WAEC Further Mathematics 2024 past questions

Checked by the Skolra question team · Last updated 23 October 2026

Ten questions from the 2024 WAEC Further Mathematics paper, with answers and explanations. We hold 47 questions from this paper, and the Skolra app has all of them.

WAEC Further Mathematics 2024 questions and answers

1. 1.If tan x = $\frac{1}{3}$, where 180º < x < 270º evaluate $\frac{sin2 x - cos x}{2 tan x + sin 2x}$, leaving the answer in surd form (radicals)
   1. A
      (18 − 9√10)/38
   2. B
      9(2 + √10)/28
   3. C
      12(2 + √10)/25
   4. D
      (18 + 9√10)/38
      Correct
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   Answer: D

   Step 1: Find sin x and cos x.
   Since 180° < x < 270°, x lies in the third quadrant, where both sine and cosine are negative.
   tan x = 1/3 means (using a right triangle) opposite = 1, adjacent = 3, so hypotenuse = √(1²+3²) = √10.
   Therefore sin x = −1/√10 and cos x = −3/√10.

   Step 2: Find sin 2x.
   sin 2x = 2 sin x cos x = 2 × (−1/√10) × (−3/√10) = 6/10 = 3/5 (positive, since two negatives multiply to a positive).

   Step 3: Build the numerator.
   sin 2x − cos x = 3/5 − (−3/√10) = 3/5 + 3√10/10 = 6/10 + 3√10/10 = (6 + 3√10)/10.

   Step 4: Build the denominator.
   2 tan x + sin 2x = 2(1/3) + 3/5 = 2/3 + 3/5 = 10/15 + 9/15 = 19/15.

2. 2.The gradient of a curve is given by 3x$^2$ - 8x + 2. If the curve passes through P(0, 4), find the equation of the curve.
   1. A
      y = x³ - 8x² + 2x + 4
   2. B
      y = x³ - 4x² + 2x + 2
   3. C
      y = x³ - 4x² + 2x
   4. D
      y = x³ - 4x² + 2x + 4
      Correct
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   Answer: D

   Step 1: Integrate the gradient function. The gradient (dy/dx) of the curve is 3x² - 8x + 2. To get the equation of the curve, integrate term by term:

   ∫3x² dx = x³
   ∫-8x dx = -4x²
   ∫2 dx = 2x

   So y = x³ - 4x² + 2x + C, where C is the constant of integration.

   Step 2: Use the given point to find C. The curve passes through P(0, 4), meaning when x = 0, y = 4. Substituting:

   4 = (0)³ - 4(0)² + 2(0) + C
   4 = C

   So the full equation is y = x³ - 4x² + 2x + 4.

   Where the other options go wrong:

   y = x³ - 4x² + 2x (with no constant) comes from integrating correctly but forgetting that a curve's equation needs a constant of integration fixed by the given point — it assumes C = 0 instead of actually using P(0, 4).

3. 3.If $\begin{pmatrix} 6 & 4 \\ 7 & 5 \end{pmatrix}$ $\begin{pmatrix} 2 \\ m \end{pmatrix}$ = 2$\begin{pmatrix} 12 \\ 14.5 \end{pmatrix}$, find the value of m.
   1. A
      2
   2. B
      4
   3. C
      3
      Correct
   4. D
      0
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   Answer: C

   Step 1: Multiply out the right-hand side first. The equation has a scalar 2 in front of the column vector on the right, so that 2 must be applied to both entries before anything else: 2 × 12 = 24 and 2 × 14.5 = 29. The right-hand side is therefore $\begin{pmatrix}24\\29\end{pmatrix}$, not $\begin{pmatrix}12\\14.5\end{pmatrix}$.

