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POST UTME PAN-ATLANTIC UNIVERSITY 2024 Mathematics | Objective

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Question 1 View Details

Find the derivative of the function ( f(x) = \(\frac{1}{2}\)\) \log_{10} \(x^2 + 1\) ) using the chain rule.

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

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

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

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

Correct Answer: A

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Question 2 View Details

A rectangular prism has a length of 5 cm, a width of 3 cm, and a height of 2 cm. Find the surface area of the prism.

Correct A. 2\(5 \cdot 3 + 3 \cdot 2 + 5 \cdot 2\)

B. 2\(5 \cdot 3 + 3 \cdot 2 + 5 \cdot 2\) + 2\(5 \cdot 2 + 3 \cdot 2 + 5 \cdot 3\)

C. 2\(5 \cdot 3 + 3 \cdot 2 + 5 \cdot 2\) - 2\(5 \cdot 2 + 3 \cdot 2 + 5 \cdot 3\)

D. 2\(5 \cdot 3 + 3 \cdot 2 + 5 \cdot 2\) + 2\(5 \cdot 2 + 3 \cdot 2 + 5 \cdot 3\) + 2\(5 \cdot 2 + 3 \cdot 2 + 5 \cdot 3\)

Correct Answer: A

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Question 3 View Details

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

Correct A. \int_\(0^1\) \(\frac{1}{x^2 + 1}\)\)\) dx

B. \int_\(0^1\) \(\frac{1}{x^2 + 1}\)\)\) dx + \int_\(0^1\) \(\frac{1}{x^2 + 1}\)\)\) dx

C. \int_\(0^1\) \(\frac{1}{x^2 + 1}\)\)\) dx - \int_\(0^1\) \(\frac{1}{x^2 + 1}\)\)\) dx

D. \int_\(0^1\) \(\frac{1}{x^2 + 1}\)\)\) dx + \int_\(0^1\) \(\frac{1}{x^2 + 1}\)\)\) dx + \int_\(0^1\) \(\frac{1}{x^2 + 1}\)\)\) dx

Correct Answer: A

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Question 4 View Details

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

Correct A. \pi \(r^2\)

B. \pi \(r^2\) + \pi \(r^2\)

C. \pi \(r^2\) - \pi \(r^2\)

D. \pi \(r^2\) + \pi \(r^2\) + \pi \(r^2\)

Correct Answer: A

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Question 5 View Details

Find the derivative of the function ( f(x) = \\(\sin^2\) x ) using the chain rule.

Correct A. f'\((x) = 2\)\sin x \cos x

B. f'\((x) = 2\)\sin x \cos x + 2 \cos x \sin x

C. f'\((x) = 2\)\sin x \cos x - 2 \cos x \sin x

D. f'\((x) = 2\)\sin x \cos x + 2 \cos x \sin x

Correct Answer: A

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Question 6 View Details

A random sample of 25 students from a university had a mean height of 175.5 cm with a standard deviation of 5.2 cm. If the population standard deviation is unknown, calculate the 95% confidence interval for the mean height of all students in the university.

Correct A. 169.3 cm, 181.7 cm

B. 170.5 cm, 180.5 cm

C. 171.5 cm, 179.5 cm

D. 172.5 cm, 178.5 cm

Correct Answer: A

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Question 7 View Details

Solve the inequality \(\frac{x^2 - 4}{x^2 - 9} > 0\).

Correct A. \(x < -3\) or \(x > 2\)

B. \(-3 < x < 2\)

C. \(x < -3\) or \(2 < x < 3\)

D. \(x > 3\)

Correct Answer: A

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Question 8 View Details

Find the sum of the first 10 terms of the geometric series \(\sum_{n = 1}^{10} 2^n\).

Correct A. 2047

B. 2048

C. 2049

D. 2050

Correct Answer: A

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Question 9 View Details

A vector (\(\vec{a}\)\) has a magnitude of 5 units and makes an angle of 30° with the positive x-axis. Find the x and y components of \\((vec{a}\(\vec{a}\).

Correct A. \(\vec{a} = 4.33\hat{i} + 2.5\hat{j}\)

B. \(\vec{a} = 4.33\hat{i} - 2.5\hat{j}\)

C. \(\vec{a} = -4.33\hat{i} + 2.5\hat{j}\)

D. \(\vec{a} = -4.33\hat{i} - 2.5\hat{j}\)

Correct Answer: A

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Question 10 View Details

Solve the equation \(\sin^2(x) + \cos^2(x) = 1\).

