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POST UTME IMS U 2025 Mathematics | Objective

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Question 1

Let \( mathbf{a} = egin{pmatrix} 2 \ 3 \end{pmatrix} \) and \( mathbf{b} = egin{pmatrix} 1 \ -2 \end{pmatrix} \). Find the projection of ( mathbf{b} ) onto ( mathbf{a} ) u\sing the formula \( mathrm{proj}_{mathbf{a}}mathbf{b} = \frac{mathbf{a} cdot mathbf{b}}{|mathbf{a}|^2} mathbf{a} \).

Correct A. \begin{pmatrix} \frac{7}{13} \\ \frac{12}{13} \end{pmatrix}

B. \begin{pmatrix} \frac{1}{13} \\ \frac{2}{13} \end{pmatrix}

C. \begin{pmatrix} \frac{2}{13} \\ \frac{3}{13} \end{pmatrix}

D. \begin{pmatrix} \frac{3}{13} \\ \frac{4}{13} \end{pmatrix}

Correct Answer: A

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Question 2

Let ( f(x) = x^3 - 6x^2 + 11x + 2 ) and ( g(x) = x^2 - 4x + 3 ). Find the derivative of the product ( f(x)g(x) ) u\sing the product rule.

A. 3x^2 - 12x + 11

B. x^2 - 4x + 3

Correct C. 3x^2 - 12x + 11 + x^2 - 4x + 3

D. x^2 - 4x + 3 - 3x^2 + 12x - 11

Correct Answer: C

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Question 3

A random experiment consists of rolling a fair six-sided die. Let ( X ) be the random variable representing the outcome. Find the probability that ( X ) is an even number.

Correct A. \frac{1}{2}

B. \frac{2}{3}

C. \frac{3}{4}

D. \frac{4}{5}

Correct Answer: A

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Question 4

Find the sum of the infinite geometric series \( \sum_{n=1}^\infty \frac{1}{2^n} \ \).

A. 1

Correct B. \frac{1}{2}

C. \frac{2}{3}

D. \frac{3}{4}

Correct Answer: B

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Question 5

Let ( f(x) = x^2 - 4x + 3 ). Find the equation of the \tangent line to the graph of ( f(x) ) at the point ( (1, f(1)) ).

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

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

C. y - 1 = x - 1

D. y - 1 = -x + 1

Correct Answer: A

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Question 6

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 7

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

A. 10

B. 100

C. 1000

Correct D. 10000

Correct Answer: D

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Question 8

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. 2

D. 3

Correct Answer: A

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Question 9

A sequence is defined by the recurrence relation \( a_n = 2a_{n-1} + 1 \) with initial term \( a_1 = 3 \). Find the value of \( a_{10} \).

Correct A. 1023

B. 1024

C. 1025

D. 1026

Correct Answer: A

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Question 10

In the circuit below, find the current flowing through the 2 ohm resistor.

A. 1 A

Correct B. 2 A

C. 3 A

D. 4 A

Correct Answer: B

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Question 11

Find the volume of the frustum of a cone with height 8 cm, lower base radius 4 cm, and upper base radius 2 cm.

A. 64\pi\text{ cm}^3

B. 128\pi\text{ cm}^3

Correct C. 192\pi\text{ cm}^3

D. 256\pi\text{ cm}^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 not blue?

Correct A. \frac{2}{3}

B. \frac{3}{5}

C. \frac{4}{7}

D. \frac{5}{8}

Correct Answer: A

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Question 13

Solve for x: \log_{10}\( x^2 \) = 4.

A. \pm 2

Correct B. \pm 4

C. \pm 8

D. \pm 16

Correct Answer: B

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Question 14

Find the value of \sin 75^\circ.

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

B. \frac{\sqrt{5}}{2}

Correct C. \frac{\sqrt{6}}{2}

D. \frac{\sqrt{10}}{2}

Correct Answer: C

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Question 15

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

A. 1

B. 2

Correct C. 3

D. 4

Correct Answer: C

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Question 16

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 17

Find the value of \( \sin \( 2x \ \) ) given that \( \sin \( x \ \) = \frac{3}{5} ) and \( \cos \( x \ \) = \frac{4}{5} ).

Correct A. \frac{24}{25}

B. \frac{16}{25}

C. \frac{12}{25}

D. \frac{8}{25}

Correct Answer: A

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Question 18

A matrix A is given by \( A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} \). Find the determinant of A.

A. -2

Correct B. 0

C. 2

D. 4

Correct Answer: B

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Question 19

A fair six-sided die is rolled. What is the probability that the number rolled is greater than 4?

A. \frac{1}{6}

B. \frac{1}{3}

C. \frac{2}{3}

Correct D. \frac{5}{6}

Correct Answer: D

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Question 20

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

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

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

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

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

Correct Answer: A

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Question 21

Find the area under the curve \( y = \frac{1}{2}x^2 + 3x - 2 \) from \( x = 0 \) to \( x = 4 \).

Correct A. 40

B. 42

C. 44

D. 46

Correct Answer: A

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Question 22

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

A. 1

Correct B. 2

C. 3

D. 4

Correct Answer: B

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Question 23

A histogram of exam scores has a mean of 60 and a s\tandard deviation of 10. What is the probability that a randomly selected score is between 50 and 70?

A. 0.34

Correct B. 0.45

C. 0.56

D. 0.67

Correct Answer: B

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Question 24

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{-x}{\( x^2 + 1 \)^2}

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

Correct Answer: A

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Question 25

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

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

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

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

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

Correct Answer: A

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