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POST UTME NILE UNIVERSITY 2025 Mathematics | Objective

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Question 1 View Details

Let X be a random variable with probability density function (pdf) given by f(x) = egin{cases} 2x & 0 leq x leq 1 \ 0 & \text{otherwise} \end{cases}. Find the probability that X is greater than 0.5.

A. 0.25

Correct B. 0.5

C. 0.75

D. 1

Correct Answer: B

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Question 2 View Details

Find the area under the curve $y = x\^2$ from $x = 0$ to $x = 2$.

Correct A. 4

B. 8

C. 16

D. 32

Correct Answer: A

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Question 3 View Details

Solve the vector equation mathbf{a} \cdot \mathbf{b} = 0 where \\(mathbf{a} cdot \mathbf{a} = \begin{pmatrix} 1 \ 2 \ 3 \end{pmatrix} and \mathbf{b} = \begin{pmatrix} x \ y \ z \end{pmatrix}.

Correct A. \begin{pmatrix} 0 \ 0 \ 0 \end{pmatrix}

B. \begin{pmatrix} 1 \ 2 \ 3 \end{pmatrix}

C. \begin{pmatrix} x \ y \ z \end{pmatrix}

D. \begin{pmatrix} 2 \ 3 \ 4 \end{pmatrix}

Correct Answer: A

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Question 4 View Details

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

Correct A. $\frac{1}{x^3}$\)\) - $\frac{2$1x}{x^3}$\)\)

B. $\frac{2$1x}{x^3}$\)\) - $\frac{1}{x^3}$\)\)

C. $\frac{1}{x^3}$\)\) + $\frac{2$1x}{x^3}$\)\)

D. $\frac{2}{x^3}$\)\) - $\frac{$1x}{x^3}$\)\)

Correct Answer: A

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Question 5 View Details

Solve the inequality $\frac{x}{x+1}$\$) > 0.$

Correct A. $-\infty, -1$ \cup $0, \infty$

B. $-\infty, 0$ \cup $1, \infty$

C. $-\infty, 0$ \cup $1, \infty$

D. $-\infty, -1$ \cup (0, 1)

Correct Answer: A

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Question 6 View Details

Find the derivative of the function ( f(x) = $\frac{x^2 + 2x - 3}{x^2 - 4}$\)\) ) using the quotient rule.

Correct A. f'$(x) =$$\frac{x^2 - 4$ 2x + 2 - $x^2 + 2x - 3$ (2x)}{ $x^2 - 4$ ^2}\)\)

B. f'$(x) =$$\frac{x^2 + 2x - 3$ 2x + 2 - $x^2 - 4$ (2x)}{ $x^2 - 4$ ^2}\)\)

C. f'$(x) =$$\frac{x^2 - 4$ 2x + 2 + $x^2 + 2x - 3$ (2x)}{ $x^2 - 4$ ^2}\)\)

D. f'$(x) =$$\frac{x^2 + 2x - 3$ 2x - 2 - $x^2 - 4$ (2x)}{ $x^2 - 4$ ^2}\)\)

Correct Answer: A

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Question 7 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 × 3) + 2(3 × 2) + 2(5 × 2)

B. 2(5 × 3) + 2(3 × 2) + 2(5 × 2) + 2(3 × 5)

C. 2(5 × 3) + 2(3 × 2) + 2(5 × 2) + 2(3 × 5) + 2(5 × 2)

D. 2(5 × 3) + 2(3 × 2) + 2(5 × 2) + 2(3 × 5) + 2(5 × 2) + 2(3 × 5)

Correct Answer: A

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Question 8 View Details

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

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

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

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

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

Correct Answer: A

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Question 9 View Details

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

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

B. $x - 2$^2 + $y - 3$^$2 = 4$

C. $x - 3$^2 + $y - 2$^$2 = 16$

D. $x - 3$^2 + $y - 2$^$2 = 4$

Correct Answer: A

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Question 10 View Details

Find the vector ( mathbf{v} ) such that $\mathbf{v} \cdot \mathbf{i} = 3$ and $\mathbf{v} \cdot \mathbf{j} = -2$.

Correct A. \mathbf{v} = \langle 3, -2 \rangle

B. \mathbf{v} = \langle -3, 2 \rangle

C. \mathbf{v} = \langle 3, 2 \rangle

D. \mathbf{v} = \langle -3, -2 \rangle

Correct Answer: A

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Question 11 View Details

Let X and Y be independent random variables with probability density functions f_$X(x) = 2x$, $0 < x$$< 1$, and f_$Y(y) =$$3y^2$,$(0 < y$\) < 1. Find the probability that $X + Y < 1\\((0 < y) < 1. Find the probability that$.

