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POST UTME OAU 2024 Mathematics | Objective

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Question 1 View Details

In the diagram below, O is the centre of the circle, AB is a chord of length 8 cm, and the line through B tangent to the circle meets the extension of OA at point C. If OC = 10 cm, find the radius of the circle (in cm).

Correct A. 4.7

B. 4.5

C. 5.5

D. 5.0

Correct Answer: A

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Question 2 View Details

Evaluate \(\sin 75^{\circ}\) in simplest radical form.

A. \sqrt{3}/2

B. (\sqrt{6}-\sqrt{2})/4

Correct C. (\sqrt{6}+\sqrt{2})/4

D. \sqrt{2}/2

Correct Answer: C

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Question 3 View Details

In the diagram below, vector \( \vec{OA}=3\mathbf{i}+4\mathbf{j} \)and vector \( \vec{OB}=-\mathbf{i}+2\mathbf{j} \). Vector \( \vec{OC} \)is the sum \( \vec{OA}+\vec{OB} \). Find the magnitude of\( \vec{OC} \).

A. 6

B. 8

C. 5

Correct D. 2\sqrt{10}

Correct Answer: D

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Question 4 View Details

Find the remainder when\( P(x)=2x^{4}-3x^{3}+x^{2}-5x+7 \)is divided by\( x^{2}-x+1 \).

A. 6x-9

Correct B. -6x+9

C. 3x+5

D. -3x+5

Correct Answer: B

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Question 5 View Details

Calculate the determinant of the matrix\( \begin{pmatrix}2&-1&3\\0&4&5\\1&2&6\end{pmatrix} \).

Correct A. 11

B. 12

C. 9

D. 10

Correct Answer: A

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Question 6 View Details

The histogram below shows the frequencies of scores obtained by 40 students in a mathematics test. The class intervals are 0‑9, 10‑19, 20‑29, 30‑39, 40‑49 with frequencies 4, 10, 12, 8, 6 respectively. What is the mean score (to the nearest integer)?

Correct A. 25

B. 27

C. 24

D. 26

Correct Answer: A

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Question 7 View Details

Let S = { n ∈ ℕ | the base‑5 representation of n uses only the digits 0, 1, 2 }. Which of the following numbers belongs to S?

A. 28

Correct B. 7

C. 13

D. 22

Correct Answer: B

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Question 8 View Details

In the diagram below, point P on the unit circle corresponds to angle θ measured counter‑clockwise from the positive x‑axis. If the coordinates of P are (-√3/2, 1/2), what is the exact value of tan θ?

A. \sqrt{3}

B. -\sqrt{3}

Correct C. -\frac{1}{\sqrt{3}}

D. \frac{1}{\sqrt{3}}

Correct Answer: C

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Question 9 View Details

In the diagram below, circle C has centre at the origin and radius 5. Line ℓ passes through points A(-3, 4) and B(6, -2). Determine the length of the chord of circle C cut by line ℓ.

A. 6

Correct B. 9.4

C. 8

D. 12

Correct Answer: B

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Question 10 View Details

Express the fraction 5/12 in base‑3 notation up to three fractional digits.

Correct A. 0.102₃

B. 0.110₃

C. 0.112₃

D. 0.101₃

Correct Answer: A

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Question 11 View Details

In the diagram below, O is the centre of a circle of radius 5 cm. Points A and B lie on the circle such that AB = 6 cm. The tangent at A meets the line OB at T. Find the length OT (in cm).

A. 25/7

B. 35/7

Correct C. 125/7

D. 45/7

Correct Answer: C

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Question 12 View Details

In the diagram below, vectors \( \mathbf{u} \) and \( \mathbf{v} \) originate from O with magnitudes \( |\mathbf{u}|=5 \) and \( |\mathbf{v}|=4 \). The angle between them is \( 60^{\circ} \). Vector \( \mathbf{w}=2\mathbf{u}-3\mathbf{v} \). What is \( |\mathbf{w}| \)?

