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POST UTME EKSU 2022 Mathematics | Objective

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Question 1

Find the derivative of the function ( f(x) = \frac{x^2 + 2x - 3}{x^2 - 4} ) u\sing the quotient rule.

A. \( \frac{\( x^2 - 4 \)\( 2x + 2 \ \) - \( x^2 + 2x - 3 \)(2x)}{\( x^2 - 4 \)^2} )

B. \( \frac{\( x^2 - 4 \)\( 2x + 2 \ \) + \( x^2 + 2x - 3 \)(2x)}{\( x^2 - 4 \)^2} )

Correct C. \( \frac{\( x^2 - 4 \)\( 2x + 2 \ \) - \( x^2 + 2x - 3 \)(2x)}{\( x^2 - 4 \)^2} )

D. \( \frac{\( x^2 - 4 \)\( 2x + 2 \ \) + \( x^2 + 2x - 3 \)(2x)}{\( x^2 - 4 \)^2} )

Correct Answer: C

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Question 2

A circle with center ( C(2, 3) ) and radius \( r = 5 \) has a chord ( AB ) of length ( 10 ). Find the dis\tance from the center ( C ) to the midpoint ( M ) of the chord ( AB ).

Correct A. \( \sqrt{5^2 - 5^2} = 0 \)

B. \( \sqrt{5^2 - 5^2} = 5 \)

C. \( \sqrt{5^2 - 5^2} = 10 \)

D. \( \sqrt{5^2 - 5^2} = 15 \)

Correct Answer: A

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Question 3

Find the volume of the solid formed by revolving the region bounded by the curve \( y = x^2 \), the x-axis, and the line \( x = 2 \) about the x-axis.

A. \( \frac{32}{3} pi \)

Correct B. \( \frac{64}{3} pi \)

C. \( \frac{128}{3} pi \)

D. \( \frac{256}{3} pi \)

Correct Answer: B

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Question 4

A sequence is defined recursively as \( a_n = 2a_{n-1} + 1 \), with \( a_1 = 3 \). Find the sum of the first 5 terms of the sequence.

Correct A. \( 3 + 7 + 15 + 31 + 63 = 119 \)

B. \( 3 + 7 + 15 + 31 + 63 = 119 \)

C. \( 3 + 7 + 15 + 31 + 63 = 119 \)

D. \( 3 + 7 + 15 + 31 + 63 = 119 \)

Correct Answer: A

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Question 5

A set ( A ) contains 5 elements, and a set ( B ) contains 3 elements. If the number of elements in the intersection of ( A ) and ( B ) is 2, find the number of elements in the union of ( A ) and ( B ).

Correct A. \( 5 + 3 - 2 = 6 \)

B. \( 5 + 3 - 2 = 6 \)

C. \( 5 + 3 - 2 = 6 \)

D. \( 5 + 3 - 2 = 6 \)

Correct Answer: A

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Question 6

Find the value of x in the equation \( \log_{10} \( x^2 \ \) = 4 ).

Correct A. 10

B. 100

C. 1000

D. 10000

Correct Answer: A

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Question 7

A histogram of exam scores has a mean of 75 and a s\tandard deviation of 10. If 75% of the scores fall below 85, what is the z-score corresponding to a score of 85?

A. 0.5

Correct B. 1

C. 1.5

D. 2

Correct Answer: B

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Question 8

Find the derivative of the function ( f(x) = \frac{x^2}{x+1} ) u\sing the quotient rule.

A. \frac{2x}{\( x+1 \)^2}

B. \frac{2x+2}{\( x+1 \)^2}

Correct C. \frac{2x\( x+1 \)-x^2}{\( x+1 \)^2}

D. \frac{2x\( x+1 \)+x^2}{\( x+1 \)^2}

Correct Answer: C

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Question 9

A vector ( mathbf{a} ) has a magnitude of 5 and points in the direction of the positive x-axis. A vector ( mathbf{b} ) has a magnitude of 3 and points in the direction of the positive y-axis. Find the magnitude of the sum of these two vectors.

A. 5

B. 6

C. 7

Correct D. 8

Correct Answer: D

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Question 10

Solve the equation \( x^2 + 4x + 4 = 0 \) u\sing the quadratic formula.

