will yield an annual profit of P100,000 for 6 years, at the end of which time the timber will be exhausted. The land itself will then have an anticipated value of P40,000. If a prospective purchaser desires a return of 8% on his investment and can deposit money in a sinking fund at 4%, what is the maximum price he should pay for the tract? A. P459,480.00 C. P578,987.00 B. P467,456.00 D. P589,908.00 Problems – Recent Board Exams, Selected Problems Set 32 Problem 29 A mine is purchased for P1,000,000.00 and it is anticipated that it will be exhausted at the end of 20 years. If the sinking-fund rate is 4%, what must be the annual return from the mine to realize a return of 7% on the investment? A. P108,350 C. P130,850 B. P150,832 D. P103,582 Problem 30 A syndicate wishes to purchase an oil well which, estimates indicate, will produce a net income of P2M per year for 30 years. What should the syndicate pay for the well if, out of this net income, a return of 10% of the investment is desired and a sinking fund is to be established at 3% interest to recover this investment? A. P16,526,295 C. P12,566,295 B. P15,626,245 D. P16,652,245 Problem 31 (CE May 1995) An investor pays P1,100,000 for a mine which will yield a net income of P200,000 at the end of each year for 10 years and then will become useless. He accumulates a replacement fund to recover his capital by annual investments at 4.5%. At what rate (%) does he receive interest on his investment at the end of each year? A. 10.04 C. 11.5 B. 8.5 D. 17.5 Problem 32 (CE May 1997) Problem 1 (CE November 2000) A line in a map was drawn at a scale of 1:25000. An error of 0.02 mm in the drawing is equivalent to how many meters in actual? A. 5 m C. 0.05 m B. 0.5 m D. 50 m Problem 2 (ME October 2000) One day a Celsius thermometer and a Fahrenheit thermometer registered exactly the same numerical value for the temperature. What was the temperature that day? A. -20 C. 40 B. 20 D. -40 Problem 3 (CE May 2000) Convert 405° to mils. A. 2,800 mils 7,200 mils B. 10,200 mils 6,200 mils C. D. Problem 4 (CE May 2000) Rationalize the following: A. B. √𝑎 √𝑎−√𝑎𝑏 1−√𝑏 1+𝑏 𝑎−𝑎√𝑏 𝑎−𝑏 𝑎+𝑎√𝑏 𝑎−𝑏 1+√𝑏 1−𝑏 Problem 5 (CE November 2000) Solve for B in the given partial fraction: C. D. 𝑥 4 − 15𝑥 3 − 32𝑥 2 − 12𝑥 − 14 (𝑥 + 1)(𝑥 − 2)(𝑥 + 3)(𝑥 2 + 2) 𝐴 𝐵 𝐶 𝐷 = + + + 2 𝑥+1 𝑥−2 𝑥+3 𝑥 +2 A. -3 C. -4 B. 3 D. 2 Problem 6 (ME October 2000) Solve for the given equation, 7.4𝑥10−4 = 𝑒 −9.7𝑥 . A. 0.7432 C. 0.7243 B. 0.7342 D. 0.4732 Problem 7 (CE May 2000) Log8 975 = x. Find x. A. 3.31 B. 4.12 C. 5.17 D. 2.87 Problem 8 There are 9 arithmetic means between 6 and 18. What is the common difference? A. 1.2 C. 5.17 B. 1 D .0.8 Problem 9 (CE May 2000) There are four geometric means between 3 and 729. Find the fourth term. A. 81 C. 243 B. 27 D. 9 Problem 10 (CE November 2000) The geometric mean of two numbers is 8 and their arithmetic mean is 17. What is the first number? A. 45 C. 32 B. 36 D. 48 Problem 11 (CE November 