A. \(x = \frac{\pi}{2}\)

Correct B. \(x = \frac{\pi}{4}\)

C. \(x = \frac{3\pi}{4}\)

D. \(x = \frac{5\pi}{4}\)

Correct Answer: B

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Question 11 View Details

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

Correct A. 10

B. 100

C. 1000

D. 10000

Correct Answer: A

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Question 12 View Details

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

Correct A. 50.24

B. 100.48

C. 200.96

D. 400.96

Correct Answer: A

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Question 13 View Details

A particle moves in a straight line with an initial velocity of 5 \((m/s\)\). If the acceleration is 2 \(m/s\\((m/s). If the acceleration is 2 \^2\), find the velocity after 3 seconds.

A. 13

B. 15

Correct C. 17

D. 19

Correct Answer: C

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Question 14 View Details

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

A. \(\frac{1}{3}\)\)

Correct B. \(\frac{2}{3}\)\)

C. \(\frac{1}{2}\)\)

D. \(\frac{2}{5}\)\)

Correct Answer: B

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Question 15 View Details

Solve the inequality \(x^2 - 4x + 4 \geq 0\).

A. x \leq 2

Correct B. x \geq 2

C. \(x < 2\)

D. \(x > 2\)

Correct Answer: B

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Question 16 View Details

Find the sum of the first 10 terms of the geometric series with first term 2 and common ratio 3.

Correct A. 2 \cdot \(\frac{3^{10} - 1}{3 - 1}\)\)

B. 2 \cdot \(\frac{3^{11} - 1}{3 - 1}\)\)

C. 2 \cdot \(\frac{3^{9} - 1}{3 - 1}\)\)

D. 2 \cdot \(\frac{3^{8} - 1}{3 - 1}\)\)

Correct Answer: A

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Question 17 View Details

A circle has equation \(x - 1 \\)^2 + \(y - 2)^\(2 = 4\) ). Find the equation of the \\\\((()^(2\) = 4\)\tangent line at point ( (3, 5) ).

Correct A. \(y - 5 = -\)\(\frac{1}{2}\)\)\(x - 3\)

B. \(y - 5 =\)\(\frac{1}{2}\)\)\(x - 3\)

C. \(y - 5 =\)\(x - 3\)

D. \(y - 5 = -\)\(x - 3\)

Correct Answer: A

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Question 18 View Details

Find the area under the curve \(y = x^2\) from \(x = 0\) to \(x = 4\).

Correct A. \(\frac{64}{3}\)\)

B. \(\frac{32}{3}\)\)

C. \(\frac{16}{3}\)\)

D. \(\frac{8}{3}\)\)

Correct Answer: A

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Question 19 View Details

Solve the system of equations: \(\begin{cases} x + y = 4 \ x - 2y = -3 \end{cases}\).

Correct A. \begin{cases} \(x = 7\) \\(y = -3\)\end{cases}

B. \begin{cases} \(x = 1\) \ \(y = 3\) \end{cases}

C. \begin{cases} \(x = 3\) \ \(y = 1\) \end{cases}

D. \begin{cases}\(x = -1\)\ \(y = 5\) \end{cases}

Correct Answer: A

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Question 20 View Details

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

Correct A. 0

B. 1

C. 2

D. 3

Correct Answer: A

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Question 21 View Details

Find the value of \(\int_0^1 2x^3 - 5x^2 + 3x - 1\) dx\).

A. 1

Correct B. 2

C. 3

D. 4

Correct Answer: B

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Question 22 View Details

Solve the inequality \(|x - 2| > 3\).

Correct A. \(-∞, -1\) ∪ (4, ∞)

B. \(-∞, 1\) ∪ (4, ∞)

C. \(-∞, -1\) ∪ (1, ∞)

D. \(-∞, 4\) ∪ (1, ∞)

Correct Answer: A

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Question 23 View Details

Find the equation of the tangent line to the curve \(y = x^2 + 2x - 3\) at the point \\(tangent line to the curve (y = x^2 + 2x - 3\((1, 2)\).

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

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

C. \(y = x - 1\)

D. \(y = x + 1\)

Correct Answer: A

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Question 24 View Details

Find the area of the region bounded by the curves \(y = x^2\) and \(y = 2x\).

Correct A. \(4/3\)

B. \(2/3\)

C. \(1/3\)

D. \(1/2\)

Correct Answer: A

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Question 25 View Details

Solve the system of equations \(x + y = 2\) and \(xy = 1\).

Correct A. (1, 1)

B. \(1, -1\)

C. \(-1, 1\)

D. \(-1, -1\)

Correct Answer: A

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