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