A. $1/4$

Correct B. $1/2$

C. $3/4$

D. 1

Correct Answer: B

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Question 12 View Details

Solve the system of equations: $\begin{align*} x^2 + y^2 + z^2 & = 4 \ x + y + z & = 0 \ x^2y^2z^2 & = 1 \end{align*}$

Correct A. $(x, y, z) = -1, 1, 0$\)

B. $(x, y, z) = (0, 0, 2)$

C. $(x, y, z) = -2, 1, 1$\)

D. $(x, y, z) = 1, -1, 0$\)

Correct Answer: A

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Question 13 View Details

A histogram of exam scores has a mean of 75 and a standard deviation of 10. If the scores are normally distributed, what is the probability that a randomly selected score is greater than 85?

Correct A. 0.1587

B. 0.3413

C. 0.4772

D. 0.6915

Correct Answer: A

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Question 14 View Details

Find the derivative of the function $f(x) = \frac{x^2 + 3x + 2}{x^2 - 4x + 3}$

Correct A. $f'(x) = \frac{x^2 - 4x + 3 2x + 3 - x^2 + 3x + 2 2x - 4}{x^2 - 4x + 3 ^2}$

B. $f'(x) = \frac{x^2 - 4x + 3 2x + 3 + x^2 + 3x + 2 2x - 4}{x^2 - 4x + 3 ^2}$

C. $f'(x) = \frac{x^2 - 4x + 3 2x + 3 - x^2 + 3x + 2 2x + 4}{x^2 - 4x + 3 ^2}$

D. $f'(x) = \frac{x^2 - 4x + 3 2x + 3 + x^2 + 3x + 2 2x + 4}{x^2 - 4x + 3 ^2}$

Correct Answer: A

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Question 15 View Details

A 3x3 matrix A has the following elements: $A = \begin{bmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \end{bmatrix}$

Correct A. $|A| = 0$

B. $|A| = 1$

C. $|A| = -1$

D. $|A| = 2$

Correct Answer: A

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Question 16 View Details

Find the sum of the infinite geometric series: $sum_{n = 1}^{\infty} \frac{1}{2^n}$

Correct A. 1

B. 2

C. 3

D. 4

Correct Answer: A

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Question 17 View Details

Solve the trigonometric equation: $\sin^2 x + \cos^2 x = 1 \$

Correct A. $x =$$\frac{\pi}{4}$\)

B. $x =$$\frac{\pi}{2}$\)

C. $x =$$\frac{\pi}{3}$\)

D. $x =$$\frac{\pi}{6}$\)

Correct Answer: A

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Question 18 View Details

Find the derivative of the function:$( f(x) =$$\frac{1}{x^2 + 1}$\)\) \)

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

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

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

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

Correct Answer: A

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Question 19 View Details

Solve the inequality: $\frac{x}{x + 1} > 0 \$

A. $x < -1$

Correct B. $x > -1$

C. $x < 0$

D. $x > 0$

Correct Answer: B

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Question 20 View Details

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. $x - 4$^2 + $y - 3$^$2 = 16$

D. $x - 3$^2 + $y - 4$^$2 = 16$

Correct Answer: A

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Question 21 View Details

In the diagram below, the equation of the circle is given by $x - 2 \$^2 + $y - 3)^\(2 = 16$ ). Find the equation of the \\\$(()^(2$ = 16\)\tangent line to the circle at the point ( (5, 7) ).

Correct A. $y = -x + 12$

B. $y = x + 5$

C. $y = -x - 3$

D. $y = x - 2$

Correct Answer: A

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Question 22 View Details

A random variable ( X ) has a probability distribution given by $P X = x \$ = $frac{1}{2}$\) ) for \$frac{1}{2})\(x = 1, 2, 3$. Find the probability that ( X ) is greater than 2.

A. 0.5

Correct B. 0.75

C. 0.25

D. 0.33

Correct Answer: B

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Question 23 View Details

Simplify the expression $\frac{\sqrt[3]{64} + \sqrt[3]{216}}{\sqrt[3]{8} - \sqrt[3]{27}}$.

Correct A. 4

B. 2

C. 3

D. 1

Correct Answer: A

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Question 24 View Details

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

A. $16/3$

Correct B. $32/3$

C. $64/3$

D. $128/3$

Correct Answer: B

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Question 25 View Details

A function ( f(x) = $\frac{1}{x^2 + 1}$\)\) ) has a local maximum at $x = a$. Find the value of ( a ).

Correct A. 0

B. 1

C. -1

D. 2

Correct Answer: A

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