A. \sqrt{120}

B. 12

C. 11

Correct D. 2\sqrt{31}

Correct Answer: D

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Question 13 View Details

The graph below represents \( y=(x^{2}+1)^{3} \). Using the graph, determine \( \frac{dy}{dx} \) at the point where \( x=1 \).

A. 12

B. 36

C. 48

Correct D. 24

Correct Answer: D

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Question 14 View Details

Solve for \( x \): \( 2^{x}\,5^{x+1}=10^{2x-1} \). Express your answer in simplest logarithmic form.

A. \log_{10}2

Correct B. 2-\log_{10}2

C. 2+\log_{10}2

D. 1

Correct Answer: B

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Question 15 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 the two balls are of different colours given that at least one of the drawn balls is red?

Correct A. 22/25

B. 4/5

C. 3/4

D. 7/10

Correct Answer: A

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Question 16 View Details

In the diagram below, the curve represents \( y = f(x) \), a cubic polynomial with turning points at A(1, -2) and B(3, 4). Which of the following statements about \( f'(x) \) is correct?

A. f'(x) > 0 for \( x < 1 \)

B. f'(x) < 0 for \( 1 < x < 3 \)

Correct C. f'(x) > 0 for \( 1 < x < 3 \)

D. f'(x) = 0 for \( x > 3 \)

Correct Answer: C

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Question 17 View Details

The number (242)_b expressed in base ten equals 98. What is the base b?

A. 8

B. 5

C. 7

Correct D. 6

Correct Answer: D

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Question 18 View Details

In the diagram below, the curve represents \( y = x^2 - 4 \) and the straight line represents \( y = 2x - 1 \). How many distinct real solutions does the system have?

Correct A. 2

B. 1

C. 0

D. 3

Correct Answer: A

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Question 19 View Details

If \( f(x)= \sqrt{3x^2+5} \) and \( g(x)=e^{f(x)} \), find g'(2).

A. \( \frac{6e^{\sqrt{17}}}{17} \)

Correct B. \( \frac{6e^{\sqrt{17}}}{\sqrt{17}} \)

C. \( e^{\sqrt{17}} \)

D. \( \frac{3e^{\sqrt{17}}}{\sqrt{17}} \)

Correct Answer: B

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Question 20 View Details

In the Venn diagram below, \( P(A)=0.4 \), \( P(B)=0.5 \) and \( P(A\cap B)=0.2 \). What is the probability that a randomly selected element belongs to exactly one of the events?

A. 0.8

B. 0.7

Correct C. 0.5

D. 0.6

Correct Answer: C

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Question 21 View Details

In the Venn diagram below, the universal set contains 120 elements. Set A has 70 elements, set B has 65 elements, and set C has 50 elements. The number of elements that belong to exactly two of the sets is 30, and the number that belong to none of the sets is 5. How many elements belong to all three sets?

Correct A. 20

B. 25

C. 30

D. 15

Correct Answer: A

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Question 22 View Details

The histogram below shows the frequency distribution of a data set. Intervals and frequencies are: 0‑9 : 4, 10‑19 : 7, 20‑29 : 5, 30‑39 : 2. Assuming each observation equals the midpoint of its interval, what is the mean of the data set (to one decimal place)?

A. 16.5

B. 19.2

C. 18.0

Correct D. 17.3

Correct Answer: D

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Question 23 View Details

The graph below plots the first four terms of an arithmetic progression on a number line. The points are at positions 2, 5, 8 and 11. What is the common difference of the progression?

Correct A. 3

B. 5

C. 4

D. 2

Correct Answer: A

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Question 24 View Details

If a=4, find the value of b that makes the determinant of the matrix \n\[ \begin{pmatrix}2&-1&3\\0&a&4\\5&2&b\end{pmatrix} \]equal to zero.

Correct A. 12

B. 11

C. 10

D. 13

Correct Answer: A

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Question 25 View Details

Solve the equation (13)_5 x - (24)_5 = (31)_5 and express the solution as a fraction in lowest terms (base 10).

A. 13/3

B. 9/2

Correct C. 15/4

D. 7/2

Correct Answer: C

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