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POST UTME EKSU 2024 Mathematics | Objective

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Question 1

Let ( X ) be a random variable with probability density function ( f(x) = egin{cases} 2x, & 0 leq x leq 1 \ 0, & \text{otherwise} \end{cases} ). Find the probability that ( X ) takes a value between 0.5 and 1.

A. \frac{1}{2}

Correct B. \frac{3}{4}

C. \frac{5}{8}

D. \frac{7}{16}

Correct Answer: B

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Question 2

Find the area under the curve \( y = x^2 \sin\( x \ \) ) from \( x = 0 \) to \( x = \frac{\pi}{2} \).

A. \frac{1}{2}

B. \frac{\pi}{4}

Correct C. \frac{\pi}{2}

D. \frac{3\pi}{4}

Correct Answer: C

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Question 3

Solve the system of linear equations: \[ \begin{cases} x + y + z = 6 \ x + 2y + 3z = 14 \ 2x + 3y + 4z = 20 \end{cases} \]

Correct A. \{ (1, 2, 3) \}

B. \{ (2, 3, 1) \}

C. \{ (3, 1, 2) \}

D. \{ (1, 3, 2) \}

Correct Answer: A

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Question 4

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

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

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

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

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

Correct Answer: A

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Question 5

Find the sum of the first 10 terms of the geometric series with first term \( a = 2 \) and common ratio \( r = \frac{1}{2} \).

Correct A. \frac{1023}{512}

B. \frac{1024}{512}

C. \frac{1025}{512}

D. \frac{1026}{512}

Correct Answer: A

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Question 6

Find the determinant of the matrix [ egin{pmatrix} 2 & 3 & 1 \ 4 & 1 & 2 \ 3 & 2 & 4 \end{pmatrix} ].

A. 0

Correct B. 1

C. 2

D. 3

Correct Answer: B

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Question 7

A sequence is defined as [ a_n = 2n^2 + 3n - 1 ]. Find the sum of the first 5 terms of the sequence.

Correct A. 105

B. 115

C. 125

D. 135

Correct Answer: A

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Question 8

Find the derivative of the function [ f(x) = 3x^2 \sin (2x) ] u\sing the product rule.

Correct A. 6x^2 \sin (2x) + 12x \cos (2x)

B. 6x^2 \sin (2x) - 12x \cos (2x)

C. 12x^2 \sin (2x) + 6x \cos (2x)

D. 12x^2 \sin (2x) - 6x \cos (2x)

Correct Answer: A

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Question 9

In the diagram below, [ egin{pmatrix} 2 & 3 & 1 \ 4 & 1 & 2 \ 3 & 2 & 4 \end{pmatrix} ] is a matrix. Find the determinant of the matrix.

A. 0

Correct B. 1

C. 2

D. 3

Correct Answer: B

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Question 10

A right-angled triangle has sides of length 3, 4, and 5. Find the area of the triangle.

Correct A. 6

B. 8

C. 10

D. 12

Correct Answer: A

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Question 11

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

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

B. \( \frac{1}{2x^{3/2}} \)

C. \( \frac{1}{x^{3/2}} \)

D. \( \frac{-1}{x^{3/2}} \)

Correct Answer: A

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Question 12

Solve the matrix equation \( egin{bmatrix} 2 & 1 \ 1 & 2 \end{bmatrix} egin{bmatrix} x \ y \end{bmatrix} = egin{bmatrix} 3 \ 4 \end{bmatrix} \) for ( x ) and ( y ).

A. \( x = 1, y = 2 \)

Correct B. \( x = 2, y = 1 \)

C. \( x = 1, y = 1 \)

D. \( x = 2, y = 2 \)

Correct Answer: B

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Question 13

A random variable ( X ) has a probability distribution given by \( P\( X = x \ \) = \frac{1}{2} ) for \( x = 1, 2, 3, 4 \). Find the probability that ( X ) is greater than 2.

A. \( \frac{1}{4} \)

B. \( \frac{1}{2} \)

Correct C. \( \frac{3}{4} \)

D. \( \frac{1}{2} \)

Correct Answer: C

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Question 14

Solve the inequality \( \log_{10} \( x^2 \ \) > 4 ) for ( x ).

Correct A. \( x > 10 \)

B. \( x < -10 \)

C. \( x > -10 \)

D. \( x < 10 \)

Correct Answer: A

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Question 15

Find the derivative of the function ( f(x) = \sin \( x^2 \) ) u\sing the chain rule.

Correct A. \( 2x \cos \( x^2 \ \) )

B. \( 2x \sin \( x^2 \ \) )

C. \( \cos \( x^2 \ \) )

D. \( \sin \( x^2 \ \) )

Correct Answer: A

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Question 16

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

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

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

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

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

Correct Answer: A

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Question 17

Solve the equation \( \log_{10} \( x^2 \ \) = 4 ) for ( x ).

Correct A. \( x = 10^4 \)

B. \( x = 10^{-4} \)

C. \( x = 10^2 \)

D. \( x = 10^{-2} \)

Correct Answer: A

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Question 18

Find the area of the triangle with vertices ( (0, 0) ), ( (3, 0) ), and ( (0, 2) ).

Correct A. 6

B. 12

C. 18

D. 24

Correct Answer: A

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Question 19

Solve the inequality \( x^2 - 4x + 4 geq 0 \).

Correct A. ( x leq 2 )

B. ( x geq 2 )

C. \( x < 2 \)

D. \( x > 2 \)

Correct Answer: A

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Question 20

Find the derivative of the function ( f(x) = \sin^2 x ) u\sing the chain rule.

Correct A. \( 2 \sin x \cos x \)

B. \( \cos^2 x \)

C. \( \sin^2 x \)

D. \( \cos x \)

Correct Answer: A

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Question 21

Solve the inequality \( \frac{x^2 - 4x - 5}{x^2 - 4x + 4} > 0 \) for ( x in mathbb{R} ).

Correct A. \( -∞, -1 \) ∪ (1, ∞)

B. \( -∞, -1 \) ∪ (1, 2)

C. \( -∞, -1 \) ∪ (2, ∞)

D. \( -∞, 1 \) ∪ (2, ∞)

Correct Answer: A

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Question 22

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

Correct A. 16π/3

B. 16π/5

C. 16π/7

D. 16π/9

Correct Answer: A

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Question 23

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

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

B. \( x - 2 \)^2 + \( y - 3 \)^2 = 20

C. \( x - 2 \)^2 + \( y - 3 \)^2 = 24

D. \( x - 2 \)^2 + \( y - 3 \)^2 = 28

Correct Answer: A

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Question 24

Find the determinant of the matrix \( egin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \end{bmatrix} \).

Correct A. 0

B. 1

C. -1

D. 2

Correct Answer: A

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Question 25

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

Correct A. 16π

B. 32π

C. 64π

D. 128π

Correct Answer: A

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