FUNDAMENTALS IN ALGEBRA
Problem 1: ECE Board April 1999
If 16 is 4 more than 4x, find 5x – 1.
A. 14
B. 3
C. 12
D. 5
Problem 2: EE Board October 1992
Find the value of x in
𝑥+1
3
+
2𝑥
4
= 47 − 2𝑥
A. 16.47
B. 12.87
C. 18.27
D. 20.17
Problem 3: EE Board October 1991
Find the value of x in the equations:
𝐴
𝐴
10 �𝑥 + 𝑦� = 𝐴
3𝐴
2� 𝑥 −
4𝐴
�
𝑦
=𝐴
A. 50/9
B. 80/9
C. 70/9
D. 60/9
Problem 4: EE Board October 1997
Find the values of x and y from the equations:
x - 4y + 2 = 0
2x + y – 4 = 0
A. 11/7, -5/7
B. 14/9, 8/9
C. 4/9, 8/9
D. 3/2, 5/3
Problem 5: ME Board October 1995
Solve for the value of x and y.
4x + 2y = 5 13x -3y = 2
A. y = 1/2, x = 3/2
B. y = 3/2, x = 1/2
C. y = 2, x = 1
D. y = 3, x =1
Problem 6: ME Board October 1996
Solve the simultaneous equations:
2x2 - 3y2 = 6
3x2 + 2y2 = 35
A. x = 3 or -3; y = 2 or -2
B. x = 3 or -3, y = -2 or 1
C. x = 3 or -3, y = -2 or -1
D. x = 3 or -3, y = 2 or -3
Problem 7: CE Board May 1997
Find the value of w in the following equations:
3x - 2y + w = 11
3
1
x + 5y - 2w = -9
2x + y -3w = -6
A. 3
B. 2
C. 4
D. -2
Problem 8: EE Board October 1993
Solve for the value of x.
2x – y + z = 6
x- 3y - 2z = 13
2x - 3y - 3z = 16
A. 4
B. 3
C. 2
D. 1
Problem 9: ME Board October 1996
Solve the simultaneous equations:
x+y=-4
x+z-1=0
y+z+1=0
A. x = -1, y = -5, z = 3
B. x = 1, y = 2, z = -3
C. x = -1, y = -3, z = 2
D. x = -2, y = -3, z = -1
Problem 10: EE Board April 1997
Multiply the following: (2x + 5y)(5x - 2y)
A. 10x2 – 21xy + 10y2
B. -10x2 + 21xy + 10y2
C. 10x2 + 21xy - 10y2
D. -10x2 - 21xy - 10y2
Problem 11: EE Board March 1998
Determine the sum of the positive valued solution
to the simultaneous equations:
xy = 15, yz = 35, zx = 21.
A. 16
B. 13
C. 17
D. 19
Problem 12: ECE Board April 1991
Simplify:
�𝑥 2 𝑦 3 𝑧−2 �
1
−3
�𝑥 −3 𝑦𝑧3 �
5
(𝑥𝑦𝑧 −3 )− 2
−
1
2
𝑥 2 𝑦 7 𝑧5
𝟏
B. 𝒙𝟐 𝒚𝟕 𝒛𝟑
1
A.
C.
D.
𝑥 2 𝑦 5 𝑧7
1
𝑥 5 𝑦 7 𝑧2
Problem 13: ECE Board November 1993
Simplify the following equation
3
1
5𝑥
2𝑥 2 +
𝟒 −
7𝑥
3
A.+𝒙+𝟑
2
B.
𝑥−3
4
𝑥+3
𝑥 22𝑥++
𝑥1
−6
+
2
2𝑥 − 3𝑥 − 2
C. 𝑥−3
2
D. 𝑥+3
Problem 14: ECE Board April 1991
Simplify:�𝑥
A.
B.
2
3
1
1
2
1
6
2
�𝑥 − 3 𝑦 −2 (𝑥 2 𝑦 −2 )−3 � �
𝟓
𝒚𝟐
𝒙
3
𝑦2
𝑥
5
𝑦2
C. 𝑥 2
D.
3
𝑦2
𝑥2
Problem 15: ECE Board April 1991
Simplify: 7𝑎+2 − 8(7)𝑎+1 + 5(7)𝑎 + 49(7)𝑎−2
A. -5a
B. -3a
C. -7a
D. -4a
Problem 16:
Solve for x:
𝑥=
�𝑏2 −4𝑏+16��𝑏2 −16�
(𝑏3 +64)
A. b + 4
B.
C.
