FE Review-Math

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FE Review-Math

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Problem 01

What is the length of a line segment with a slope of 4/3, measured from the y-axis to a point (6,4)?

A. 10 B. 25 C. 50 D. 75

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Problem 02

What is the general form of the equation for a line whose x-intercept is 4 and y-intercept is -6?

A. 2x - 3y - 18 = 0

B. 2x + 3y + 18 = 0

C. 3x - 2y - 12 = 0

D. 3x + 2y + 12 = 0

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Problem 03

Simplify and evaluate the following expression.

log3 ( %)+ log3 (12)- log3 (2)

A. 0.95 B. 1.33 C. 2.00 D. 2.20

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Problem 04

For some angle 8, csc(8) = -8/5. What is cos(28)?

A. 7/32 B. 1/4

C. 3/8

D. 5/8

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Problem 05

The expression ...

csc(x )cos3 (x) tan ( x)

... is equivalent to which of the following?

A. sin(x)

B. cos(x)

C . 1 - sin2(x) • • D. 1 + sin2(x)

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Problem 06

What is the value of x (less than 360°) that will satisfy the following equation?

sin2 (x)+ 4sin(x)+3 = 0

A. 45°

B. 90°

C. 180° D. 270°

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Problem 07

If the rectangu lar coordinates of a point are (-3, -5.2), what are its polar coordinates?

A. (-6, -120°)

B. (6, -120°)

C. (6, 120°)

D. (6, -150°)

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Problem 08

Determine the {x, y) coordinates of the center of the ci rcle defined by the following equation?

x2 - 8x + l - IO y + 25 = 0

A. (3, 2) B. (3, 4) C. (4, 5) D . (5, 4)

• •

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Problem 09

What is the slope of the following curve when it crosses the positive part of the x-axis?

A. 3/20 B. 1/5 c. 1/3

D. 7

y = I Ox2 - 3x - I

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Problem 10

What is the maximum value of the following function on the interval x < O?

f (x) = 2x3 + 12x2 - 30x+ 10

A. -210 B. -36 C. -5 D. 210

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Problem 11

What is the value of the following limit?

lim[-sin ...... (5__,_x)] <->0 X

A. 0

B. 0.87 C. 1 D. 5

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Problem 12

What is the partial derivative with respect to "x" of the following function?

f(x)= x2/ +x/ +sin(x) +cos2 (x)+sin3 (y)

A. (2x+ y)/ +3sin2 (y) cos(y)

B. ( 4x-3/)xy2 + 3sin2 (y)cos(y)

C. ( 3x + 4/ ).ry+3sin2 (y )cos(y)

• • D . (2x+ y)y3 +[t-2sin(x)]cos (x)

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Problem 13

What is the result of the following indefinite integral?

f ( 15x4 -8xJ + ~ + 7 )dx

A. 3x5 - 2x4 + In l,~+7x+ C

B. 60x3 - 24x2 - x-2 + C

C. 60x4 -24x3 -x-2 + 7x+ C

•• D. (15/ 4)x5 - (8/ 3)x4 - x-2 + 7x+C

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Problem 14

What is the result of the following indefinite integral?

f cos2 (x)sin(x) dx

A. -2/3[sin3 (x)] +c

B. - 1/3 [ COS3 ( X)] + C

C. l/3[sin3(x)] +c

• • D . l/2[sin2 (x )cos2 (x)] + C

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Problem 15

What is the total area of the region bounded by the x-axis and the curve y = sin (x) on the interval between x = n/2 and x = 2n?

A. 1

B. 2 C. 3 D. 4

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Problem 16

What is the general solution to the following differential equation?

A.

B. C. D.

y"-8y' + l6y=0

y= C,e4x

y=(C1 +C2x)e4.•

Y - Ce-4x+Ce4"

- I 2

y = C1e2

x + C2e4x

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Problem 17

For the three vectors what is the value of the following product?

A ·(Bx C)

- A = 6i + 8j + 1 Ok - B = i + 2j + 3k - C = 3i + 4j + 5k

A. 0 B. 64 C. 80

• D. 216

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Problem 18

What is the angle between the two given vectors?

- A = 4i + 12j + 6k - B = 24i - 8j + 6k

A. -84.32° B. 84.32° C. 101 .20° D. 122.36°

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Problem 19

What is the determinant of the following matrix?

[2 3 4]

A ; 5 6 7

7 8 9

A. -8 B. -4 C. 0 D. 4

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Problem 20

The second and sixth terms of a geometric progression are 3/10 and 243/160, respectively. What is the first term of this sequence?

