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UTME 2023

JAMB Mathematics past questions and answers

10 questions from the most recent UTME Mathematics papers in our bank, across different topics. Try each one, then open the answer to see a worked solution.

  1. Question 1 · UTME 2023 · Progression

    What is the general term of the sequence 3, 8, 13, 18, …?

    1. A5n − 2
    2. B5n + 2
    3. C5
    4. D5n
    Show answer and solution

    Answer: A. 5n − 2

    What the question is asking: Find the nth term of the A.P.

    Step 1 – a = 3, d = 5

    Step 2 – Tₙ = 3 + 5(n − 1) = 5n − 2

  2. Question 2 · UTME 2023 · Probability

    A bag contains 8 red balls and some white balls. The probability of drawing a white ball is half the probability of drawing a red ball. Two balls are drawn one after the other without replacement. Find the probability that the first is red and the second is white.

    1. A⅓
    2. B2/9
    3. C⅔
    4. D8/33
    Show answer and solution

    Answer: D. 8/33

    What the question is asking: Find the number of white balls, then P(red then white).

    Step 1 – P(white) = ½P(red), so white = ½ × 8 = 4 balls; total 12

    Step 2 – P(red first) = 8/12

    Step 3 – P(white second) = 4/11

    Step 4 – 8/12 × 4/11 = 32/132 = 8/33

  3. Question 3 · UTME 2023 · Indices, Logarithms, Surds

    Solve the logarithmic equation log₂ (6 − x) = 3 − log₂ x.

    1. Ax = 4 or 2
    2. Bx = −4 or −2
    3. Cx = −4 or 2
    4. Dx = 4 or −2
    Show answer and solution

    Answer: A. x = 4 or 2

    What the question is asking: Combine the logs and solve.

    Step 1 – log₂ (6 − x) + log₂ x = 3 → x(6 − x) = 8

    Step 2 – x² − 6x + 8 = 0 → (x − 2)(x − 4) = 0

    Step 3 – Both make 6 − x and x positive, so x = 2 or 4

  4. Question 4 · UTME 2023 · Matrices and Determinants

    Find the matrix A if A × (0 1 ; 2 −1) = (2 −1 ; 1 0). [Matrices are written row by row; ";" separates the rows.]

    1. A(2 1 ; −½ −½)
    2. B(0 1 ; ½ ½)
    3. C(2 1 ; 0 −1)
    4. D(2 1 ; ½ −2)
    Show answer and solution

    Answer: B. (0 1 ; ½ ½)

    What the question is asking: Multiply both sides on the right by the inverse of (0 1 ; 2 −1).

    Step 1 – det = 0 × (−1) − 1 × 2 = −2

    Step 2 – Inverse = (1/−2)(−1 −1 ; −2 0) = (½ ½ ; 1 0)

    Step 3 – A = (2 −1 ; 1 0)(½ ½ ; 1 0) = (1 − 1 1 + 0 ; ½ ½) = (0 1 ; ½ ½)

    Check: (0 1 ; ½ ½)(0 1 ; 2 −1) = (2 −1 ; 1 0) ✓

  5. Question 5 · UTME 2023 · Trigonometry

    A boat sails 8 km north from P to Q and then 6 km west from Q to R. Calculate the bearing of R from P, to the nearest degree.

    1. A217°
    2. B323°
    3. C037°
    4. D053°
    Show answer and solution

    Answer: B. 323°

    What the question is asking: Standing at P, in which direction is R?

    Step 1 – tan θ = 6/8 → θ ≈ 36.87° west of north

    Step 2 – Bearing = 360° − 36.87° ≈ 323°

  6. Question 6 · UTME 2023 · Fractions, Decimals, Percentages

    An article sold for N230.00 makes a 15% profit. Find the profit or loss per cent if it is sold for N180.00.

    1. A10% gain
    2. B10% loss
    3. C12% loss
    4. D12% gain
    Show answer and solution

    Answer: B. 10% loss

    What the question is asking: Find the cost price, then the new percentage.

    Step 1 – 1.15C = 230 → C = 200

    Step 2 – 180 < 200, so a loss of 20

    Step 3 – 20/200 × 100% = 10% loss

  7. Question 7 · UTME 2023 · Permutation and Combination

    How many different 8-letter arrangements are possible using the letters of the word SYLLABUS?

    1. A(8 − 1)!
    2. B8!/2!
    3. C8!/(2!2!)
    4. D8!
    Show answer and solution

    Answer: C. 8!/(2!2!)

    What the question is asking: Count arrangements with repeated letters.

    Step 1 – S appears twice and L appears twice

    Step 2 – 8!/(2!2!)

  8. Question 8 · UTME 2023 · Quadratic and Simultaneous Equations

    A rectangle has one side that is 6 cm shorter than the other. The area of the rectangle will increase by 68 cm² if 2 cm is added to each side. Find the length of the shorter side.

    1. A15 cm
    2. B19 cm
    3. C13 cm
    4. D21 cm
    Show answer and solution

    Answer: C. 13 cm

    What the question is asking: Set up an equation for the change in area.

    Step 1 – Shorter side x, longer x + 6

    Step 2 – (x + 2)(x + 8) − x(x + 6) = 68 → 4x + 16 = 68

    Step 3 – x = 13 cm

  9. Question 9 · UTME 2023 · Coordinate Geometry

    Find the value of t if the distance between the points P(−3, −14) and Q(t, −5) is 9 units.

    1. A3
    2. B2
    3. C−3
    4. D−2
    Show answer and solution

    Answer: C. −3

    What the question is asking: Use the distance formula.

    Step 1 – (t + 3)² + 9² = 81

    Step 2 – (t + 3)² = 0 → t = −3

  10. Question 10 · UTME 2023 · Binary Operations

    Let * and ^ be two binary operations such that a * b = a²b and a ^ b = 2a + b. Find (−4 * 2) ^ (7 * −1).

    1. A−49
    2. B64
    3. C113
    4. D15
    Show answer and solution

    Answer: D. 15

    What the question is asking: Work out each bracket, then apply ^.

    Step 1 – −4 * 2 = 16 × 2 = 32

    Step 2 – 7 * −1 = 49 × (−1) = −49

    Step 3 – 32 ^ −49 = 64 − 49 = 15

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