𝑏−4
𝑏+2
𝑏2 −4
𝑏+2
D. b – 4
Problem 17: ECE Board April 1993
Solve for y:
𝑥
𝑧
𝑦
=
=
(𝑏 − 𝑐) (𝑎 − 𝑐) (𝑎 − 𝑏)
A. x – z
B. x + z
C. a + b
D. a – b
Problem 18: ME Board October 1996
Resolve
𝑥+2
𝑥 2 −7𝑥+12
2
6
A. 𝑥−4 − 𝑥−3
5
3
B. 𝑥−4 − 𝑥−3
𝟓
𝟔
C. 𝒙−𝟒 − 𝒙−𝟑
5
7
D. 𝑥−4 − 𝑥−3
into partial fraction.
3
1
Problem 19: CE Board May 1996
Find the value of A in the equation:
𝑥 2 + 4𝑥 + 10
𝐴
𝐶
𝐵(2𝑥 + 2)
= + 2
+ 𝑥 2 + 2𝑥 + 5
3
2
𝑥 + 2𝑥A.+-25𝑥 𝑥 𝑥 + 2𝑥 + 5
B. 1/2
C. -1/2
D. 2
Problem 20: ME Board October 1996
The value of (3 to 2.5 power) square is equal to:
A. 729
B. 140
C. 243
D. 81
Problem 21:
Evaluate: 64x • 4y
A. 256xy
B. 4x + 3y
C. 64x + 3y
D. 43x + y
Problem 22: ECE Board April 1993
Solve for x in the following equations.
27x = 9y
81y3-x = 243
A. 1
B. 1.5
C. 2
D. 2.5
Problem 23: ECE Board April 1993
Evaluate:
4(52𝑛+1 ) − 10(52𝑛−1 )
𝑦=
2(52𝑛 )
A. y = 5n
B. y = 9
C. y = 52n
D. y = 18
Problem 24: ECE Board April 1990
Given: (an)( am) = 100,000 ; anm = 1,000,000 ;
Find a:
𝑎𝑛
𝑎𝑚
= 10
A. 12
B. 9
C. 11
D. 10
Problem 25: ECE Board November 1991
Give the factors of a2 – x2
A. 2a – 2x
B. (a + x)(a – x)
C. (a + x)(a – x)
D. 2x – 2a
3
1
Problem 26: ME Board October 1996
Factor the expression x2 + 6x + 8 as completely possible.
A. (x + 4)(x + 2)
B. (x – 4)(x + 2)
C. (x – 4)(x – 2)
D. (x + 6)(x + 2)
Problem 27: ECE Board November 1990
Expand: (a – b)3 = ?
A. a3 – 3a2b + 3ab2 + b3
B. a3 – 3a2b – 3ab2 – b3
C. a3 + 3a2b + 3ab2 – b3
D. a3 – 3a2b + 3ab2 – b3
Problem 28:
Find the value of k so that 4x2 + 6x + k is a perfect square.
A. 36
B. 2.5
C. 9
D. 2.25
Problem 29: ME Board April 1995
Factor the expression 3x3 – 3x2 – 18x
A. 3x(x – 3)(x + 2)
B. 3x(x + 3)(x + 2)
C. 3x(x + 3)(x – 2)
D. 3x(x – 3)(x – 2)
Problem 30:
If p – q = 5 and pq = k/2, then p2 + q2 equals,
A. k
B. 25k
C. 25 + k
D. k/25
Problem 31: ME Board April 1995
𝑚
Simplify: 𝑏 𝑛
A.
√𝑏𝑛
D.
𝑏𝑚
𝑛
B. 𝑏 𝑚+𝑛
𝒏 𝒎
C. √𝒃
𝑛
Problem 32: ME Board April 1998
Find the value of x which will satisfy the following expression:
√𝑥 − 2 = √𝑥 + 2
A. 3/2
B. 9/4
C. 18/6
D. None of these
Problem 33:
Simplify: �
𝑎𝑏
3
√𝑎𝑏
A. √𝒂𝒃
B. √𝑎𝑏
𝟑
3
1
𝑎𝑏
C. √𝑎𝑏
D.
𝑎𝑏
3
√𝑎𝑏
Problem 34: ME Board April 1996
If x to the ¾ power equals 8, x equals
A. -9
B. 6
C. 9
D. 16
Problem 35:
Solve for x: �𝑥 + 2√𝑥 + 3 − 3 = 0
A. 3
B. 23
C. 3 and 23
D. 20
Problem 36: CE Board November 1991
3
Solve for x from the given equation: �8 �2√8𝑥
4
=2
A. 4
B. 2
C. 3
D. 5
Problem 37: EE Board October 1997
If f(x) = 2x2 + 2x + 4, what is f(2)?
