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POST UTME MOUNTAIN TOP UNIVERSITY 2018 Mathematics | Objective

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Question 1

Solve the inequality \( \frac{x}{x-2} > 1 \) for \( x > 2 \).

Correct A. \( x > 3 \ \)

B. \( x < 3 \ \)

C. \( x = 3 \ \)

D. \( x = 2 \ \)

Correct Answer: A

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Question 2

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

Correct A. \( \frac{64}{3} \ \)

B. \( 16 \ \)

C. \( 32 \ \)

D. \( 64 \ \)

Correct Answer: A

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Question 3

Solve the system of equations \( \begin{cases} x + y = 4 \ x - 2y = -2 \end{cases} \).

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

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

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

D. \( x = 4, y = 0 \ \)

Correct Answer: A

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Question 4

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

Correct A. \( \frac{\pi}{12} \ \)

B. \( \frac{\pi}{6} \ \)

C. \( \frac{\pi}{4} \ \)

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

Correct Answer: A

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Question 5

Solve the matrix equation \( \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 3 \ 7 \end{bmatrix} \).

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

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

C. \( x = 3, y = 4 \ \)

D. \( x = 4, y = 3 \ \)

Correct Answer: A

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Question 6

A random variable X has a probability distribution given by P\( X = 1 \) = 0.3, P\( X = 2 \) = 0.4, P\( X = 3 \) = 0.3. If Y is an indep\endent random variable with P\( Y = 1 \) = 0.5, P\( Y = 2 \) = 0.5, find the probability that X + Y = 3.

A. 0.35

B. 0.45

Correct C. 0.55

D. 0.65

Correct Answer: C

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Question 7

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

Correct A. \text{Equation of circle: } \( x - 2 \)^2 + \( y - 3 \)^2 = 4^2

B. \text{Equation of circle: } \( x + 2 \)^2 + \( y - 3 \)^2 = 4^2

C. \text{Equation of circle: } \( x - 2 \)^2 + \( y + 3 \)^2 = 4^2

D. \text{Equation of circle: } \( x + 2 \)^2 + \( y + 3 \)^2 = 4^2

Correct Answer: A

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Question 8

Solve the equation \frac{x^2 - 4}{x + 2} = 2.

A. x = 0

Correct B. x = 2

C. x = -2

D. x = 4

Correct Answer: B

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Question 9

Find the determinant of the matrix \begin{bmatrix} 2 & 3 & 1 \ 4 & 1 & 2 \ 3 & 2 & 1 \end{bmatrix}.

Correct A. 0

B. 2

C. 4

D. 6

Correct Answer: A

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Question 10

Solve the trigonometric equation \sin^2 x + \cos^2 x = 1.

A. x = \frac{\pi}{2}

Correct B. x = \frac{\pi}{4}

C. x = \frac{3\pi}{4}

D. x = \frac{5\pi}{4}

Correct Answer: B

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Question 11

Let \( A = egin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \) and \( B = egin{bmatrix} 5 & 6 \ 7 & 8 \end{bmatrix} \). Find the determinant of the matrix ( AB ) if it exists.

A. 0

B. 1

Correct C. 2

D. 3

Correct Answer: C

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Question 12

A box contains 5 red balls and 3 blue balls. If a ball is drawn at random, what is the probability that it is blue?

Correct A. 1/2

B. 2/5

C. 3/8

D. 1/4

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

Let \( S = \{ 1, 2, 3, 4, 5 \} \). Find the number of subsets of ( S ) that contain exactly 3 elements.

A. 5

B. 10

Correct C. 15

D. 20

Correct Answer: C

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Question 15

Find the sum of the first 5 terms of the geometric progression ( 2, 6, 18, 54, ... ).

Correct A. 120

B. 130

C. 140

D. 150

Correct Answer: A

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Question 16

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

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

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

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

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

Correct Answer: C

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Question 17

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

A. x = 2

Correct B. x = 3

C. x = 4

D. x = 5

Correct Answer: B

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Question 18

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

Correct A. 6

B. 8

C. 10

D. 12

Correct Answer: A

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Question 19

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

A. x^2 \cos x + 2x \sin x

B. x^2 \cos x - 2x \sin x

Correct C. x^2 \cos x + 2x \sin x

D. x^2 \cos x - 2x \sin x

Correct Answer: C

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Question 20

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

Correct A. x < -3

B. x > -3

C. x < 3

D. x > 3

Correct Answer: A

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Question 21

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 = 4

C. \( x - 2 \)^2 + \( y - 3 \)^2 = 9

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

Correct Answer: A

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Question 22

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

A. 2 \cos (2x)

B. 2 \sin (2x)

Correct C. 2 \cos (2x)

D. 2 \sin (2x)

Correct Answer: C

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Question 23

Solve the system of equations \( x + y = 4 \) and \( x - y = 2 \).

A. x = 3, y = 1

B. x = 1, y = 3

C. x = 2, y = 2

D. x = 4, y = 0

Correct Answer: VIEW ANSWER

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Question 24

Let X and Y be indep\endent events with P(X) = 0.4 and P(Y) = 0.6. What is the probability that both events occur?

Correct A. 0.24

B. 0.36

C. 0.48

D. 0.64

Correct Answer: A

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Question 25

Find the sum of the first 10 terms of the geometric series 2 + 6 + 18 + ...

A. 3120

B. 3240

Correct C. 3360

D. 3480

Correct Answer: C

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