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POST UTME OAU 2021 Mathematics | Objective

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Question 1

Let \( mathbf{a} = egin{pmatrix} 2 \ 3 \end{pmatrix} \) and \( mathbf{b} = egin{pmatrix} -1 \ 4 \end{pmatrix} \). Find the projection of ( mathbf{b} ) onto ( mathbf{a} ).

A. \( egin{pmatrix} 0 \ 0 \end{pmatrix} \)

Correct B. \( egin{pmatrix} 1 \ 2 \end{pmatrix} \)

C. \( egin{pmatrix} -1 \ 2 \end{pmatrix} \)

D. \( egin{pmatrix} 1 \ -2 \end{pmatrix} \)

Correct Answer: B

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Question 2

In the diagram below, ( AB ) is a diameter of the circle with center ( O ). If \( angle AOB = 60^{circ} \), find the measure of ( angle ACB ).

A. 30^{circ}

B. 45^{circ}

Correct C. 60^{circ}

D. 90^{circ}

Correct Answer: C

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Question 3

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 4

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

Correct A. -1

B. 1

C. 2

D. 3

Correct Answer: A

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Question 5

In the sequence ( 1, 2, 4, 8, 16, ... ), find the next term.

A. 32

Correct B. 64

C. 128

D. 256

Correct Answer: B

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Question 6

Determine the value of x in the equation \( \frac{x}{2} + \frac{3}{4} = \frac{7}{8} \).

A. 1

B. 2

C. 3

Correct D. 4

Correct Answer: D

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Question 7

A histogram is constructed with the following data: 5, 7, 9, 11, 13. What is the mean of the data?

A. 9

Correct B. 10

C. 11

D. 12

Correct Answer: B

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Question 8

Find the volume of the sphere \( x^2 + y^2 + z^2 = 4 \) u\sing the method of cylindrical shells.

A. 16\pi

B. 32\pi

Correct C. 48\pi

D. 64\pi

Correct Answer: C

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Question 9

A vector \( \vec{a} = 2\hat{i} + 3\hat{j} \) is rotated by an angle of 30 degrees. What is the magnitude of the rotated vector?

A. 3.61

B. 4.24

Correct C. 4.47

D. 4.58

Correct Answer: C

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Question 10

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{1}{\( x^2 + 1 \)^2}

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

Correct Answer: A

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Question 11

Find the volume of the frustum of a cone with height 6 cm, lower base radius 4 cm, and upper base radius 2 cm.

A. 24\pi cm^3

Correct B. 48\pi cm^3

C. 96\pi cm^3

D. 192\pi cm^3

Correct Answer: B

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Question 12

Solve for x in the equation \( x^4 - 6x^2 + 8 = 0 \).

Correct A. x = \pm 2

B. x = \pm \sqrt{2}

C. x = \pm 2\sqrt{2}

D. x = \pm 4

Correct Answer: A

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Question 13

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 = 25

C. \( x - 2 \)^2 + \( y - 3 \)^2 = 36

D. \( x - 2 \)^2 + \( y - 3 \)^2 = 49

Correct Answer: A

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Question 14

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

Correct A. x \leq -1 \text{ or } x \geq \frac{3}{2}

B. x \leq -1 \text{ or } x \geq 1

C. x \leq -1 \text{ or } x \geq 2

D. x \leq 1 \text{ or } x \geq 2

Correct Answer: A

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Question 15

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

A. 64

B. 128

Correct C. 192

D. 256

Correct Answer: C

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Question 16

Find the area under the curve y = x^2 + 2x - 3 from x = 0 to x = 2.

A. 14

Correct B. 16

C. 18

D. 20

Correct Answer: B

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Question 17

In the diagram below, 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 18

Two events A and B are indep\endent. If P(A) = 0.4 and P(B) = 0.6, find P(A ∩ B).

Correct A. 0.24

B. 0.36

C. 0.48

D. 0.64

Correct Answer: A

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Question 19

Simplify the expression: \frac{2^3 \cdot 3^2 \cdot 4^2}{2^2 \cdot 3 \cdot 4^3}.

Correct A. \frac{1}{4}

B. \frac{1}{2}

C. 1

D. 2

Correct Answer: A

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Question 20

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 21

Solve the quadratic equation \( x^2 + 4x - 5 = 0 \) u\sing the quadratic formula. What is the value of ( x )?

A. 1

B. -5

Correct C. 2

D. -2

Correct Answer: C

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Question 22

Find the volume of the solid formed by revolving the region bounded by the curves \( y = x^2 \) and \( y = 4 - x \) about the x-axis.

A. 16π

Correct B. 32π

C. 64π

D. 128π

Correct Answer: B

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Question 23

A circle has a diameter of 10 cm. Find the area of the circle.

Correct A. 25π

B. 50π

C. 100π

D. 200π

Correct Answer: A

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Question 24

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

Correct A. x < -3 or x > 1

B. x < -1 or x > 3

C. x < 1 or x > 3

D. x < -3 or x < 1

Correct Answer: A

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Question 25

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

Correct A. 6x + 2

B. 6x - 2

C. 3x^2 + 2

D. 3x^2 - 2

Correct Answer: A

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