A. 4x + 2
B. 16
C. x2 + x + 2
d. 8
Problem 38: EE Board April 1997
If n is any positive integer, when
(n – 1) (n – 2) (n – 3)… (3)(2)(1) =
A. e(n – 1)
B. (n – 1)!
C. n!
D. (n – 1)n
Problem 39:
What is the least common multiple of 15 and 18?
A. 3
B. 5
C. 90
D. 270
Problem 40: ECE Board April 1998
What is the lowest common factor of 10 and 32?
A. 320
B. 2
C. 180
D. 90
Problem 41:
The numbers 12 and 16 has the greatest common divisor of
3
1
A. 2
B. 4
C. 6
D. 192
Problem 42: EE Board April 1996, EE Board March 1998
The Polynomial x3 + 4x2 – 3x + 8 is divided by (x – 5), then the remainder is,
A. 175
B. 140
C. 218
D. 200
Problem 43:
Find the quotient of 3x5 – 4x3 + 2x2 + 36x + 48 divided by x3 – 2x2 + 6.
A. 3x2 – 4x – 8
B. 3x2 + 4x + 8
C. 3x2 – 6x – 8
D. 3x2 + 6x + 8
Problem 44: CE Board November 1997
Find the remainder if we divide 4y3 + 18y2 + 8y – 4 by (2y + 3).
A. 10
B. 11
C. 15
D. 13
Problem 45: ECE Board November 1999
Given: f(x) = (x + 3)(x – 4) + 4. When f(x) is divided by (x – k), the remainder is k. Find k.
A. 2
B. 4
C. 6
D. 8
Problem 46:
The expression x4 + ax3 + 5x2 + bx + 6 when divided by (x – 2) leaves a remainder of 16 and
when divided by (x + 1) leaves a remainder of 10. Find a and b.
A. a = 5, b = 7
B. a = -5, b = 7
C. a = -5, b = -7
D. a = 5, b = -7
Problem 47:
The mean of x and y is a, the mean of y and z is b and the mean of x and z is c. What is the
mean of x, y, and z?
A.
B.
C.
D.
𝒂+𝒃+𝒄
𝟑
𝑎+𝑏+𝑐
2
𝑎+𝑏+𝑐
𝑎𝑏𝑐
𝑎𝑏𝑐
𝑎+𝑏+𝑐
Problem 48: ECE Board April 1999
Find the mean proportional of 4 and 36.
A. 72
B. 24
C. 12
D. 20
3
1
Problem 49: ECE Board 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. 50
C. 38.62
D. 57.12
Problem 50: ECE Board April 1998
The arithmetic mean of 6 numbers is 17. If two numbers are added to the progression, the
new set of numbers will have an arithmetic mean of 19. What are the two numbers if their
difference is 4?
A. 21, 25
B. 23, 27
C. 8, 12
D. 16, 20
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 factorial?
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
3
1
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)
3
1
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)
Solve for x: 4 + ((x + 3)/(x – 3)) – ((4x2)/(x2 – 9)) = ((x + 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
3
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
3
1
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
Simplify the following expression: ((5x)/(2x2 + 7x + 3)) – ((x + 3)/(2x2 – 3x – 2)) + ((2x +
1)/(x2 + 6 – 6)).
A. 2/(x-3)
B. (x-3)/5
C. (x+3)/(x-1)
D. 4/(x+3)
Problem 85
If 3x = 4y then ((3x2)/(4y2)) is equal to:
A. ¾
B. 4/3
C. 2/3
D. 3/2
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
3
1
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. ((x2 = 4x + 10)/(x3 + 2x2 + 5x)) = A/x + ((B(2x + 2))/(x2 +
2x + 5)) + (C/(x2 + 2x + 5))
A. 2
B. -2
C. -1/2
D. ½
Problem 90
Find A and B such that ((x + 10)/(x2 – 4)) = (A/(x – 2)) + (B/(x + 2))
A. A = -3; B = 2
B. A = -3; B = -2
C. A = 3; B = 2
D. A = 3; B = 2
Problem 91 (ME October 1996)
Resolve ((x + 2)/(x2 – 7x + 12) into partial fraction.
A. (6/(x – 4)) – (2/(x – 3))
B. (6/(x – 4)) + (7/(x – 3))
C. (6/(x – 4)) – (5/(x – 3))
D. (6/(x – 4)) + (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
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
3
1
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 98 (ECE November 1997)
If (x + 3):10 = (3x – 2) : 8, find (2x –1)
A. 1
B. 2
C. 3
D. 4
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 100
Solve for x: x2 + 4x > 12.
A. -6 > x > 2
B. 6 > x > -2
C. -6 > x > -2
D. 6 > x > 2
3
1