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UI Post-UTME Mathematics 2025 (Model questions) past questions and answers

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

Question 1

Algebraic Expressions

If x + 1/x = 3, find the value of x3 + 1/x3

  1. A.18
  2. B.21
  3. C.27
  4. D.30
Show the answer and explanation

Answer: A (18)

Explanation

Cube both sides: x³ + 1/x³ + 3(x + 1/x) = 27, so x³ + 1/x³ = 27 − 3(3) = 18.

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Question 2

Logarithms

Solve for x: log2(x + 1) + log2(x − 2) = 2

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

Explanation

(x + 1)(x − 2) = 2² = 4, so x² − x − 6 = 0, giving x = 3 or −2. Logs need x − 2 > 0, so x = 3.

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Question 4

Progressions

A geometric progression has first term 12 and third term 27. Find the common ratio

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

Explanation

ar² = 27 with a = 12, so r² = 27/12 = 9/4 and r = ±3/2 (both signs give third term 27).

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Question 5

Quadratic Equations

If α and β are the roots of the equation 2x2 − 5x + 1 = 0. Find the value of α2 + β2

  1. A.17/4
  2. B.19/14
  3. C.21/4
  4. D.19/4
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Answer: C (21/4)

Explanation

α + β = 5/2 and αβ = 1/2, so α² + β² = (α + β)² − 2αβ = 25/4 − 1 = 21/4.

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Question 6

Coordinate Geometry

A line passes through the point (2, −1) and is perpendicular to the line 3x − 2y + 7 = 0. Find its equation

  1. A.2x + 3y − 1 = 0
  2. B.2x + 3y − 7 = 0
  3. C.3x + 2y − 4 = 0
  4. D.2x + 3y − 1 = 3
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Answer: A (2x + 3y − 1 = 0)

Explanation

The given line has gradient 3/2, so the perpendicular has gradient −2/3. y + 1 = −2/3(x − 2) gives 3y + 3 = −2x + 4, i.e. 2x + 3y − 1 = 0.

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Question 7

Progressions

The first 10 terms of an arithmetic progression sum to 185 while the first 5 terms sum to 65. Find the common difference

  1. A.2/3
  2. B.11/2
  3. C.11/5
  4. D.3
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Answer: C (11/5)

Explanation

S₅ = 65 gives a + 2d = 13; S₁₀ = 185 gives 2a + 9d = 37. Then 2(13 − 2d) + 9d = 37, so 5d = 11 and d = 11/5.

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Question 8

Probability

A bag contains 5 red balls, 4 blue balls and 3 green balls. Two balls are drawn successively without replacement. What is the probability that the two balls are of different colours?

  1. A.7/11
  2. B.35/66
  3. C.47/66
  4. D.19/33
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Answer: C (47/66)

Explanation

P(same colour) = (5×4 + 4×3 + 3×2)/(12×11) = 38/132 = 19/66, so P(different) = 1 − 19/66 = 47/66.

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Question 9

Permutation and Combination

How many 4-digit even numbers can be formed from the digits 1, 2, 3, 4, 5, 6 without repetition?

  1. A.300
  2. B.240
  3. C.200
  4. D.180
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Answer: D (180)

Explanation

The last digit must be 2, 4 or 6 (3 ways); the other three places can be filled in 5 × 4 × 3 = 60 ways. Total = 3 × 60 = 180.

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Question 10

Measures of Central Tendency

The mean of 12 observations is 18. When one observation is recorded as 11 instead of 23, what will the mean be?

  1. A.17
  2. B.19
  3. C.21
  4. D.25
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Answer: A (17)

Explanation

The correct total is 12 × 18 = 216. Recording 23 as 11 makes the total 12 less, 204, so the mean becomes 204 ÷ 12 = 17.

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Question 11

Circle Theorems

A circle has centre O and radius 5cm. From an external point P, a tangent PT is drawn to touch the circle at T. If OP = 13cm, find the length PT

  1. A.6cm
  2. B.14cm
  3. C.12cm
  4. D.11cm
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Answer: C (12cm)

Explanation

The radius OT is perpendicular to the tangent, so PT² = 13² − 5² = 144 and PT = 12 cm.

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Question 12

Trigonometry

If tan θ + cot θ = 10/3, where 0° < θ < 90°, find sin 2θ

  1. A.4/5
  2. B.3/10
  3. C.3/5
  4. D.4/10
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Answer: C (3/5)

Explanation

tan θ + cot θ = 1/(sin θ cos θ) = 10/3, so sin θ cos θ = 3/10 and sin 2θ = 2 sin θ cos θ = 3/5.

