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

2x+ 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 = {α , β } and the

relation R from X to Y given by R = { ( 5 , β ) , ( 7 , β ) , ( 9 , α ) , ( 9 , β ) } .

What is the matrix of R ?

(a)[

0 1 0 1 1 1]

(b)

0 10 11 1

(c)

[0 0 11 1 1

](d)

0 11 00 1

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 = 1.5x + 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 = 2x2 + x − 4 ?

(a)

(− 1 +

√33

4, 0

)(b) (−1 , 0 ) (c)

(− 1−

√33

4, 0

)(d)

(−1 +

√33

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

3π (b) 4 (c) 64π (d)

64

3π

MENSURATION OF AREAS AND VOLUMES

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Mathematics 8 of 37

6. The exact value of cos

(7 π

12

)is most nearly

(a) 0.9995 (b)

√3 + 1

2√

2(c)

1−√

3

2√

2(d) −

√3

4

TRIGONOMETRY

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Mathematics 9 of 37

7. Consider the complex numbers z1 = 2 + 2 j and z2 = 2 ∠π

6. What is the value of

the product z1 z2 ?

(a) 2√

3− 2 +(

2 + 2√

3)j (b) 4

√2 ∠

5 π

12(c) 2

√3 + 2 +

(2 + 2

√3)j (d) 4

√2 ∠

π

10

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Mathematics 10 of 37

7 (continued)... Consider the complex numbers z1 = 2 + 2 j and z2 = 2 ∠π

6.

What is the value of the product z1 z2 ?

(a) 2√

3− 2 +(

2 + 2√

3)j (b) 4

√2 ∠

5 π

12(c) 2

√3 + 2 +

(2 + 2

√3)j (d) 4

√2 ∠

π

10

ALGEBRA OF COMPLEX NUMBERS

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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 θ , 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 θ , 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− 6x+ 10 y+ 14 = 0 ?

(a) 2√

5 (b) 20 (c) 4√

3 (d) 4

CONIC SECTIONS

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11. The roots of the cubic equation given by x3 − 4x2 + 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 − 4x2 + 6 ?

(a) −8 (b) 0 (c) 6 (d) no maximum exists

DIFFERENTIAL CALCULUS

DERIVATIVES AND INDEFINITE INTEGRALS

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13. What is∂f (x, y)

∂yof f (x, y) = 4 ln(y)− sec(x) cos

(√y)

+ 15 x − π ?

(a)4

y+ sec(x) sin

(√y)

(b)4

y+

1

2

1√y

sec(x) sin(√

y)

(c)4

y− 1

2

1√y

sec(x) sin(√

y)

(d)4

y+

1

2

1√y

sec(x) sin(√

y)

+ 15x ln(15)− 1


E. Brown


13 (continued)... f (x, y) = 4 ln(y)− sec(x) cos(√

y)

+ 15 x − π


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14. The value of the limit limx→0

x2

sin(x)is

(a) does not exist (b) 0 (c) ∞ (d) 2

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15. The indefinite integral of f (x) = x sin(2x)

is

(a) −1

2x cos

(2x)

+1

4sin(2x)

(b) −1

2x cos

(2x)

+1

4sin(2x)

+ C

(c) −1

4x2 cos

(2x)

+ C (d) −1

2x cos

(2x)

+1

2sin(2x)

+ C


E. Brown


15 (continued)... The indefinite integral of f (x) = x sin(2x)

is

(a) −1

2x cos

(2x)

+1

4sin(2x)

(b) −1

2x cos

(2x)

+1

4sin(2x)

+ C

(c) −1

4x2 cos

(2x)

+ C (d) −1

2x cos

(2x)

+1

2sin(2x)

+ 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 = 2x2 , the line y = 9 , and the y-axis?

(a)9√2

(b) 486 (c)27√

2(d)

18√2


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 = 2x2 , the line y = 9 , and the y-axis?

(a)9√2

(b) 486 (c)27√

2(d)

18√2


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

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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) =

∞∑n=0

an xn for coefficients an, n = 0, 1, 2, . . ., what series given below is equal to y ′(x) ?

(a)

∞∑n=0

ann + 1

xn+1 (b)

∞∑n=0

n an xn (c)

∞∑n=1

n an xn−1 (d)

∞∑n=0

n an xn−1

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19. What is the Maclaurin series expansion of e 3x ?

(a)

∞∑n=0

n en−1 (b)

∞∑n=0

3

n!xn (c)

∞∑n=0

0 (d)

∞∑n=0

3n

n!xn


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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) 2π f (t) (b) 2 π f (−t) (c) 2 π f (−ω) (d) 2 π f (ω)

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

(a) 2π f (t) (b) 2 π f (−t) (c) 2 π f (−ω) (d) 2 π f (ω)

E. Brown


22. What is the Fourier series of f (t) = 3 cos(4 t)

on the interval[

0 ,π

2

]?

(a) 3 cos(4 t)

(b)

∞∑n=1

[n2 cos(4n t) + (n− 1) sin(4n t)

]

(c)

∞∑n=1

3 cos(4n t) (d)

∞∑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

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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)[−1 −13

](b) does not exist (c)

[14 −3

](d)

[14

−3

]

E. Brown


26. What is the curl of the vector field ~F =⟨− x y3 z , x3 , −z3

⟩?

(a) −x y3 j +(

3x2 + 3x y2 z)

k (b) −x y3 i + 3x y2 z j

(c)⟨

0 , x y3 , 3x2 + 3x y2 z⟩

(d)⟨

0 , −x y3 , 3x2 + 3x 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. Differential equations - see Differential 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.pdfmathematics and...

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