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OAU Post-UTME Mathematics 2023 past questions and answers

17 questions. Try each one, then open the answer to check yourself and read the explanation.

Question 1

Probability

The table shows the outcome when a die is thrown a number of times.
Score: 1, 2, 3, 4, 5, 6
Frequency: 25, 30, 45, Y, 40, 32
If the probability of obtaining a 3 is 0.225. How many times was the die thrown?

  1. A.172
  2. B.45
  3. C.200
  4. D.155
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Answer: C (200)

Explanation

P(3) = 45 ÷ total throws = 0.225, so total = 45 ÷ 0.225 = 200 throws.

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Question 2

Progressions

The 3rd and 6th terms of a Geometric Progression G.P. are 48 and 14 2/9 respectively. What is the first term of the G.P.?

  1. A.2/3
  2. B.48
  3. C.32
  4. D.108
Show the answer and explanation

Answer: D (108)

Explanation

ar5 ÷ ar2 = r3 = (128/9) ÷ 48 = 8/27, so r = 2/3. Then a = 48 ÷ (2/3)2 = 48 × 9/4 = 108.

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Question 3

Progressions

If 1/2, 1/x, 1/3 are successive terms of an arithmetic progression, find the value of (2 − x)/(x − 3).

  1. A.−1/2
  2. B.2/5
  3. C.2/3
  4. D.1/4
Show the answer and explanation

Answer: C (2/3)

Explanation

In an A.P. the middle term is the average: 2/x = 1/2 + 1/3 = 5/6, so x = 12/5. Then (2 − 12/5)/(12/5 − 3) = (−2/5)/(−3/5) = 2/3.

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Question 4

Sets

The sets A = {1, 3, 5, 7, 9, 11}, B = {2, 3, 5, 7, 11, 15}, C = {3, 6, 9, 12, 15} are subsets of U = {1, 2, 3, …, 15}. What is the value of A' ∩ C (B ∪ C)

  1. A.{6, 12, 15}
  2. B.{12, 15}
  3. C.{6}
  4. D.{2, 6, 12, 15}
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Answer: A ({6, 12, 15})

Explanation

A' is everything in U not in A, so A' ∩ C = {6, 12, 15}. All of these are in B ∪ C (which contains C), so the result stays {6, 12, 15}.

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Question 7

Differentiation

Differentiate 1/√x with respect to x

  1. A.−1/(2(√x)3)
  2. B.−1/(2√x)
  3. C.−1/(2x3)
  4. D.1/(2√x)
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Answer: A (−1/(2(√x)3))

Explanation

1/√x = x−1/2. Its derivative is −½x−3/2 = −1/(2(√x)3).

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Question 8

Differentiation

What is the value of dy/dx when x = 4 for the function y = x ln (x − 3)?

  1. A.0
  2. B.4
  3. C.1
  4. D.½
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Answer: B (4)

Explanation

By the product rule dy/dx = ln(x − 3) + x/(x − 3). At x = 4: ln 1 + 4/1 = 0 + 4 = 4.

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Question 9

Integration

Evaluate ∫06 1/(x + 2) dx.

  1. A.2
  2. B.2 ln 2
  3. C.ln 2
  4. D.ln 3
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Answer: B (2 ln 2)

Explanation

The integral of 1/(x + 2) is ln(x + 2). From 0 to 6: ln 8 − ln 2 = ln 4 = 2 ln 2.

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Question 10

Matrices and Determinants

Given that A = [2 3; 4 1] and B = [3 5; 6 2] compute 2B − A

  1. A.[4 8; 3 1]
  2. B.[3 2; 1 4]
  3. C.[4 7; 8 2]
  4. D.[4 7; 8 3]
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Answer: D ([4 7; 8 3])

Explanation

2B = [6 10; 12 4]. Subtract A entry by entry: [6−2 10−3; 12−4 4−1] = [4 7; 8 3].

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Question 11

Trigonometry

Given that sin x = 2/3, evaluate, leaving your answer in surd form, tan x − cos x

  1. A.√3/5
  2. B.2√3/15
  3. C.√5/15
  4. D.2√5/15
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Answer: C (√5/15)

Explanation

With opposite 2 and hypotenuse 3, the adjacent side is √5. So tan x = 2/√5 = 2√5/5 and cos x = √5/3. 2√5/5 − √5/3 = (6√5 − 5√5)/15 = √5/15.

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Question 12

Trigonometry

If 9 cos x − 7 = 2 and 0° ≤ x ≤ 90°, find x

  1. A.0°
  2. B.90°
  3. C.30°
  4. D.45°
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Answer: A (0°)

Explanation

9 cos x = 9, so cos x = 1, which gives x = 0°.

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Question 13

Indices

If 2x+y = 16 and 4x−y = 1/32, find the value of x/y

  1. A.3/4
  2. B.13/4
  3. C.4/13
  4. D.3/13
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Answer: D (3/13)

Explanation

2x+y = 24 gives x + y = 4. 4x−y = 22(x−y) = 2−5 gives x − y = −5/2. Solving: x = 3/4, y = 13/4, so x/y = 3/13.

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Question 14

Indices

If 2n = x, find 2(2 + n/3)

  1. A.2x
  2. B.2x1/3
  3. C.x1/3
  4. D.4x1/3
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Answer: D (4x1/3)

Explanation

22 + n/3 = 22 × (2n)1/3 = 4x1/3.

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Question 15

Logarithms

Given that log 2 = 0.3010, log 3 = 0.4771, find the value of log100.24.

  1. A.0.6199
  2. B.1̄.3801
  3. C.1.3801
  4. D.2̄.3801
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Answer: B (1̄.3801)

Explanation

0.24 = (23 × 3)/100, so log 0.24 = 3(0.3010) + 0.4771 − 2 = 1.3801 − 2 = 1̄.3801 (which is −0.6199).

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Question 16

Logarithms

Given that log105 = 0.699, log103 = 0.477, find the value of x if x0.8265 = 45

  1. A.0.1
  2. B.10
  3. C.100
  4. D.1
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Answer: C (100)

Explanation

Take logs: 0.8265 log x = log 45 = 2 log 3 + log 5 = 0.954 + 0.699 = 1.653. So log x = 2 and x = 100.

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Question 17

Logarithms

Evaluate 4 log327 + log281/3.

  1. A.11
  2. B.12
  3. C.13
  4. D.18
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Answer: C (13)

Explanation

log327 = 3, so 4 × 3 = 12. 81/3 = 2 and log22 = 1. Total 13.

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Question 19

Surds

Simplify (√a + 1/√a)2 − (√a − 1/√a)2

  1. A.√a
  2. B.2/√a
  3. C.4
  4. D.2
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Answer: C (4)

Explanation

Use p² − q² = (p + q)(p − q): (2√a)(2/√a) = 4.

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Question 20

Coordinate Geometry

Find the coordinate of the point at which the perpendicular bisector of the line joining (2, 7) to (10, 3) meet the x-axis

  1. A.(−7/2, 0)
  2. B.(0, 7/2)
  3. C.(0, −7/2)
  4. D.(7/2, 0)
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Answer: D ((7/2, 0))

Explanation

Midpoint is (6, 5) and the segment's gradient is −1/2, so the bisector has gradient 2: y − 5 = 2(x − 6). Put y = 0: x = 7/2, giving (7/2, 0).

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