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Mathematics and
Advanced Engineering Mathematics
Dr. Elisabeth Brown
c� 2019
1
Mathematics 2 of 37
Fundamentals of Engineering (FE)
Other Disciplines Computer-Based Test (CBT)
Exam Specifications
Mathematics 3 of 37
1. What is the value of x in the equation given by log3�2 x+4
�� log3
�x� 2
�= 1 ?
(a) 10 (b) �1 (c) �3 (d) 5
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Mathematics 4 of 37
2. Consider the sets X and Y given by X = { 5 , 7 , 9 } and Y = {↵ , � } and the
relation R from X to Y given by R = { ( 5 , � ) , ( 7 , � ) , ( 9 , ↵ ) , ( 9 , � ) } .What is the matrix of R ?
(a)h0 1 0 1 1 1
i(b)
2
640 10 11 1
3
75 (c)
"0 0 11 1 1
#(d)
2
640 11 00 1
3
75
DISCRETE MATH
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Mathematics 5 of 37
3. What is the x-intercept of the straight line that passes through the point ( 0 , 3 )
and is perpendicular to the line given by y = 1.5 x + 4 ?
(a)�0 , 3
�(b)
�2 , 0
�(c)
�� 2 , 0
�(d)
✓9
2, 0
◆
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Mathematics 6 of 37
4. What is the smallest x-intercept of the parabola given by y = 2 x2 + x � 4 ?
(a)
� 1 +
p33
4, 0
!(b) (�1 , 0 ) (c)
� 1�
p33
4, 0
!(d)
�1 +
p33
4, 0
!
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Mathematics 7 of 37
5. What is the volume of the largest sphere with center�5 , 4 , 9
�that is contained in
the first octant?
(a)256
3⇡ (b) 4 (c) 64⇡ (d)
64
3⇡
MENSURATION OF AREAS AND VOLUMES
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Mathematics 8 of 37
6. The exact value of cos
✓7 ⇡
12
◆is most nearly
(a) 0.9995 (b)
p3 + 1
2p2
(c)1�
p3
2p2
(d) �p3
4
TRIGONOMETRY
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Mathematics 9 of 37
7. Consider the complex numbers z1 = 2 + 2 j and z2 = 2 \ ⇡
6. What is the value of
the product z1 z2 ?
(a) 2p3� 2 +
�2 + 2
p3�j (b) 4
p2 \ 5 ⇡
12(c) 2
p3 + 2 +
�2 + 2
p3�j (d) 4
p2 \ ⇡
10
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Mathematics 10 of 37
7 (continued)... Consider the complex numbers z1 = 2 + 2 j and z2 = 2 \ ⇡
6.
What is the value of the product z1 z2 ?
(a) 2p3� 2 +
�2 + 2
p3�j (b) 4
p2 \ 5 ⇡
12(c) 2
p3 + 2 +
�2 + 2
p3�j (d) 4
p2 \ ⇡
10
ALGEBRA OF COMPLEX NUMBERS
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Mathematics 11 of 37
8. What are the real numbers a and b such that the complex number z =1� 2 j
3 + jcan be written as z = a + b j ?
