mathematics and advanced engineering mathematics review 2019/fe exam review math...mathematics and...

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

E. Brown

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

E. Brown

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

◆

E. Brown

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

!

E. Brown

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

E. Brown

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

E. Brown

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

E. Brown

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

E. Brown


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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9. The value of the angle ✓ , shown below, is most nearly

(a) 29.7� (b) 55.9� (c) 50.3� (d) 81.6�

E. Brown


9 (continued)... The value of the angle ✓ , shown below, is most nearly

(a) 29.7� (b) 55.9� (c) 50.3� (d) 81.6�

E. Brown


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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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

E. Brown


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

E. Brown


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


E. Brown


13 (continued)... f (x, y) = 4 ln(y)� sec(x) cos�p

y�+ 15 x � ⇡


E. Brown


14. The value of the limit limx!0

x2

sin(x)is

(a) does not exist (b) 0 (c) 1 (d) 2

E. Brown


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


E. Brown


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


E. Brown


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


E. Brown


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


E. Brown


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

E. Brown


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

E. Brown


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

E. Brown


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


E. Brown


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

E. Brown


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


E. Brown


21. What is the Fourier transform of F (t) ?

(a) 2 ⇡ f (t) (b) 2 ⇡ f (�t) (c) 2 ⇡ f (�!) (d) 2 ⇡ f (!)

E. Brown


21 (continued)... What is the Fourier transform of F (t) ?

(a) 2 ⇡ f (t) (b) 2 ⇡ f (�t) (c) 2 ⇡ f (�!) (d) 2 ⇡ f (!)

E. Brown


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)

�

E. Brown


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

E. Brown


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

E. Brown


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

#

E. Brown


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

E. Brown


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

mathematics and advanced engineering mathematics review 2019/fe exam review math...mathematics and...

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