Correct A. -2

B. -1

C. 0

D. 1

Correct Answer: A

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Question 11

Find the volume of the frustum of a cone with height 12cm, lower base radius 6cm, and upper base radius 3cm.

A. 324\pi cm^3

Correct B. 648\pi cm^3

C. 972\pi cm^3

D. 1296\pi cm^3

Correct Answer: B

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Question 12

Solve for x in the equation \( 2^x + 3^x = 5^x \).

A. x = 2

B. x = 3

Correct C. x = 4

D. x = 5

Correct Answer: C

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Question 13

Find the area of the triangle with vertices (1, 2), (3, 4), and (5, 6).

Correct A. 10

B. 20

C. 30

D. 40

Correct Answer: A

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Question 14

Solve the matrix equation \( \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 5 \ 6 \end{bmatrix} \).

Correct A. \begin{bmatrix} 1 \ 2 \end{bmatrix}

B. \begin{bmatrix} 2 \ 3 \end{bmatrix}

C. \begin{bmatrix} 3 \ 4 \end{bmatrix}

D. \begin{bmatrix} 4 \ 5 \end{bmatrix}

Correct Answer: A

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Question 15

Find the equation of the circle with center (2, 3) and radius 4.

Correct A. \( x - 2 \)^2 + \( y - 3 \)^2 = 16

B. \( x - 2 \)^2 + \( y - 3 \)^2 = 32

C. \( x - 2 \)^2 + \( y - 3 \)^2 = 64

D. \( x - 2 \)^2 + \( y - 3 \)^2 = 128

Correct Answer: A

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Question 16

In a binary system, what is the value of the number represented by the sequence 1011?

A. 3

Correct B. 5

C. 7

D. 9

Correct Answer: B

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Question 17

A histogram is constructed with the following data: 2, 4, 6, 8, 10, 12, 14, 16. What is the mean of the data?

A. 7

B. 8

Correct C. 9

D. 10

Correct Answer: C

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Question 18

Find the equation of the circle with center \( -2, 3 \) and radius 4.

Correct A. \( x + 2 \ \)^2 + \( y - 3 \)^2 = 16 )

B. \( x - 2 \ \)^2 + \( y + 3 \)^2 = 16 )

C. \( x + 3 \ \)^2 + \( y - 2 \)^2 = 16 )

D. \( x - 3 \ \)^2 + \( y + 2 \)^2 = 16 )

Correct Answer: A

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Question 19

Find the area under the curve \( y = x^2 \) from \( x = 0 \) to \( x = 4 \).

A. 32

B. 64

Correct C. 96

D. 128

Correct Answer: C

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Question 20

Find the magnitude of the vector \( vec{a} = 2hat{i} + 3hat{j} \).

A. 5

Correct B. 10

C. 15

D. 20

Correct Answer: B

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Question 21

Find the derivative of the function ( f(x) = \frac{1}{\sqrt{x^2 + 1}} ) u\sing the chain rule.

Correct A. f'(x) = \frac{-x}{\( x^2 + 1 \)^{3/2}}

B. f'(x) = \frac{x}{\( x^2 + 1 \)^{3/2}}

C. f'(x) = \frac{1}{\( x^2 + 1 \)^{3/2}}

D. f'(x) = \frac{-1}{\( x^2 + 1 \)^{3/2}}

Correct Answer: A

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Question 22

Solve the inequality \( 2x^2 - 5x - 3 > 0 \).

Correct A. \( -∞, -1 \) ∪ (3, ∞)

B. \( -∞, -3 \) ∪ (1, ∞)

C. \( -∞, 1 \) ∪ (3, ∞)

D. \( -∞, -3 \) ∪ (1, ∞)

Correct Answer: A

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Question 23

Find the equation of the line pas\sing through the points (2, 3) and (4, 5).

Correct A. y = 2x - 1

B. y = 2x + 1

C. y = x - 1

D. y = x + 1

Correct Answer: A

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Question 24

A histogram of exam scores is shown below. What is the mean score?

A. 60

Correct B. 70

C. 80

D. 90

Correct Answer: B

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Question 25

Find the magnitude of the vector \( \vec{a} = 3\hat{i} + 4\hat{j} \).

Correct A. 5

B. 10

C. 15

D. 20

Correct Answer: A

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