(a) a =1
3, b = �2 (b) a = �1
4, b = 0 (c) a =
1
10, b =
7
10(d) a =
1
10, b = � 7
10
ALGEBRA OF COMPLEX NUMBERS
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Mathematics 12 of 37
9. The value of the angle ✓ , shown below, is most nearly
(a) 29.7� (b) 55.9� (c) 50.3� (d) 81.6�
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Mathematics 13 of 37
9 (continued)... The value of the angle ✓ , shown below, is most nearly
(a) 29.7� (b) 55.9� (c) 50.3� (d) 81.6�
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Mathematics 14 of 37
10. What is the radius of the circle given by the equation x2+ y2� 6 x+10 y+14 = 0 ?
(a) 2p5 (b) 20 (c) 4
p3 (d) 4
CONIC SECTIONS
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Mathematics 15 of 37
11. The roots of the cubic equation given by x3 � 4 x2 + 6 = 0 are most nearly
(a) x = �0.5, 1.2, 2.6 (b) x = �3.514, 0, 3.514
(c) no solutions exist (d) x = �1.086, 1.572, 3.514
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Mathematics 16 of 37
12. What is the maximum value of the function f (x) = x3 � 4 x2 + 6 ?
(a) �8 (b) 0 (c) 6 (d) no maximum exists
DIFFERENTIAL CALCULUS
DERIVATIVES AND INDEFINITE INTEGRALS
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Mathematics 17 of 37
13. What is@f (x, y)
@yof f (x, y) = 4 ln(y)� sec(x) cos
�py�+ 15 x � ⇡ ?
(a)4
y+ sec(x) sin
�py�
(b)4
y+
1
2
1py
sec(x) sin�p
y�
(c)4
y� 1
2
1py
sec(x) sin�p
y�
(d)4
y+
1
2
1py
sec(x) sin�p
y�+ 15x ln(15)� 1
DERIVATIVES AND INDEFINITE INTEGRALS
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Mathematics 18 of 37
13 (continued)... f (x, y) = 4 ln(y)� sec(x) cos�p
y�+ 15 x � ⇡
DERIVATIVES AND INDEFINITE INTEGRALS
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Mathematics 19 of 37
14. The value of the limit limx!0
x2
sin(x)is
(a) does not exist (b) 0 (c) 1 (d) 2
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Mathematics 20 of 37
15. The indefinite integral of f (x) = x sin�2 x�is
(a) �1
2x cos
�2 x�+
1
4sin�2 x�
(b) �1
2x cos
�2 x�+
1
4sin�2 x�+ C
(c) �1
4x2 cos
�2 x�+ C (d) �1
2x cos
�2 x�+
1
2sin�2 x�+ C
DERIVATIVES AND INDEFINITE INTEGRALS
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Mathematics 21 of 37
15 (continued)... The indefinite integral of f (x) = x sin�2 x�is
(a) �1
2x cos
�2 x�+
1
4sin�2 x�
(b) �1
2x cos
�2 x�+
1
4sin�2 x�+ C
(c) �1
4x2 cos
�2 x�+ C (d) �1
2x cos
�2 x�+
1
2sin�2 x�+ C
DERIVATIVES AND INDEFINITE INTEGRALS
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Mathematics 22 of 37
16. What is the area of the region of the first quadrant of the xy-plane that is bounded
by the curve y = 2 x2 , the line y = 9 , and the y-axis?
(a)9p2
(b) 486 (c)27p2
(d)18p2
DERIVATIVES AND INDEFINITE INTEGRALS
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Mathematics 23 of 37
16 (continued)... What is the area of the region of the first quadrant of the xy-plane
that is bounded by the curve y = 2 x2 , the line y = 9 , and the y-axis?
(a)9p2
(b) 486 (c)27p2
(d)18p2
DERIVATIVES AND INDEFINITE INTEGRALS
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Mathematics 24 of 37
17. What is the first moment of area with respect to the y-axis for the area in the first
quadrant bounded by the curve y = x2 , the line y = 9 , and the y-axis?
(a)486
5(b)
81
2(c)
81
4(d) 27
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Mathematics 25 of 37
17 (continued)... What is the first moment of area with respect to the y-axis for the area in the first
quadrant bounded by the curve y = x2 , the line y = 9 , and the y-axis?
(a)486
5(b)
81
2(c)
81
4(d) 27
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Mathematics 26 of 37
18. If y(x) =1X
n=0
an xn for coe�cients an, n = 0, 1, 2, . . ., what series given below is equal to y 0(x) ?
