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POST UTME COAL CITY UNIVERSITY 2017 Mathematics | Objective

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Question 1

Find the equation of the circle pas\sing through the points (1, 2), (3, 4), and $ -2, 3 $.

Correct A. $ x^2 + y^2 + 4x - 8y + 5 = 0 $

B. $ x^2 + y^2 - 4x + 8y + 5 = 0 $

C. $ x^2 + y^2 + 8x - 4y + 5 = 0 $

D. $ x^2 + y^2 - 8x + 4y + 5 = 0 $

Correct Answer: A

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Question 2

If ( f(x) = \frac{1}{x^2 + 1} ), find ( f'(x) ) u\sing the chain rule.

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

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

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

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

Correct Answer: A

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Question 3

A fair six-sided die is rolled. If the outcome is an even number, a second die is rolled. Find the probability that the sum of the two dice is 7.

Correct A. $ \frac{1}{12} $

B. $ \frac{1}{6} $

C. $ \frac{1}{4} $

D. $ \frac{1}{3} $

Correct Answer: A

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Question 4

Find the vector ( mathbf{v} ) such that $ mathbf{v} cdot mathbf{i} = 3 $ and $ mathbf{v} cdot mathbf{j} = -2 $.

Correct A. $ mathbf{v} = 3mathbf{i} - 2mathbf{j} $

B. $ mathbf{v} = -3mathbf{i} + 2mathbf{j} $

C. $ mathbf{v} = 3mathbf{i} + 2mathbf{j} $

D. $ mathbf{v} = -3mathbf{i} - 2mathbf{j} $

Correct Answer: A

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Question 5

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

Correct A. ( 0 )

B. ( 1 )

C. $ -1 $

D. ( 2 )

Correct Answer: A

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Question 6

Find the area under the curve of ( f(x) = \frac{1}{x^2} ) from $ x = 1 $ to $ x = 2 $.

Correct A. $ \frac{1}{1} - \frac{1}{4} $

B. $ \frac{1}{2} - \frac{1}{4} $

C. $ \frac{1}{4} - \frac{1}{2} $

D. $ \frac{1}{4} - \frac{1}{1} $

Correct Answer: A

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Question 7

In the diagram below, find the length of ( AB ) if $ AC = 10 $ cm, $ BC = 6 $ cm, and $ angle ABC = 90^circ $.

Correct A. 8 cm

B. 6 cm

C. 10 cm

D. 12 cm

Correct Answer: A

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Question 8

Solve for ( x ) in the equation $ \log_{10} \( x^2 \ $ = 4 ).

Correct A. $ 10^4 $

B. $ 10^{-4} $

C. $ 10^2 $

D. $ 10^{-2} $

Correct Answer: A

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Question 9

Find the volume of the sphere with radius $ r = 4 $ cm.

Correct A. $ \frac{4}{3} pi \( 4 \ $^3 )

B. $ \frac{4}{3} pi \( 4 \ $^2 )

C. $ \frac{4}{3} pi \( 4 \ $ )

D. $ \frac{4}{3} pi \( 4 \ $^2 )

Correct Answer: A

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Question 10

Find the equation of the line pas\sing through the points ( (2,3) ) and ( (4,5) ).

Correct A. $ y = \frac{1}{2} x + 1 $

B. $ y = \frac{1}{2} x - 1 $

C. $ y = 2x + 1 $

D. $ y = 2x - 1 $

Correct Answer: A

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Question 11

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} ).

Correct A. $ egin{pmatrix} -\frac{11}{13} \ \frac{24}{13} \end{pmatrix} $

B. $ egin{pmatrix} \frac{11}{13} \ -\frac{24}{13} \end{pmatrix} $

C. $ egin{pmatrix} -\frac{24}{13} \ \frac{11}{13} \end{pmatrix} $

D. $ egin{pmatrix} \frac{24}{13} \ -\frac{11}{13} \end{pmatrix} $

Correct Answer: A

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Question 12

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

Correct A. 14

B. -14

C. 21

D. -21

Correct Answer: A

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Question 13

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

A. 10

Correct B. 15

C. 20

D. 25

Correct Answer: B

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Question 14

Find the sum of the first five terms of the geometric series $ 2 + 6 + 18 + 54 + ldots $.

Correct A. 242

B. 242

C. 242

D. 242

Correct Answer: A

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Question 15

Solve the equation $ \sin^2 x + \cos^2 x = 1 $ for ( x ) in the interval ( [0, 2pi] ).

Correct A. $ x = 0, pi, 2pi $

B. $ x = \frac{pi}{2}, \frac{3pi}{2} $

C. $ x = \frac{pi}{4}, \frac{3pi}{4}, \frac{5pi}{4}, \frac{7pi}{4} $

D. $ x = \frac{pi}{6}, \frac{5pi}{6}, \frac{7pi}{6}, \frac{11pi}{6} $

Correct Answer: A

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Question 16

Find the mean of the data set ( { 2, 4, 6, 8, 10 } ).

Correct A. 6

B. 8

C. 10

D. 12

Correct Answer: A

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Question 17

Find the median of the data set ( { 1, 3, 5, 7, 9 } ).

A. 5

Correct B. 7

C. 9

D. 11

Correct Answer: B

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Question 18

Find the mode of the data set ( { 2, 4, 6, 8, 10 } ).

A. 2

B. 4

C. 6

D. 8

Correct Answer: VIEW ANSWER

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Question 19

Find the value of $\frac{d}{dx}\left$ \frac{1}{x^2}\right $$.

Correct A. -\frac{2}{x^3}

B. +\frac{2}{x^3}

C. -\frac{1}{x^3}

D. +\frac{1}{x^3}

Correct Answer: A

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Question 20

A histogram of exam scores has a mean of 70 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 80?

Correct A. 0.135

B. 0.135

C. 0.135

D. 0.135

Correct Answer: A

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Question 21

Solve the equation $x^2 + 5x + 6 = 0$ u\sing the quadratic formula.

Correct A. x = -2 or x = -3

B. x = 2 or x = 3

C. x = -3 or x = 2

D. x = 3 or x = -2

Correct Answer: A

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Question 22

A firm produces two products, A and B. Product A requires 2 hours of labor and 3 hours of capital, while product B requires 3 hours of labor and 2 hours of capital. If the firm has 120 hours of labor and 100 hours of capital, how many units of product A and product B should the firm produce to maximize profit?

Correct A. (20, 30)

B. (30, 20)

C. (25, 25)

D. (35, 15)

Correct Answer: A

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Question 23

A particle moves along the x-axis with a velocity given by $v(t) = 2t^2 - 5t + 1$. Find the acceleration of the particle at time $t = 2$ seconds.

Correct A. 4

B. -10

C. -6

D. 6

Correct Answer: A

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Question 24

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

A. 0

Correct B. 10

C. 20

D. 30

Correct Answer: B

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Question 25

Solve the system of linear equations u\sing matrices: [ \begin{cases} 2x + 3y = 7 \\ x - 2y = -3 \end{cases} \].

Correct A. \begin{pmatrix} x = 1 \\ y = 2 \end{pmatrix}

B. \begin{pmatrix} x = 2 \\ y = 1 \end{pmatrix}

C. \begin{pmatrix} x = 3 \\ y = 0 \end{pmatrix}

D. \begin{pmatrix} x = 0 \\ y = 3 \end{pmatrix}

Correct Answer: A

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