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POST UTME CHRISTOPHER UNIVERSITY 2022 Mathematics | Objective

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Question 1

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

Correct A. f'(x) = \frac{-x}{\( 1 + x^2 \)^{3/2}}

B. f'(x) = \frac{x}{\( 1 + x^2 \)^{3/2}}

C. f'(x) = \frac{1}{\( 1 + x^2 \)^{3/2}}

D. f'(x) = \frac{-1}{\( 1 + x^2 \)^{3/2}}

Correct Answer: A

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Question 2

A random experiment has two indep\endent events, A and B. If P(A) = 0.4 and P(B) = 0.6, find the probability that both events occur.

Correct A. 0.24

B. 0.48

C. 0.64

D. 0.8

Correct Answer: A

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Question 3

Simplify the expression \( \frac{2x^2 \sqrt{3x}}{x^3 \sqrt{2x}} \).

Correct A. 2 \sqrt{\frac{3}{2}}

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

C. 2 \sqrt{\frac{3}{2}} \sqrt{x}

D. 2 \sqrt{\frac{2}{3}} \sqrt{x}

Correct Answer: A

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Question 4

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

Correct A. y = 2x - 1

B. y = 2x + 1

C. y = x + 2

D. y = x - 2

Correct Answer: A

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Question 5

A binary operation \( * \) on the set of real numbers is defined as \( a * b = a^2 + b^2 \). Find the value of \( 2 * 3 \).

Correct A. 13

B. 5

C. 7

D. 9

Correct Answer: A

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Question 6

Let ( X ) and ( Y ) be indep\endent random variables with probability density functions \( f_X\( x \ \) = egin{cases} 2x, & 0 leq x leq 1 \ 0, & \text{otherwise} \end{cases} ) and \( f_Y\( y \ \) = egin{cases} 3y^2, & 0 leq y leq 1 \ 0, & \text{otherwise} \end{cases} ). Find the probability that \( X + Y leq 1 \).

Correct A. \frac{1}{2}

B. \frac{1}{3}

C. \frac{2}{3}

D. \frac{3}{4}

Correct Answer: A

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Question 7

Find the equation of the circle with center ( (2, 3) ) and radius ( 4 ) in the form \( x - h \ \)^2 + \( y - k \)^2 = r^2 ).

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 8

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

A. x = 2

Correct B. x = 3

C. x = 4

D. x = 5

Correct Answer: B

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Question 9

Find the sum of the first 10 terms of the geometric series \( 2, 4, 8, 16, \ldots \).

A. 1023

Correct B. 1024

C. 1025

D. 1026

Correct Answer: B

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Question 10

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

Correct A. y = 2x - 1

B. y = 2x + 1

C. y = 2x - 2

D. y = 2x + 2

Correct Answer: A

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Question 11

Let X be a random variable with probability density function f(x) = \( \frac{1}{2}e^{-|x|} \) for -∞ < x < ∞. Find the probability that X is greater than 1.

Correct A. 0.5

B. 0.6

C. 0.7

D. 0.8

Correct Answer: A

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Question 12

Solve for x in the equation \( x^3 - 6x^2 + 11x - 6 = 0 \).

A. 1

B. 2

Correct C. 3

D. 4

Correct Answer: C

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Question 13

Find the surface area of the solid formed by rotating the region bounded by y = x^2 and y = 4 - x^2 about the x-axis.

A. 16π

Correct B. 32π

C. 48π

D. 64π

Correct Answer: B

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Question 14

Let A be a 3x3 matrix with elements a_{ij}. Find the determinant of A.

Correct A. a_{11}a_{22}a_{33} - a_{12}a_{23}a_{31} + a_{13}a_{21}a_{32}

B. a_{11}a_{22}a_{33} - a_{12}a_{23}a_{31} - a_{13}a_{21}a_{32}

C. a_{11}a_{22}a_{33} + a_{12}a_{23}a_{31} + a_{13}a_{21}a_{32}

D. a_{11}a_{22}a_{33} + a_{12}a_{23}a_{31} - a_{13}a_{21}a_{32}

Correct Answer: A

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Question 15

Find the sum of the infinite geometric series with first term 1 and common ratio \( \frac{1}{2} \).

A. 1

Correct B. 2

C. 3

D. 4

Correct Answer: B

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Question 16

Find the determinant of the matrix [ egin{array}{ccc} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \end{array} ].

A. -3

Correct B. 0

C. 3

D. 6

Correct Answer: B

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Question 17

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 18

Find the volume of the sphere with radius 5 cm.

A. 500\pi

B. 1000\pi

Correct C. 2000\pi

D. 5000\pi

Correct Answer: C

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Question 19

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

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

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

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

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

Correct Answer: C

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Question 20

Find the surface area of the cylinder with radius 4 cm and height 10 cm.

A. 160\pi

B. 320\pi

Correct C. 400\pi

D. 800\pi

Correct Answer: C

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Question 21

A rec\tangular prism has a length of 5 cm, a width of 3 cm, and a height of 2 cm. Find the volume of the prism in cubic centimeters.

Correct A. 30

B. 40

C. 50

D. 60

Correct Answer: A

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Question 22

Solve for x in the equation: \( 2^x = 64 \).

A. 5

Correct B. 6

C. 7

D. 8

Correct Answer: B

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Question 23

A circle has a radius of 4 cm. Find the area of the circle in square centimeters.

Correct A. 50.24

B. 50.26

C. 50.28

D. 50.30

Correct Answer: A

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Question 24

A rec\tangular prism has a length of 6 cm, a width of 4 cm, and a height of 3 cm. Find the surface area of the prism in square centimeters.

A. 104

Correct B. 108

C. 112

D. 116

Correct Answer: B

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Question 25

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

Correct A. -2

B. -1

C. 0

D. 1

Correct Answer: A

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