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POST UTME PAN-ATLANTIC UNIVERSITY 2020 Mathematics | Objective

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Question 1

Determine the value of the determinant of the matrix [ egin{pmatrix} 2 & 3 \ 4 & 5 \end{pmatrix} ].

Correct A. -1

B. 1

C. 2

D. 3

Correct Answer: A

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Question 2

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

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

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

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

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

Correct Answer: B

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Question 3

A set of exam scores has a mean of 80 and a s\tandard deviation of 10. If a new score of 90 is added to the set, what is the new mean?

A. 79.5

Correct B. 80.5

C. 81.5

D. 82.5

Correct Answer: B

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Question 4

Solve the equation [ 2^x + 3^x = 5^x ].

Correct A. x = 1

B. x = 2

C. x = 3

D. x = 4

Correct Answer: A

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Question 5

A sequence is defined by the formula [ a_n = 2n + 1 ]. Find the sum of the first 5 terms of the sequence.

A. 15

B. 20

Correct C. 25

D. 30

Correct Answer: C

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Question 6

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

Correct A. \( -infty, -2 \ \) cup [0, infty) )

B. \( -infty, -2 \ \) cup \( -2, 0 \) cup (0, infty) )

C. \( -infty, -2 \ \) cup (0, infty) )

D. \( -infty, -2 \ \) cup [0, 2] )

Correct Answer: A

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Question 7

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

Correct A. -120

B. 120

C. -60

D. 60

Correct Answer: A

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Question 8

Solve the system of linear equations \( egin{cases} x + y + z = 6 \ x + 2y + 3z = 12 \ 2x + 3y + 4z = 20 \end{cases} \).

Correct A. \( x = 1, y = 2, z = 3 \)

B. \( x = 2, y = 1, z = 3 \)

C. \( x = 1, y = 3, z = 2 \)

D. \( x = 2, y = 3, z = 1 \)

Correct Answer: A

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Question 9

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

Correct A. ( f'(x) = \frac{\( x^2 - 4 \)\( 2x + 2 \) - \( x^2 + 2x - 3 \)(2x)}{\( x^2 - 4 \)^2} )

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

C. ( f'(x) = \frac{\( x^2 - 4 \)\( 2x + 2 \) - \( x^2 + 2x - 3 \)(2x)}{\( x^2 - 4 \)^2} )

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

Correct Answer: A

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Question 10

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

Correct A. \( x = -2 pm 0i \)

B. \( x = -2 pm 2i \)

C. \( x = -2 pm 4i \)

D. \( x = -2 pm 6i \)

Correct Answer: A

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Question 11

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

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

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

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

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

Correct Answer: C

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Question 12

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

A. \left\( -\frac{5}{4}, \frac{3}{2} \right \)

Correct B. \left\( -\infty, -\frac{5}{4} \right \) \cup \left\( \frac{3}{2}, \infty \right \)

C. \left\( -\infty, -\frac{5}{4} \right \) \cup \left\( -\frac{3}{2}, \infty \right \)

D. \left\( -\infty, \frac{3}{2} \right \)

Correct Answer: B

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Question 13

Find the sum of the first 5 terms of the geometric series \( 2 + 6 + 18 + \cdots \).

Correct A. 120

B. 150

C. 180

D. 200

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} 3 \ 4 \end{bmatrix}

C. \begin{bmatrix} 5 \ 6 \end{bmatrix}

D. \begin{bmatrix} 7 \ 8 \end{bmatrix}

Correct Answer: A

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Question 15

Find the probability that at least one of the events A and B occurs, given that P(A) = 0.4, P(B) = 0.3, and P\( A \cap B \) = 0.1.

A. 0.5

Correct B. 0.6

C. 0.7

D. 0.8

Correct Answer: B

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Question 16

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

A. 10

B. 100

C. 1000

Correct D. 10000

Correct Answer: D

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Question 17

Solve for y in the equation \( 2y^2 + 5y - 3 = 0 \).

A. -1

Correct B. 1

C. 3

D. -3

Correct Answer: B

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Question 18

Find the sum of the first 5 terms of the geometric sequence with first term 2 and common ratio 3.

A. 10

B. 20

Correct C. 30

D. 40

Correct Answer: C

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Question 19

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

A. \( -∞, 1 \) ∪ (3, ∞)

Correct B. \( -∞, 3 \) ∪ (1, ∞)

C. \( -∞, 1 \) ∪ (1, 3)

D. \( -∞, 3 \) ∪ (3, ∞)

Correct Answer: B

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Question 20

Find the area of the triangle with vertices (0, 0), (3, 0), and (0, 4).

A. 6

B. 12

Correct C. 18

D. 24

Correct Answer: C

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Question 21

A right circular cone has a height of 15 cm and a base radius of 8 cm. Find the volume of the cone in terms of π.

Correct A. \frac{4000\pi}{3}

B. \frac{800\pi}{3}

C. \frac{1600\pi}{3}

D. \frac{3200\pi}{3}

Correct Answer: A

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Question 22

Solve the equation: \( x^2 + 4x + 4 = 0 \).

Correct A. \( x = -2 \)

B. \( x = 2 \)

C. \( x = -1 \)

D. \( x = 1 \)

Correct Answer: A

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Question 23

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

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

Correct Answer: A

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Question 24

A set A contains 3 elements. Find the number of subsets of A.

Correct A. 8

B. 16

C. 32

D. 64

Correct Answer: A

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Question 25

Find the sum of the first 5 terms of the geometric progression 2, 6, 18, ...

Correct A. 312

B. 3120

C. 31200

D. 312000

Correct Answer: A

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