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POST UTME PAN-ATLANTIC UNIVERSITY 2017 Mathematics | Objective

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Question 1

Find the determinant of the matrix \( egin{bmatrix} 2 & 3 \ 4 & 5 \end{bmatrix} \)

A. 1

Correct B. -1

C. 2

D. 3

Correct Answer: B

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Question 2

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

Correct A. x = -2

B. x = -1

C. x = 1

D. x = 2

Correct Answer: A

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Question 3

Find the derivative of the function ( f(x) = 3x^2 + 2x - 5 )

Correct A. 6x + 2

B. 3x^2 + 2

C. 6x - 2

D. 3x^2 - 2

Correct Answer: A

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Question 4

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

Correct A. x > 4

B. x < 4

C. x > 2

D. x < 2

Correct Answer: A

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Question 5

Find the vector ( mathbf{a} ) such that \( mathbf{a} cdot mathbf{b} = 6 \) and \( mathbf{a} cdot mathbf{c} = 3 \), where \( mathbf{b} = egin{bmatrix} 2 \ 3 \end{bmatrix} \) and \( mathbf{c} = egin{bmatrix} 1 \ 4 \end{bmatrix} \)

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

B. \begin{bmatrix} 2 \ 1 \end{bmatrix}

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

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

Correct Answer: A

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Question 6

In a certain number base, the representation of the number 27 is 101. If the base is denoted by 'b', find the value of 'b' u\sing the equation \( b^2 - 5b - 6 = 0 \).

A. 4

Correct B. 6

C. 8

D. 10

Correct Answer: B

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Question 7

A random experiment consists of rolling a fair six-sided die and then flipping a fair coin. If the number on the die is even and the coin lands heads up, find the probability that the number on the die is greater than 3.

Correct A. 1/4

B. 1/6

C. 1/2

D. 2/3

Correct Answer: A

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Question 8

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

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

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

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

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

Correct Answer: A

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Question 9

A rec\tangular box has dimensions 5 cm, 8 cm, and 3 cm. Find the volume of the box in cubic centimeters.

Correct A. 120

B. 150

C. 180

D. 200

Correct Answer: A

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Question 10

Find the magnitude of the vector \( \vec{a} = \langle 3, 4 \rangle \ \) and the vector \( \vec{b} = \langle -2, 1 \rangle \ \).

Correct A. \( \sqrt{29} \)

B. \( \sqrt{5} \)

C. \( \sqrt{2} \)

D. \( \sqrt{3} \)

Correct Answer: A

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Question 11

Find the area under the curve \( y = \frac{1}{2}x^2 + 3x - 2 \) from \( x = 0 \) to \( x = 4 \).

Correct A. 40

B. 42

C. 44

D. 46

Correct Answer: A

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Question 12

Solve for ( x ) in the equation \( \left| x - 2 \right| + \left| x - 5 \right| = 3 \).

A. 1

B. 3

Correct C. 4

D. 6

Correct Answer: C

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Question 13

A histogram of exam scores has a mean of 75 and a s\tandard deviation of 10. If the scores are normally distributed, what is the probability that a randomly selected score is between 60 and 90?

A. 0.5

B. 0.6

Correct C. 0.7

D. 0.8

Correct Answer: C

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Question 14

In a right-angled triangle, the length of the hypotenuse is 10 cm and one of the other sides is 6 cm. Find the length of the third side.

A. 8

Correct B. 9

C. 10

D. 12

Correct Answer: B

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Question 15

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

A. 1

Correct B. 2

C. 3

D. 4

Correct Answer: B

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Question 16

Find the equation of the circle with center at ((2,3)) and radius (5).

Correct A. \( x-2 \ \)^2 + \( y-3 \)^2 = 25 )

B. \( x+2 \ \)^2 + \( y-3 \)^2 = 25 )

C. \( x-2 \ \)^2 + \( y+3 \)^2 = 25 )

D. \( x+2 \ \)^2 + \( y+3 \)^2 = 25 )

Correct Answer: A

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Question 17

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

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

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

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

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

Correct Answer: A

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Question 18

A rec\tangular prism has a length of ( 10 ) cm, a width of ( 5 ) cm, and a height of ( 3 ) cm. Find its volume.

A. ( 150 ) cm\( ^3 \)

Correct B. ( 300 ) cm\( ^3 \)

C. ( 500 ) cm\( ^3 \)

D. ( 1000 ) cm\( ^3 \)

Correct Answer: B

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Question 19

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

Correct A. \( x = 2 \)

B. \( x = 3 \)

C. \( x = 4 \)

D. \( x = 5 \)

Correct Answer: A

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Question 20

A circle has an equation of the form \( x^2 + y^2 + Dx + Ey + F = 0 \). If the circle passes through the points ( (1,2) ) and ( (3,4) ), find its equation.

Correct A. \( x^2 + y^2 - 4x - 6y + 5 = 0 \)

B. \( x^2 + y^2 + 4x + 6y + 5 = 0 \)

C. \( x^2 + y^2 - 6x - 4y + 5 = 0 \)

D. \( x^2 + y^2 + 6x + 4y + 5 = 0 \)

Correct Answer: A

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Question 21

Find the determinant of the matrix [ egin{array}{ccc} 2 & 3 & 5 \ 1 & 2 & 3 \ 4 & 1 & 2 \end{array} ].

A. 0

Correct B. 1

C. 2

D. 3

Correct Answer: B

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Question 22

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

A. \left\( x \leq -2 \right \)

Correct B. \left\( x \geq -2 \right \)

C. \left\( x < -2 \right \)

D. \left\( x > -2 \right \)

Correct Answer: B

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Question 23

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

Correct A. 16

B. 32

C. 64

D. 128

Correct Answer: A

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Question 24

Solve the equation \( \log_2 (x) = 3 \).

Correct A. 8

B. 16

C. 32

D. 64

Correct Answer: A

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Question 25

Find the value of \( \sin (2x) \) when \( x = \frac{\pi}{6} \).

A. \frac{1}{2}

Correct B. \frac{\sqrt{3}}{2}

C. \frac{\sqrt{2}}{2}

D. \frac{\sqrt{6}}{2}

Correct Answer: B

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