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POST UTME IMS U 2025 Mathematics | Objective

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Question 1 View Details

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} ) using the formula \(\\(end{pmatrix}). Find the projection of ( mathbf{b} ) onto ( mathbf{a} ) using the formula \mathrm{proj}_{\mathbf{a}}\mathbf{b} = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}|^2} \mathbf{a}\).

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 View Details

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) ) using 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 View Details

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 View Details

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 View Details

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 View Details

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 View Details

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 View Details

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 View Details

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 View Details

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 View Details

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\pitext{ cm}^3

B. 128\pitext{ cm}^3

Correct C. 192\pitext{ cm}^3

D. 256\pitext{ cm}^3

Correct Answer: C

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Question 12 View Details

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 View Details

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 View Details

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 View Details

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 View Details

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 View Details

Find the value of (\(12x \\) ) given that \\((\)12x \(\(1x \\) = \(\frac{3}{5}\)\) ) and 1x \\) = \(\frac{4}{5}\)\) ).

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 View Details

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 View Details

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 View Details

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 View Details

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 View Details

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 View Details

A histogram of exam scores has a mean of 60 and a standard 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 View Details

Find the derivative of the function ( f(x) = \(\frac{1}{x^2 + 1}\)\)\) ) using 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 View Details

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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