A. 1/10 B. 1/5

C. 3/5

D. 3/2

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1. To find the width of a river, a surveyor sets up a transit at point C on one river bank and sights directly across to point B on the other bank. The surveyor then walks along the bank for a distance of 275 m to point A. The angle CAB is 57° 28'.

B -

-57°28'

275 m

What is the approximate width of the river?

(A) 150 m

(B) 230 m

(C) 330 m

(D) 430 m

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2. In the following illustration, angles 2 and 5 are 90° , AD = 15, DC = 20, and AC= 25.

D

What are the lengths BC and BD, respectively?

(A) 12 and 16

(B) 13 and 17

(C) 16 and 12

(D) 18 and 13

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3. What is the length of the line segment with slope 4/3 that extends from the point (6, 4) to the y-axis?

(A) 10

(B) 25

(C) 50

(D) 75

4. Which of the following expressions is equivalent to sin 20?

(A) 2 sin 0 cos 0

(B) cos2 0 - sin2 0

(C) sin 0 cos 0

(D) 1 - cos 20 2

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5. Which of the following equations describes a circle with center at (2, 3) and passing through the point (-3, - 4)?

(A) (x+ 3) 2 + (y+ 4) 2 = 85

(B) ( X + 3) 2 + ( y + 2) 2 = J74 (C) (x-3)2+(y-2)2=74

(D) ( x - 2 )2 + ( y - 3) 2 = 7 4

6. The equation for a circle is x2 + 4x+ y2 + 8y = 0. What are the coordinates of the circle's center?

(A) (-4, - 8)

(B) (-4, - 2)

(C) (-2, -4)

(D) (2 , - 4)

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7. Which of the following statements is FALSE for all noncircular ellipses?

(A) The eccentricity, e, is less than one.

(B) The ellipse has two foci.

( C) The sum of the two distances from the two foci to any point on the ellipse is 2 a (i.e., twice the semimajor distance).

(D) The coefficients A and C preceding the x2 and i/ terms in the general form of the equation are equal.

8. What is the area of the shaded portion of the circle shown?

(A) 5n - 1 6

(B) (~~) (5n - 3)

(C) 50n 3

(D) 49n - v'3

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9 . A pipe with a 20 cm inner diameter is filled to a depth equal to one-third of its diameter. What is the approximate area in flow?

(A) 33 cm2

(B) 60 cm2

(C) 92 cm2

(D) 100 cm2

1 O. The equation y = a1 + ~x is an algebraic expression for which of the following?

(A) a cosine expansion series

(B) projectile motion

( C) a circle in polar form

(D) a straight line

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11. For the right triangle shown, x = 18 cm and y = 13 cm.

~v~13cm

x = 18 cm

Most nearly, what is csc 0?

(A) 0.98

(B) 1.2

(C) 1.7

(D) 15

12. A circular sector has a radius of 8 cm and an arc length of 13 cm. Most nearly, what is its area?

(A) 48 cm2

(B) 50 cm2

(C) 52 cm2

(D) 60 cm2

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13. The equation -3x2 - 4y2 = 1 defines

(A) a circle

(B) an ellipse

(C) a hyperbola

(D) a parabola

14. What is the approximate surface area (including both side and base) of a 4 m high right circular cone with a base 3 min diameter?

(A) 24 m2

(B) 27 m2

(C) 32 m2

(D) 36 m2

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15. A particle moves in the x-y plane. After t s, the x­and y-coordinates of the particle's location are x = 8 sin t and y = 6 cos t. Which of the following equa­tions describes the path of the particle?

(A) 36x2 + 64y2 = 2304

(B) 36x2 - 64y2 = 2304

(C) 64x2 + 36y2 = 2304

(D) 64x2 - 36y2 = 2304

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1. What is the name for a vector that represents the sum of two vectors?

(A) scalar ' ' (B) resultant

(C) tensor

(D) moment

2. The second and sixth terms of a geometric progres­sion are 3/10 and 243/160, respectively. What is the first term of this sequence?

(A) 1/10

(B) 1/5

(C) 3/5

(D) 3/2

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3. Using logarithmic identities, what is most nearly the numerical value for the following expression?

(A) 0.95

(B) 1.33

(C) 2.00

(D) 2.20

4. Which of the following statements is true for a power series with the general term aixi?

I. An infinite power series converges for x < l .

II. Power series can be added together or subtracted within their interval of convergence.

III. Power series can be integrated within their inter-val of convergence.

(A) I only

(B) II only

(C) I and III

(D) II and III

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5. What is most nearly the length of the resultant of the following vectors?

