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POST UTME OAU 2025 Mathematics | Objective

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Question 1 View Details

A factory operates two machines, M1 and M2. Machine M1 produces 70% of the total output and has a defect rate of 2%; Machine M2 produces 30% of the output and has a defect rate of 5%. An item selected at random from the total output is found to be defective. What is the probability that it was produced by Machine M2?

A. 0.600

B. 0.483

C. 0.500

Correct D. 0.517

Correct Answer: D

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Question 2 View Details

Solve the inequality \( \frac{x^{2}-5x+6}{x^{2}-4}\le 0 \). Choose the correct solution set.

A. (-\infty,-2)\cup(2,3]

Correct B. (-2,2)\cup(2,3]

C. (-2,2)\cup(3,\infty)

D. (2,3]

Correct Answer: B

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Question 3 View Details

A circle has its centre on the line \( y=2x+1 \) and passes through the point \( (3,4) \). The circle is tangent to the x‑axis. What is the smaller possible radius of the circle?

A. 23+4\sqrt{26}

B. 11-2\sqrt{26}

Correct C. 23-4\sqrt{26}

D. 11+2\sqrt{26}

Correct Answer: C

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Question 4 View Details

Vectors \( \mathbf{a} \) and \( \mathbf{b} \) satisfy \( |\mathbf{a}|=5 \), \( |\mathbf{b}|=3 \) and \( \mathbf{a}\cdot\mathbf{b}=12 \). Find the magnitude of \( 2\mathbf{a}-\mathbf{b} \).

A. 7

Correct B. \sqrt{61}

C. \sqrt{65}

D. \sqrt{55}

Correct Answer: B

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Question 5 View Details

A solid consists of a right circular cylinder of radius 5 cm and height 12 cm attached to a cone of the same base radius and height 8 cm on top. Find the total surface area of the solid (including the base of the cylinder but excluding the common circular face).

A. \pi(140+5\sqrt{89})

B. \pi(150+5\sqrt{89})

C. \pi(145+\sqrt{89})

Correct D. \pi(145+5\sqrt{89})

Correct Answer: D

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Question 6 View Details

Solve the inequality\( \frac{2x^{2}-5x-3}{x-2}>0 \).

A. \( x>3 \)

B. \( -\frac{1}{2}

C. No solution

Correct D. \( -\frac{1}{2}3 \)

Correct Answer: D

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Question 7 View Details

A right circular frustum has lower radius 6 cm, upper radius 3 cm and height 10 cm. What is its volume?

A. \( 105\pi \text{ cm}^{3} \)

B. \( 630\pi \text{ cm}^{3} \)

Correct C. \( 210\pi \text{ cm}^{3} \)

D. \( 420\pi \text{ cm}^{3} \)

Correct Answer: C

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Question 8 View Details

Find the determinant of the matrix\( \begin{pmatrix}2&-1&3\\0&4&5\\1&0&-2\end{pmatrix} \).

A. -21

B. 21

C. 33

Correct D. -33

Correct Answer: D

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Question 9 View Details

For the data set {5,7,8,12,13,14,16,20}, compute the mean and the population standard deviation (rounded to two decimal places).

A. Mean = 11.5, SD = 4.00

B. Mean = 12.5, SD = 4.50

Correct C. Mean = 11.875, SD \approx 4.68

D. Mean = 12, SD = 5.00

Correct Answer: C

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Question 10 View Details

Determine all real values of\( k \)for which the line\( y=kx+1 \)is tangent to the circle\( (x-2)^{2}+(y+3)^{2}=25 \).

Correct A. No real value of\( k \)

B. \( k=-\frac{4}{5} \)

C. \( k=\frac{2}{3} \)

D. \( k=-2 \)

Correct Answer: A

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Question 11 View Details

Find the area bounded by the curve y = x^3 - 3x and the x‑axis between x = -2 and x = 1.

A. 2.50

B. 4.00

Correct C. 3.75

D. 5.25

Correct Answer: C

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Question 12 View Details

The sum of the first n terms of an arithmetic progression is given by S_n = 3n^2 + 5n. What is the value of the 10th term of the progression?