2000) Twenty-eight persons can do a job in 60 days. They all start complete. Five persons quitted the job at the beginning of the 6th day. They were reinforced with 10 persons at the beginning of the 45th day. How many days was the job delayed? A. 5.75 days days B. 1.14 days days C. 1.97 D. 2.45 Problem 12 Twenty men can finish a job in 20 days. Twentyfive men started the job. If ten men quitted the job after 18 days, find the total number of days to finish the job. A. 27 C. 26 B. 28 D. 29 Problem 13 Twelve workers could do a job in 20 days. Six workers started the job. How many workers should be reinforced at the beginning of the 7th day to finish the job for a total of 18 days from the start? A. 10 C. 9 B. 13 D. 11 Problem 14 (ME October 2000) Box A has 4 white balls, 3 balls, and 3 orange balls. Box B has 2 white balls, 4 blue balls, and 4 orange balls. If one ball is drawn from each box, what is the probability that one of the two balls will be orange? A. 27⁄50 C. 23⁄50 B. 27⁄50 D. 7⁄25 Problem 15 Twelve books consisting of six mathematics books, 2 hydraulics books and four structural books are arranged on a shelf at random. Determine the probability that books of the same kind are all together. A. 1/2310 C. 1/3810 B. 1/5620 D. 1/1860 Problem 16 (ME October 16 2000) What is the angle between two vectors A and B? 𝐴 = 4𝑖 + 12𝑗 + 6𝑘 𝐵 = 24𝑖 − 8𝑗 + 6𝑘 A. 175.4° 84.3° B. -84.9° 86.3° Problem 17 (ME October 2000) The expression [𝑡𝑎𝑛𝜃𝑠𝑒𝑐𝜃(1 − sin2 𝜃)] / (𝑐𝑜𝑠𝜃) simplifies to: A. 𝑠𝑖𝑛𝜃 𝑡𝑎𝑛𝜃 B. 𝑠𝑒𝑐𝜃 𝑐𝑜𝑠𝜃 C. D. C. D. Problem 18 (CE November 2000) Given that 𝑡𝑎𝑛𝐴 = 4/5, what is the value 4𝑠𝑖𝑛𝐴−𝑐𝑜𝑠𝐴 of 3𝑐𝑜𝑠𝐴+𝑠𝑖𝑛𝐴 ? A. 0.579 C. 0.654 B. 0.752 D. 0.925 Problem 19 (CE November 2000) A flagpole 3 m high stands at the top of a pedestal 2 m high located at one side of a pathway. At the opposite side of the pathway directly facing the flagpole, the flagpole subtends the same angle as the pedestal. What is the width of the pathway? A. 4.47 m C. 6.28 m B. 3.21 m m D. 8.1 Problem 20 (CE May 2000) Find the area in sq. m. of a spherical triangle of whose angles are 123°, 84°, and 73°. The radius of the sphere is 30 m. A. 1863.3 C. 1958.6 B. 1570.8 D. 1480.2 Problem 21 (CE May 2000) Two sides of a triangle measure 18 cm and 6 cm. The third side may be: A. 12 C. 10 B. 13 D. 11 Problem 22 (CE May 2000) The perimeter of an ellipse is 28.448 units. If the major axis is 5 units, what is the length of the minor axis? A. 9 C. 8 B. 7 D. 6 Problem 23 (CE November 2000) A right regular hexagonal prism is inscribed in a right circular cylinder whose height is 20 cm. The difference between the circumference of the circle and the perimeter of the hexagon is 4 cm. Determine the volume of the prism. A. 9756 cc C. 10857 cc B. 114752 cc D. 10367 cc Problem 24 (ME October 2000) Find the area bounded by the x-axis, the line 𝑥 = 4, and the parabola𝑦 2 = 4𝑥. A. 64/2 C. 32/4 B. 32/3 D. 32/2 Problem 25 (CE November 2000) What is the area bounded by