(A) 3

(B) 4

(C) 10

(D) 14

3i + 4j - 5k

7i + 2j + 3k

-16i - 14j + 2k

6. What is the solution to the following system of simul­taneous linear equations?

lOx + 3y + lOz = 5

8x- 2y+ 9z= 3

8x + y - lOz = 7

(A) X= 0.326; y = - 0.192; Z= 0.586

(B) X= 0.148; y= 1.203; Z= 0.099

(C) X= 0.625; y = 0.186; Z= -0.181

(D) X= 0.282; y = - 1.337; Z= - 0.131

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7. What is the inverse of matrix A?

(A) [ ~ : l (B) [ : ~ l (C) [ _ : - : l (D) r-: _: l

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8. If the determinant of matrix A is -40, what is the determinant of matrix B?

4 3 2 1 2 1.5 1 0.5

0 1 2 -1 0 1 2 -1 A= B=

2 3 -1 1 2 3 -1 1

1 1 1 2 1 1 1 2

(A) -80

(B) - 40

(C) -20

(D) 0.5

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9. Given the origin-based vector A = i + 2j + k, what is most nearly the angle between A and the x-axis?

(A) 22°

(B) 24°

(C) 66°

(D) 80°

1 O. Which is a true statement about these two vectors?

A = i+2j +k

B = i + 3j - 7k

(A) Both vectors pass through the point (0, -1, 6) .

(B) The vectors are parallel.

( C) The vectors are orthogonal.

(D) The angle between the vectors is 1 7.4 °.

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11. What is most nearly the acute angle between vec­tors A = (3, 2, 1) and B = (2, 3, 2), both based at the origin?

(A) 25°

(B) 33°

(C) 35°

(D) 59°

12. Force vectors A, B , and C are applied at a single point.

A = i+3j +4k

B = 2i + 7j - k

C= -i+4j + 2k

What is most nearly the magnitude of the resultant force vector, R?

(A) p (B) 14

(C) 15

(D) 16

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13. What is the sum of 12 + 13j and 7 - 9j?

(A) 19 - 22j

(B) 19 + 4j

(C) 25 - 22j

(D) 25 + 4j

14. What is the product of the complex numbers 3 + 4j and 7 - 2j?

(A) 10+2j

(B) 13 + 22j

(C) 13 + 34j

(D) 29 + 22j

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1. Which of the following is NOT a correct derivative? ',

(A) -1:.._ cos x = - sin x dx

(B) -1:.._ (1 - x) 3 = - 3(1 - x)2 dx

(C) d l 1 dxx x2

(D) d dx csc x = - cot x

2. What is the derivative, dy/dx, of the expression x2 y- e2x = sin y?

(A) 2e2x

x2 - cosy

(B) 2e2x - 2xy

x2 - cos y

(C) 2 e2x - 2xy

(D) x2 - cos y

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3 . What is the approximate area bounded by the curves y = 8 - x2 and y = - 2 + af'?

(A) 22

(B) 27

(C) 30

(D) 45

4. What are the mm1mum and maximum values, respectively, of the equation f(x) = 5x3 - 2af' + 1 on the interval [- 2, 2]?

(A) - 47, 33

(B) - 4, 4

(C) 0.95, 1

(D) 0, 0.27

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5. In vector calculus, a gradient is a

I. vector that points in the direction of the rate of change of a scalar field

II. vector whose magnitude indicates the maximum rate of change of a scalar field

III. scalar that indicates the magnitude of the rate of change of a vector field in a particular direction

IV. scalar that indicates the maximum magnitude of the rate of change of a vector field in any partic­ular direction

(A) I only

(B) III only

(C) I and II

(D) III and IV

6. Which of the illustrations shown represents the vec­tor field , F(x, y) = - yi + xj?

(A)

* /

X

(B)

(C)

(D)

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7. A peach grower estimates that if he picks his crop now, he will obtain 1000 lugs of peaches, which he can sell at $1.00 per lug. However, he estimates that his crop will increase by an additional 60 lugs of peaches for each week that he delays picking, but the price will drop at a rate of $0.025 per lug per week. In addition, he will experience a spoilage rate of approximately 10 lugs for each week he delays. In order to maximize his revenue, how many weeks should he wait before picking the peaches?

(A) 2 weeks

(B) 5 weeks

(C) 7 weeks

(D) 10 weeks

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8. Determine the following indefinite integral.