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Question 13

Sets

In a class of 80 students, 42 offer Mathematics, 35 offer Physics and 18 offer both mathematics and physics. How many offer neither?

  1. A.21
  2. B.11
  3. C.25
  4. D.18
Show the answer and explanation

Answer: A (21)

Explanation

Number offering at least one = 42 + 35 − 18 = 59, so neither = 80 − 59 = 21.

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Question 14

Matrices and Determinants

If the matrix [x 2; 3 x] (first row x, 2; second row 3, x) has a determinant of 16, find x

  1. A.±4
  2. B.±5
  3. C.±√22
  4. D.±22
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Answer: C (±√22)

Explanation

Determinant = x·x − 2·3 = x² − 6 = 16, so x² = 22 and x = ±√22.

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Question 15

Progressions

If U1 = 2, Un+1 = 2Un + 1, find U4

  1. A.15
  2. B.47
  3. C.23
  4. D.31
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Answer: C (23)

Explanation

U₂ = 2(2) + 1 = 5, U₃ = 2(5) + 1 = 11, U₄ = 2(11) + 1 = 23.

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Question 16

Algebraic Expressions

If x + y = 10, xy = 21. Find x2y + xy2

  1. A.231
  2. B.210
  3. C.147
  4. D.189
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Answer: B (210)

Explanation

x²y + xy² = xy(x + y) = 21 × 10 = 210.

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Question 17

Variation

If y varies directly as the square of x and inversely as (x + 1) and x = 2 when y = 8/3, find y when x = 5

  1. A.20/3
  2. B.25/3
  3. C.100/9
  4. D.50/9
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Answer: B (25/3)

Explanation

y = kx²/(x + 1). 8/3 = 4k/3 gives k = 2. When x = 5, y = 2 × 25/6 = 25/3.

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Question 18

Polynomials

Find the coefficient of x3 in the expression (2x − 1/x)5

  1. A.40
  2. B.−80
  3. C.80
  4. D.−40
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Answer: B (−80)

Explanation

The general term is ⁵Cᵣ(2x)⁵⁻ʳ(−1/x)ʳ, with power of x = 5 − 2r. For x³, r = 1: 5 × 2⁴ × (−1) = −80.

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Question 19

Algebraic Expressions

If (x + 1)(x + 2)(x + 3) = x3 + ax2 + bx + 6, find the value of a + b.

  1. A.16
  2. B.11
  3. C.17
  4. D.10
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Answer: C (17)

Explanation

Expanding gives x³ + 6x² + 11x + 6, so a = 6, b = 11 and a + b = 17.

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Question 20

Polynomials

When the polynomial P(x) = x3 + kx2 − 5x + 7 is divided by (x − 2), the remainder is 9. Find k

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

Explanation

By the remainder theorem P(2) = 8 + 4k − 10 + 7 = 9, so 4k = 4 and k = 1.

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Question 21

Coordinate Geometry

Find the value of k if the points (1, 2), (3, 5), (k, 8) are collinear

  1. A.5
  2. B.4
  3. C.6
  4. D.7
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Answer: A (5)

Explanation

Gradient from (1, 2) to (3, 5) is 3/2; from (3, 5) to (k, 8) it is 3/(k − 3). Equal gradients give k − 3 = 2, so k = 5.

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Question 22

Permutation and Combination

Six students − A, B, C, D, E and F − are to sit around a circular table. In how many distinct arrangements can they sit if A and B must sit next to each other

  1. A.48
  2. B.24
  3. C.720
  4. D.120
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Answer: A (48)

Explanation

Treat A and B as one unit: 5 units round a circle give (5 − 1)! = 24 ways, and A and B can swap: 24 × 2 = 48.

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Question 23

Indices

Solve 2x+1 + 2x = 48

  1. A.3
  2. B.6
  3. C.5
  4. D.4
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Answer: D (4)

Explanation

2ˣ(2 + 1) = 48, so 2ˣ = 16 = 2⁴ and x = 4.

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Question 24

Binary Operations

Find the identity element for an operation defined by: a * b = ab − a − b + 2

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

Explanation

We need a * e = a: ae − a − e + 2 = a, i.e. (a − 1)(e − 2) = 0 for all a, so e = 2.

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Question 25

Commercial Arithmetic

A trader marks an item 40% above cost price. He allows a discount of 25%, find his percentage profit or loss

  1. A.5% loss
  2. B.10% Profit
  3. C.15% profit
  4. D.5% profit
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Answer: D (5% profit)

Explanation

Selling price = 1.4 × 0.75 = 1.05 of cost price, a 5% profit.

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