(a)1X
n=0
ann + 1
xn+1 (b)1X
n=0
n an xn (c)
1X
n=1
n an xn�1 (d)
1X
n=0
n an xn�1
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Mathematics 27 of 37
19. What is the Maclaurin series expansion of e 3x ?
(a)1X
n=0
n en�1 (b)1X
n=0
3
n!xn (c)
1X
n=0
0 (d)1X
n=0
3n
n!xn
DERIVATIVES AND INDEFINITE INTEGRALS
E. Brown
Mathematics 28 of 37
20. The indefinite integral of5
(x + 2) (x + 1)2is
(a) 5 ln |x+ 2|� 5 ln |x+ 1|� 5
x+ 1+ C (b)
5
x+ 2� 5
x+ 1+
5
(x+ 1)2+ C
(c) 5 ln |x+ 2|� 5 ln |x+ 1|+ 5 ln�(x+ 1)2
�+ C (d) 5 ln |x+ 2|� 5
x+ 1+ C
INTEGRAL CALCULUS
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Mathematics 29 of 37
20 (continued)... The indefinite integral of5
(x + 2) (x + 1)2is
(a) 5 ln |x+ 2|� 5 ln |x+ 1|� 5
x+ 1+ C (b)
5
x+ 2� 5
x+ 1+
5
(x+ 1)2+ C
(c) 5 ln |x+ 2|� 5 ln |x+ 1|+ 5 ln�(x+ 1)2
�+ C (d) 5 ln |x+ 2|� 5
x+ 1+ C
DERIVATIVES AND INDEFINITE INTEGRALS
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Mathematics 30 of 37
21. What is the Fourier transform of F (t) ?
(a) 2 ⇡ f (t) (b) 2 ⇡ f (�t) (c) 2 ⇡ f (�!) (d) 2 ⇡ f (!)
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Mathematics 31 of 37
21 (continued)... What is the Fourier transform of F (t) ?
(a) 2 ⇡ f (t) (b) 2 ⇡ f (�t) (c) 2 ⇡ f (�!) (d) 2 ⇡ f (!)
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Mathematics 32 of 37
22. What is the Fourier series of f (t) = 3 cos�4 t�on the interval
h0 ,
⇡
2
i?
(a) 3 cos�4 t�
(b)1X
n=1
n2 cos(4n t) + (n� 1) sin(4n t)
�
(c)1X
n=1
3 cos(4n t) (d)1X
n=1
3n cos(2n t) +
n
2sin(2n t)
�
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Mathematics 33 of 37
23. Consider the curve given by the function f (x) = �x2 + 2 x . The area under the
curve for 0 x 1.5 , approximated by using the forward rectangular rule with
�x = 12 , is most nearly
(a)9
8(b)
17
8(c)
13
8(d)
7
8
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Mathematics 34 of 37
24. Consider the exact area, Ac, under the curve f (x) = �x2 + 2x for 0 x 1.5 .
Ac falls most nearly between which of the following precision limits?
(a)7
8± 1
8(b)
7
8± 1
4(c)
13
8± 1
8(d)
13
8± 1
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Mathematics 35 of 37
25. For matrices A =
"�2
3
#and B =
"12 5
0 �1
#, what is ATB ?
(a)h�1 �13
i(b) does not exist (c)
h14 �3
i(d)
"14
�3
#
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Mathematics 36 of 37
26. What is the curl of the vector field ~F =⌦� x y3 z , x3 , �z3
↵?
(a) �x y3 j +�3 x2 + 3 x y2 z
�k (b) �x y3 i + 3 x y2 z j
(c)⌦0 , x y3 , 3 x2 + 3 x y2 z
↵(d)
⌦0 , �x y3 , 3 x2 + 3 x y2 z
↵
DETERMINANTS
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Mathematics 37 of 37
Mathematics and Advanced Engineering Mathematics
Exam Specifications Topic [ Example Question(s) in this Review ]
A. Analytic geometry [ 5, 10 ]
trigonometry [ 6, 9 ]
B. Calculus [ 12, 13, 14, 15, 16, 17, 18, 19, 20 ]
C. Di↵erential equations - see Di↵erential Equations video!
D. Numerical methods - e.g., algebraic equations [ 3, 12 ]
roots of equations [ 3, 4, 11, 12 ]
approximations [ 23, 24 ]
precision limits [ 24 ]
E. Linear algebra (e.g., matrix operations) [ 25, 26 ]
Dr. Elisabeth Brown c� 2019