(A) ~ + ln I xi - j_ + C 4 X

(B)

(C)

(D)

- x 2 + log x - 8x + C

x2 2 -+lnlxl--+ C 2 x 2

x2 4 -+ ln lxl - -+ C 2 X

9. Find dy/ dx for the parametric equations given.

X = 2t2 - t

y = t3 - 2t + 1

(A) 3t2

(B) 3t2 /2

(C) 4t-1

(D) (3f - 2)/(4t - 1)

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10. A two-dimensional function, f(x, y), is defined as

f(x, y) = 2x2 - y2 + 3x - y

What is the direction of the line passing through the point (1, -2) that has the maximum slope?

(A) 4i + 2j

(B) 7i + 3j

(C) 7i + 4j

(D) 9i - 7j

' 11. Evaluate the following limit.

,:,

r (x2 - 4) " x~ x-2

(A) 0

(B) 2 (C) 4 (D) 00

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11. Evaluate the following limit.

(A) 0

(B) 2

(C) 4

(D) oo

r (x2 - 4) x~ x - 2

12. If f( x, y) = x2 y3 + xy4 + sin x+ cos2 x+ sin3 y, what is of/ox?

(A) (2x + y)y3 + 3 sin2 ycos y

(B) ( 4x- 3y2 )xy2 + 3 sin2 ycos y

(C) (3x+ 4y2 )xy+ 3 sin2 ycos y

(D) (2x + y)y3 +(1-2sin x)cosx

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13. What is dy/dx if y= (2xY?

(A) (2xt(2 + ln 2x)

(B) 2x(l + ln 2x?

(C) (2xt(ln 2x2)

(D) (2xt(l+ln2x)

1. What is the solution to the following differential equation?

y' + 5y = 0

(A) y=5x+ C

(B) y= Ce-5x

(C) y= Ce5 x

(D) either (A) or (B)

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2. What is the solution to the following linear difference equation?

(k+ l)(y(k+ 1)) - ky(k) = 1

(A) y(k) = 12 -¼ (B) y(k) = 1 - 1f (C) y(k) = 12 + 3k

(D) y(k) = 3 + ¼

3. What is the general solution to the following differ­ential equation?

(d2 y) (dy) 2 - - 4 - +4y=O dx2 dx

(A) y = C 1 cos x + C 2 sin x

(B) y = C1 ex+ C2e-x

(C) y = e-x( C1 cos x - C2 sin x)

(D) y = ex( C1 cos x + C2 sinx)

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4. What is the general solution to the following differ­ential equation?

d2 d _Jj_ + 22 + 2y = 0 dx2 dx

(A) y = C 1 sin x - C 2 cos x

(B) y = C 1 cos x - C 2 sin x

(C) y = C1 cos x + C2 sin x

(D) y = e-x( C1 cos x + C2 sin x)

S. What is the complementary solution to the following differential equation?

y" - 4y' + 2f y = 10 cos 8x

(A) y = 2C1x + C2x - C3x

(B) Y = C 1 e2x + C 2 e1.5x

(C) y = C1e2xcosl.5x+ C2e2xsinl.5x

(D) y = C 1 ex tan x + C 2 ex cot x

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6. What is the general solution to the following differ­ential equation?

(A)

(B)

(C)

(D)

y"+y'+y=O

y = e-½x ( C 1 cos 1 x + C 2 sin 1 x) 1

y = e -2x ( C 1 cos !x + C 2 sin !x) -

y = e- 2x( C1 cos 1x + C2 sin 1x) y= e-2x(c1 cos!x+ C2sin!x)

7. What is the solution to the following differential equation if x = 1 at t= 0, and dx/ dt = 0 at t= O?

(A)

(B)

(C)

(D)

x = e- 4t + 4te- 4t

x = ie-2t( cos 2t + sin 2t) + i x = e- 4t + 4te-4t + i x = ie-4t + !te-4t + i

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8. In the following differential equation with the initial condition x(0) = 12, what is the value of x(2)?

(A) 3.4 x 10-3

(B) 4.0 X 10-3

(C) 5.1 x 10-3

(D) 6.2 X 10- 3

dx + 4x = 0 dt

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9. What are the three general Fourier coefficients for the sawtooth wave shown?

(A)

(B)

(C)

(D)

fWC 1~.---:::::::::1~

'IT 2'11" t

-1 ao = 0, an = 0, bn = -

rrn 1 -1

ao = - , an = 0, bn = -2 rrn

1 ao = 1, an = 1, bn = -

rrn 1 1 1

ao = 2) an = 2 ) bn = rrn

1 O. The values of an unknown function follow a Fibo­nacci number sequence. It is known that f(l) = 4 and f(2) = 1.3. What is f(4)?

(A) -4.1

(B) 0.33

(C) 2.7

(D) 6.6