A. 70

B. 58

C. 65

Correct D. 62

Correct Answer: D

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Question 13 View Details

Compute the determinant of the matrix A = [[2, -1, 0], [3, 4, -2], [1, 0, 5]].

A. 62

B. 45

Correct C. 57

D. 48

Correct Answer: C

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Question 14 View Details

A box contains 4 red, 5 blue, and 6 green balls. Two balls are drawn sequentially without replacement. What is the probability that both balls are red?

A. 1/7

B. 4/15

C. 3/28

Correct D. 2/35

Correct Answer: D

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Question 15 View Details

Find the greater solution of the equation \log_{2}(x^2 - 5x + 6) = 2.

A. (5 - \sqrt{17})/2

Correct B. (5 + \sqrt{17})/2

C. 5

D. 2

Correct Answer: B

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Question 16 View Details

Consider the polynomial function f(x)=2x^{3}-3x^{2}-12x+5. Determine the interval(s) on which f(x) is increasing.

A. (-\infty, 2)

B. (-1, 2)

Correct C. (-\infty, -1) \cup (2, \infty)

D. (-\infty, -2) \cup (1, \infty)

Correct Answer: C

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Question 17 View Details

Simplify the trigonometric expression \( \frac{\sin^{2}\theta}{1+\cos\theta} \) and express the result in its simplest form.

A. 1+\cos\theta

Correct B. 1-\cos\theta

C. \sin\theta

D. \tan\theta

Correct Answer: B

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Question 18 View Details

Evaluate \( \left( \frac{27^{2/3}\, \cdot\, 9^{1/2}}{3^{5}} \right)^{2} \).

A. \frac{1}{9}

B. \frac{1}{27}

C. \frac{3}{81}

Correct D. \frac{1}{81}

Correct Answer: D

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Question 19 View Details

Solve for x in the equation \( \log_{2}(x^{2}-5x+6)=2 \).

A. \frac{5+\sqrt{17}}{2}

B. \frac{5-\sqrt{17}}{2}

Correct C. Both A and B

D. No real solution

Correct Answer: C

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Question 20 View Details

A continuous random variable X has probability density function f(x)=k x^{2} for 0 \le x \le 2 and f(x)=0 otherwise. Let g(t)=e^{t^{2}}. Compute the value of E[X] + g'(1).

A. 2e-\frac{3}{2}

Correct B. \frac{3}{2}+2e

C. 3+e

D. 1.5e

Correct Answer: B

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Question 21 View Details

Let A be the 3×3 matrix [[2, -1, 3], [0, 4, 5], [-2, 1, 0]]. Compute det(A).

A. 0

B. -24

C. 12

Correct D. 24

Correct Answer: D

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Question 22 View Details

A box contains 5 red balls, 7 blue balls, and 8 green balls. Two balls are drawn at random without replacement. What is the probability that the two balls are of different colors?

A. 3/5

Correct B. 131/190

C. 13/20

D. 7/10

Correct Answer: B

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Question 23 View Details

The first term of a geometric progression is (13)_5 (in base 5) and the common ratio is 2 (in base 10). Find the 4th term expressed in base 8.

A. 200

Correct B. 100

C. 120

D. 140

Correct Answer: B

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Question 24 View Details

The histogram below shows the distribution of marks obtained by 50 students in a mathematics test. Class intervals are 0‑9, 10‑19, 20‑29, 30‑39, 40‑49, 50‑59, 60‑69, 70‑79, 80‑89, 90‑100. The frequencies are respectively: 2, 3, 5, 8, 10, 9, 5, 4, 2, 2. Compute the mean mark (to the nearest integer).

A. 45

Correct B. 47

C. 49

D. 51

Correct Answer: B

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Question 25 View Details

Evaluate \frac{(23)_6}{(5)_6} and express the result in base 10.

Correct A. 3

B. 2

C. 4

D. 5

Correct Answer: A

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