the curves 𝑦 2 = 4𝑥 and 𝑥 2 = 4𝑦 ? A. 6.0 C. 6.666 B. 7.333 D. 5.333 Problem 26 (CE November 2000) Given a regular hexagonal with consecutive corners ABCDEF. If the bearing of side AB is N 25° E, what is the bearing of side FA? A. N 15° W C. N 35° W B. N 45° W D. N 5° W Problem 27 (CE November 2000) The perimeter of a triangle is 58 cm and its area is 144 sq. Cm. What is the radius of the inscribed circle? A. 4.97 cm C. 5.52 cm B. 9.65 cm D.3.12 cm Problem 28 (ME October 2000) What is the area bounded by the curves 𝑦 = 8 − 𝑥 2 and 𝑦 = −2 + 𝑥 2 ? A. 22.4 C. 44.7 B. 26.8 D. 29.8 Problem 29 A solid sphere of radius 20 cm was placed on top of hallow circular cylinder of radius 10 cm. What volume of the sphere was inside the cylinder? A. 431 cc C. 325 cc B. 568 cc D. 542 cc Problem 30 A trough is formed by nailing together, edge two boards 130 cm in length, so that the right section is a right triangle. If 3500 cc of water is poured into the trough and if the trough is held so that right section of the water is an isosceles right triangle, how deep is the water? A. 6.32 cm C. 4.21 cm B. 5.19 cm D. 6.93 cm Problem 31 (CE May 2000) The lateral area of a right circular cone of radius 4 cm is 100.53 sq. cm. Determine the slant height. A. 8 cm C. 6 cm B. 9 cm D. 10 cm Problem 32 (CE May 2000) The frustum of a regular triangular pyramid has equilateral triangles for its bases and has an altitude of 8 m. The lower base edge is 9 m. If the volume is 135 cu. m., what is the upper base edge? A. 2 m C. 4 m B. 5 m D. 3 m Problem 33 (CE May 2000) A cylinder of radius 6 m has its axis along the xaxis. A second cylinder of the same radius has its axis along the y-axis. Find the volume, in the first octant, common to the two cylinders. A. 866 𝑚3 C. 144 𝑚3 B. 1152 𝑚3 D. 288 𝑚3 Problem 34 (CE May 2000) Find the volume of a right circular cylinder whose lateral area is 25.918 𝑚2 and base area of 7.068 𝑚2 . A. 19.44𝑚3 C. 20.53 𝑚3 B. 15.69𝑚3 D. 18.12 𝑚3 Problem 35 (CE November 2000) A solid has a circular base of base radius 20 cm. find the volume of the solid if every plane section perpendicular to a certain diameter is an isosceles right triangle with one leg in the plane of the base. A. 21333 cc C. 18667 cc B. 24155 cc D. 20433 cc Problem 36 (CE November 2000) The base diameter of a cone is 18 cm and its axis is inclined 60° with the base. If the axis is 20 cm long, what is the volume of the cone? A. 1524 cc C. 1245 cc B. 1469 cc D. 1689 cc Problem 37 (ME October 2000) The equation 9𝑥 2 + 16𝑦 2 + 54𝑥 − 64𝑦 = −1 describes: A. a circle C. a hyperbola B. a parabola D. an ellipse Problem 38 (CE May 2000) Two vertices of a triangle are (6, -1) and (7, -3). Find the ordinate of the vertex such that the centroid of the triangle will be (0, 0). A. -13 C. 13 B. 4 D. -4 Problem 39 (CE May 2000) Determine the equation of the directrix of the curve 𝑥 2 = 16𝑦? A. 𝑥 + 4 = 0 C. 𝑦 − 4 = 0 B. 𝑥 − 4 = 0 D. 