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POST UTME AL-HIKMAH UNIVERSITY 2022 Mathematics | Objective

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Question 1

A random sample of 25 students from a university had a mean height of 175 cm with a s\tandard deviation of 5 cm. If the population s\tandard deviation is 6 cm, calculate the s\tandard error of the mean.

A. 2.5 cm

Correct B. 3.2 cm

C. 3.5 cm

D. 4.1 cm

Correct Answer: B

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Question 2

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 3

A histogram of exam scores is shown below. If the mean score is 60 and the s\tandard deviation is 10, calculate the area under the curve between 50 and 70.

A. 0.5

Correct B. 0.6

C. 0.7

D. 0.8

Correct Answer: B

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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 = 2x $ about the x-axis.

Correct A. \frac{4}{3}\pi

B. \frac{2}{3}\pi

C. \frac{1}{3}\pi

D. \frac{1}{2}\pi

Correct Answer: A

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Question 5

A vector $ \vec{a} $ has a magnitude of 5 units and makes an angle of 30 degrees with the positive x-axis. Find the x and y components of the vector.

Correct A. \langle 4, 2.5 \rangle

B. \langle 3, 2.5 \rangle

C. \langle 4, 3 \rangle

D. \langle 3, 3 \rangle

Correct Answer: A

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Question 6

Let X and Y be indep\endent events with P(X) = 0.4 and P(Y) = 0.6. Find the probability that X occurs but Y does not.

Correct A. 0.24

B. 0.36

C. 0.48

D. 0.64

Correct Answer: A

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Question 7

Solve for x in the equation $ 2^x + 2^{x+1} = 3 cdot 2^{x+1} $.

A. -1

B. 0

Correct C. 1

D. 2

Correct Answer: C

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Question 8

Find the area under the curve $ y = \frac{1}{x} $ from x = 1 to x = 2.

A. 0.5

B. 1

Correct C. 1.5

D. 2

Correct Answer: C

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Question 9

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 10

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

A. 0

B. 1

Correct C. -1

D. 2

Correct Answer: C

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Question 11

A researcher conducted a study on the relationship between the mean height of a population and the s\tandard deviation. The data points are as follows: 160, 170, 180, 190, 200. If the mean height is 175 cm, what is the s\tandard deviation of the population?

A. 10 cm

B. 15 cm

Correct C. 20 cm

D. 25 cm

Correct Answer: C

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Question 12

Solve for 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 13

A company produces two products, A and B. The profit from product A is $10 per unit, and the profit from product B is $15 per unit. If the company produces 100 units of product A and 50 units of product B, what is the total profit?

A. $1500

Correct B. $2000

C. $2500

D. $3000

Correct Answer: B

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Question 14

A histogram is shown below. What is the mode of the data?

Correct A. A

B. B

C. C

D. D

Correct Answer: A

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Question 15

A 3x3 matrix is given as: \[ \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \end{bmatrix} \]. What is the determinant of the matrix?

Correct A. 0

B. 1

C. 2

D. 3

Correct Answer: A

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Question 16

Determine the value of $\sum_{n=1}^{\infty}\frac{2n}{3^n}$.

Correct A. 4

B. 6

C. 8

D. 10

Correct Answer: A

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Question 17

Solve the inequality $\frac{x^2-4x-3}{x^2-4x+3}>0$.

Correct A. $ -\infty,-1)\cup\( 3,\infty \ $

B. $ -\infty,-3)\cup\( 1,\infty \ $

C. $ -\infty,-1)\cup\( 1,\infty \ $

D. $ -\infty,-3)\cup\( 3,\infty \ $

Correct Answer: A

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Question 18

Find the equation of the \tangent line to the curve $y=x^2+2x-3$ at the point $$ 1,-2 $$.

Correct A. y=-4x+1

B. y=4x-1

C. y=-x-1

D. y=x-1

Correct Answer: A

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Question 19

A particle moves along the x-axis with the equation of motion $x(t)=2t^2-5t+1$. Find the velocity and acceleration at time $t=2$ seconds.

Correct A. (2,6)

B. $ -4,-2 $

C. (4,6)

D. $ -2,-4 $

Correct Answer: A

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Question 20

Solve the system of equations $\begin{cases}x+y+z=6\x+2y+3z=11\2x+3y+4z=14\end{cases}$.

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 21

Find the equation of the circle with center at $ -2, 3 $ and pas\sing through the point (1, 6).

Correct A. \boxed{$ x + 2 $^2 + $ y - 3 $^2 = 25}

B. $ x - 2 $^2 + $ y + 3 $^2 = 16

C. $ x + 2 $^2 + $ y - 3 $^2 = 36

D. $ x - 2 $^2 + $ y + 3 $^2 = 25

Correct Answer: A

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Question 22

Solve for x in the equation $ 2^x + 3^x = 5^x $.

Correct A. \boxed{x = 2}

B. x = 3

C. x = 4

D. x = 5

Correct Answer: A

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Question 23

Find the volume of the frustum of a cone with height 10 cm, lower base radius 6 cm, and upper base radius 4 cm.

Correct A. \boxed{\frac{400\pi}{3} \text{ cm}^3

B. \frac{500\pi}{3} \text{ cm}^3

C. \frac{600\pi}{3} \text{ cm}^3

D. \frac{700\pi}{3} \text{ cm}^3

Correct Answer: A

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Question 24

A set A contains 5 elements. If the number of subsets of A is 32, find the number of elements in the power set of A.

Correct A. \boxed{32}

B. 16

C. 64

D. 128

Correct Answer: A

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Question 25

Find the sum of the first 10 terms of the geometric progression 2, 6, 18, ...

Correct A. \boxed{59049}

B. 6144

C. 6561

D. 729

Correct Answer: A

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