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POST UTME LEAD CITY UNIVERSITY 2017 Mathematics | Objective

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Question 1

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

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

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

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

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

Correct Answer: C

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Question 2

Solve the inequality $ 2x^2 + 5x - 3 > 0 $.

Correct A. $ x < -\frac{1}{2} $ or $ x > \frac{3}{2} $

B. $ x < -\frac{3}{2} $ or $ x > \frac{1}{2} $

C. $ x < -\frac{1}{2} $ or $ x < \frac{3}{2} $

D. $ x > -\frac{1}{2} $ or $ x < \frac{3}{2} $

Correct Answer: A

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Question 3

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

Correct A. $ x - 2 \ $^2 + $ y - 3 $^2 = 16 )

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

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

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

Correct Answer: A

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Question 4

A card is drawn from a s\tandard deck of 52 cards. What is the probability that the card is a heart or a diamond?

A. $ \frac{13}{52} $

Correct B. $ \frac{26}{52} $

C. $ \frac{39}{52} $

D. $ \frac{52}{52} $

Correct Answer: B

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Question 5

Find the volume of the solid formed by revolving the region bounded by the curve $ y = x^2 $ and the line $ y = 2 $ about the x-axis.

Correct A. $ \frac{8pi}{3} $

B. $ \frac{16pi}{3} $

C. $ \frac{32pi}{3} $

D. $ \frac{64pi}{3} $

Correct Answer: A

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Question 6

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

A. 0

Correct B. -2

C. 2

D. -1

Correct Answer: B

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Question 7

Find the derivative of the function ( f(x) = 3x^2 + 2x - 5 ) u\sing the power rule.

Correct A. 6x + 2

B. 6x - 2

C. 3x^2 + 2

D. 3x^2 - 2

Correct Answer: A

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Question 8

Solve the system of linear equations u\sing matrices. Find the values of ( x ) and ( y ) in the system $ \begin{bmatrix} 2 & 1 \ 1 & 2 \ \end{bmatrix} \begin{bmatrix} x \ y \ \end{bmatrix} = \begin{bmatrix} 3 \ 4 \ \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 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 - 3 $^2 + $ y - 2 $^2 = 16 \)

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

D. $ x - 3 $^2 + $ y - 2 $^2 = 9 \)

Correct Answer: A

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Question 10

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

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 11

Find the value of $\int_{0}^{\pi} \sin^2(x) \cos^2(x) dx$.

A. 0

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

C. \frac{\pi}{2}

D. \frac{3\pi}{4}

Correct Answer: B

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Question 12

A set $A$ is defined as $A = \{x \in \mathbb{R} : x^2 + 2x + 1 > 0\}$. Find the value of $\cup A$.

Correct A. \mathbb{R}

B. $ -\infty, 0 $

C. $ 0, \infty $

D. $ -1, 1 $

Correct Answer: A

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Question 13

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

Correct A. 1

B. \frac{1}{2}

C. \frac{1}{3}

D. \frac{1}{4}

Correct Answer: A

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Question 14

A sequence $a_n$ is defined as $a_n = \frac{1}{n}$. Find the value of $\lim_{n \to \infty} a_n$.

Correct A. 0

B. 1

C. \frac{1}{2}

D. \frac{1}{3}

Correct Answer: A

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Question 15

A probability experiment consists of rolling a fair six-sided die. Find the probability that the number rolled is greater than 4.

A. \frac{1}{6}

B. \frac{1}{3}

C. \frac{1}{2}

Correct D. \frac{2}{3}

Correct Answer: D

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Question 16

Find the value of $x$ in the equation $2^x + 2^{x+2} = 2^{x+3}$.

A. 1

Correct B. 2

C. 3

D. 4

Correct Answer: B

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Question 17

Solve the inequality $\frac{1}{x-2} + \frac{1}{x+2} \leq \frac{1}{2}$.

A. x \in $ -\infty, -2 $ \cup $ 2, \infty $

B. x \in $ -\infty, -2 $ \cup (2, 4]

Correct C. x \in $ -\infty, -2 $ \cup (2, \infty]

D. x \in $ -\infty, -2] \cup \( 2, \infty \ $

Correct Answer: C

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Question 18

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

A. 10

Correct B. 12

C. 14

D. 16

Correct Answer: B

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Question 19

Solve the equation $\frac{x}{x+1} + \frac{x}{x-1} = 2$.

A. x = -1

Correct B. x = 1

C. x = -2

D. x = 2

Correct Answer: B

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Question 20

Find the value of $x$ in the equation $\log_{10} $ x^2 $ = 4$.

A. 2

Correct B. 4

C. 8

D. 16

Correct Answer: B

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Question 21

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

Correct A. -3

B. 3

C. -1

D. 1

Correct Answer: A

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Question 22

A sequence is defined by the recurrence relation [ a_n = 2a_{n-1} + 3 ] with initial term [ a_1 = 2 ]. Find the sum of the first five terms of the sequence.

A. 55

Correct B. 65

C. 75

D. 85

Correct Answer: B

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Question 23

The mean of a set of five numbers is 15. If one of the numbers is 10, find the sum of the remaining four numbers.

Correct A. 50

B. 60

C. 70

D. 80

Correct Answer: A

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Question 24

Solve the equation [ 2x^2 + 3x - 1 = 0 ].

A. x = -1/2

Correct B. x = 1/2

C. x = -1

D. x = 1

Correct Answer: B

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Question 25

Find the vector [ mathbf{a} ] such that [ mathbf{a} cdot mathbf{b} = 10 ] and [ mathbf{a} cdot mathbf{c} = 5 ], where [ mathbf{b} = egin{pmatrix} 2 \ 3 \end{pmatrix} ] and [ mathbf{c} = egin{pmatrix} 1 \ 2 \end{pmatrix} ].

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

B. \begin{pmatrix} 2 \ 3 \end{pmatrix}

C. \begin{pmatrix} 3 \ 4 \end{pmatrix}

D. \begin{pmatrix} 4 \ 5 \end{pmatrix}

Correct Answer: A

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