𝑦 + 4 = 0 Problem 40 (CE November 2000) Find the area of the curve 𝑥 2 + 𝑦 2 + 6𝑥 − 12𝑦 + 9 = 0. A. 125 sq. units C. 92 sq. units B. 113 sq. units D. 138 sq. units Problem 41 (CE November 2000) Find the distance between the foci of the curve 9𝑥 2 + 25𝑦 2 − 18𝑥 + 100𝑦 − 116 = 0. A. 7 C. 8 B. 6 D. 12 Problem 42 (CE November 2000) What is the equivalent rectangular coordinate of a point whose coordinate is (7, 38°). A. (3.56, 4.31) C. (5.52, 4.31) B. (4.31, 5.52) D. (4.31, 3.56) Problem 43 The chords of the parabola 𝑦 2 = 16𝑥 having equal slope of 2 is bisected by its diameter. What is the equation of the diameter? A. 𝑦 = 16 C. 𝑦 = 2 B. 𝑦 = 8 D. 𝑦 = 4 Problem 44 Find the slope of the line whose parametric equation is 𝑥 = 3𝑡 2 − 6 and 𝑦 = 4𝑡 2 + 7. 3 A. 4 C. 3 1 B. 2 4 D. 3 Problem 45 (ME October 2000) The first derivative with respect to y of the function 3 𝑓 (𝑦) = √9 is: A. 3√9/2 C. 9 B. 0 D. 3√9 Problem 46 (ME October 2000) Find the derivative of 𝑓 (𝑥 ) = [𝑥 to the 3rd power – (𝑥 − 1) to the 3rd power] to the 3rd power? A. 3𝑥 3 − 3(𝑥 − 1) B. 9[𝑥 to the 3rd power – (𝑥 − 1) to the 3rd power]2[𝑥 2 − (𝑥 − 1)2 ] C. 9[𝑥 to the 3rd power – (𝑥 − 1) to the 3rd power] [𝑥 2 − (𝑥 − 1)2 ] D. 9[𝑥 to the 3rd power – (𝑥 − 1) to the 3rd power] Problem 47 The derivative of sin(2𝑥 2 + 1)with respect to 𝑥 is: A. 4𝑥 cos(2𝑥 2 + 1) C. cos(2𝑥 2 + 1) B. 𝑥𝑠𝑖𝑛(2𝑥 2 + 1) D. 𝑥 cos(2𝑥 2 + 1) Problem 48 What is the second derivative of 𝑦 = 4𝑥 2 − 2𝑥 + 1 at 𝑥 = 2. A. 8 C. 1 B. 0 D. Not defined Problem 49 (CE May 2000) At what value of x will the slope of the curve 𝑥 3 − 9𝑥 − 𝑦 = 0 be 18? A. 2 C. 5 B. 4 D. 3 Problem 50 (CE November 2000) The slope of the curve at any point is given as 6𝑥 − 2 and the curve passes through (5, 3). Determine the equation of the curve. A. 3 + 2𝑥 − 𝑦 + 62 = 0 C.2𝑥 2 + 3𝑥 − 𝑦 − 62 = 0 B. 2𝑥 2 − 3𝑥 + 𝑦 + 62 = 0 D. 3𝑥 2 − 2𝑥 − 𝑦 − 62 = 0 Problem 51 (CE May 2000) The total surface area of a closed cylindrical tank is 153.94 sq. m. If the volume is to be maximum, what is its height in meters? A. 6.8 m C. 3.6 m B. 5.7 m D. 4.5 m Problem 52 (CE November 2000) A closed cylindrical tank having a volume of 71.57 𝑚3 is to be constructed. If the surface area is to e minimum, what is the required diameter of the tank? A. 4 m C. 5 m B. 5.5 m D. 4.5 m Problem 53 Two post, one 16 feet and the other 24 feet are 30 feet apart. If the post are to be supported y a cable running from the top of the first post to a stake on the ground and then back to the top of the second post, find the distance from the lower post to the stake to use the least amount of wire. A. 6 feet C. 15 feet B. 9 feet D. 12 feet Problem 54 The motion of a body moved vertically upwards is expressed as ℎ = 100𝑡 − 16.1𝑡 2 Where h is the height in feet and t is the time in seconds. What is the velocity of the body when 𝑡 = 2 seconds? A. 21.7 fps C. 24.1 fps B. 28.7 fps D. 35.6 fps Problem 55 (CE May 2000) A lighthouse is 2 km off a straight shore. A searchlight at the lighthouse focuses to a car moving along the shore. When the car is 1 km from the point nearest to the lighthouse, the searchlight rotates 0.25 rev/hour. Find the speed of the car in kph. A. 3.93 C. 2.92 B. 2.56 D. 3.87 Problem 56 (CE May 2000) 𝑥 Evaluate ∫02 3 𝑒 3𝑠𝑖𝑛𝜃 𝑐𝑜𝑠𝜃 𝑑𝜃 A. 15.421 C. 17.048 B. 19.086 D. 20.412 Problem 57 Determine the are enclosed by the curve 𝑟 2 = 4𝑎2 𝑐𝑜𝑠30 A. 𝑎2 C. 8𝑎2 B. 4𝑎2 D. 16𝑎2 Problem 58 (CE May 2000) Determine the moment of inertia about the x-axis, of the area bounded by the curve 𝑥 2 =4y, the line 𝑥 = −4, and the x-axis. A. 9.85 C. 10.17 B. 13.24 D. 12.19 Problem 59 (CE November 2000) 𝑥𝑑𝑥 Evaluate the integral of 𝑥2 +2 with limits from 0 to 1. A. 0.322 C. 0.203 B. 0.018 D. 0.247 Problem 60 (CE November 2000) The area bounded by the curve 𝑦 = 𝑠𝑖𝑛𝑥 from 𝑥 = 0 to 𝑥 = 𝜋 is revolved about the x-axis. What is the volume generated? A. 2.145 cu. units C. 3.452 cu. units B. 4.935 cu. units D. 5.214 cu. units Problem 61 (ME October 2000) If you borrow money from your friend with simple interest of 12%, find the present worth of 20,000 which is due at the end of nine months. A. P18,688.20 C. P18,518.50 B. P18,691.50 D. P18,348.60 Problem 62 (ME October 2000) Business needs to have P100, 000 in five years. How much must he put into his 10% account in the bank now? A. P72,085.6 C. P70,654.1 B. P62,092.1 D. P60,345.2 Problem 63 (ME October 2000) What is the present worth of P54, 000.00 due in five years if money is worth 11% and is compounded semi-annually? A. P30,367.12 C. P31,613.25 B. P28,654.11 D. P34,984.32 Problem 64 (CE May 2000) How long will it take for money to quadruple itself if invested at 20% compounded quarterly? A. 10.7 years C. 9.5 years B. 6.3 years D. 7.1 years Problem 65 (ME October 2000) The interest on an account is 13% compounded annually. How many years approximately will take to triple the amount? A. 8 years C. 9.5 years B. 8.5 years D. 9 years Problem 66 (ME October 2000) When will an investment of P4000 double if the effective rate is 8% per annum? A. 8.4 C. 9.01 B. 8.3 D. 10.2 Problem 67 (ME October 2000) A savings association pays 1.5% interest quarterly. What is the effective annual interest rate? A. 6.14% C. 7.32% B. 8.54% D. 6.45% Problem 68 (ME October 2000) A bank offers 0.5% effective monthly interest. What is the effective annual rate with monthly compounding? A. 6.2% C. 7.2% B. 6% D. 7% Problem 69 (ME October 2000) What nominal rate converted quarterly could be used instead of 12% compounded semi-annually? A. 10.76% C. 11.82% B. 11.43% D. 11.97% Problem 70 (CE November 2000) P1, 000,000 was invested to an account earning 8% compounded continuously. What is the amount after 20 years? A. P4,452,796.32 C. P5,356,147.25 B. P4,953,032.42 D. P3,456,254.14 Problem 71 (ME October 2000) A sum of money is deposited now in a savings account. The effective annual interest rate is 12%. How much money must be deposited to yield P500.00 at the end of 11 months? A. P153.00 C. P446.00 B. P144.00 D. P451.00