   Step 2: Carry out the matrix-vector multiplication on the left. Multiplying $\begin{pmatrix}6&4\\7&5\end{pmatrix}\begin{pmatrix}2\\m\end{pmatrix}$ gives two equations:
   Row 1: 6(2) + 4(m) = 24 → 12 + 4m = 24 → 4m = 12 → m = 3
   Row 2: 7(2) + 5(m) = 29 → 14 + 5m = 29 → 5m = 15 → m = 3

4. 4.The point P(-3, 5) lies on a line which is perpendicular to 2x - 4y + 3 = 0. Find the equation of the line.
   1. A
      2x + y - 1 = 0
   2. B
      x - 2y + 13 = 0
   3. C
      2x - y + 11 = 0
   4. D
      2x + y + 1 = 0
      Correct
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   Answer: D

   Step 1: Find the gradient of the given line.
   2x - 4y + 3 = 0 → 4y = 2x + 3 → y = ½x + ¾.
   So the gradient of this line is ½.

   Step 2: Find the gradient of the perpendicular line.
   For perpendicular lines, m₁ × m₂ = -1. Since m₁ = ½, the required gradient is m₂ = -2.

   Step 3: Use the point P(-3, 5) with gradient -2.
   y - y₁ = m(x - x₁)
   y - 5 = -2(x - (-3))
   y - 5 = -2(x + 3) = -2x - 6
   y = -2x - 1, which rearranges to 2x + y + 1 = 0. Checking: at x = -3, y = 6 - 1 = 5, which matches P. This is the correct equation.

5. 5.Using the trapezium rule with seven ordinates, evaluate, correct to three decimal places, $\int_{2.4}^{3.6} \frac{1}{\sqrt{x^2 - 2}}$dx
   1. A
      0.932
   2. B
      0.274
   3. C
      0.465
   4. D
      0.466
      Correct
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   Answer: D

   Setting up the ordinates. Seven ordinates over [2.4, 3.6] means six equal strips, so the width is h = (3.6 − 2.4) ÷ 6 = 0.2. The ordinates are taken at x = 2.4, 2.6, 2.8, 3.0, 3.2, 3.4, 3.6, and at each one we evaluate f(x) = 1/√(x² − 2):

   f(2.4) = 0.515711
   f(2.6) = 0.458349
   f(2.8) = 0.413803
   f(3.0) = 0.377964
   f(3.2) = 0.348367
   f(3.4) = 0.323423
   f(3.6) = 0.302061

   Applying the trapezium rule. The rule states I ≈ (h/2)[first + last + 2(sum of the middle ordinates)]. Here that is:

   I ≈ 0.1 × [(0.515711 + 0.302061) + 2(0.458349 + 0.413803 + 0.377964 + 0.348367 + 0.323423)]
   = 0.1 × [0.817772 + 2(1.921906)]
   = 0.1 × [0.817772 + 3.843812]
   = 0.1 × 4.661584
   = 0.466 (correct to three decimal places).

6. 6.Evaluate: $\frac{cos^2 300^{\circ} - 4sin^2 120^{\circ}}{tan^2 135^{\circ}}$
   1. A
      -5/4
   2. B
      -11/4
      Correct
   3. C
      -3/4
   4. D
      11/4
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   Answer: B

   Step 1 — Reference angles.
   300° lies in the fourth quadrant, so cos 300° = cos 60° = 1/2, giving cos²300° = 1/4.

   Step 2 — The sine term.
   120° lies in the second quadrant, so sin 120° = sin 60° = √3/2, giving sin²120° = 3/4. Multiplying by 4 gives 4sin²120° = 3.

   Step 3 — Numerator.
   cos²300° − 4sin²120° = 1/4 − 3 = −11/4.

   Step 4 — The tangent term.
   135° lies in the second quadrant, so tan 135° = −tan 45° = −1. Squaring removes the sign, so tan²135° = 1.

   Step 5 — Divide.
   −11/4 ÷ 1 = −11/4, which is the correct value.

7. 7.A fair dice is thrown twice. Find the probability that the sum obtained will be a factor of 12.
   1. A
      1/3
      Correct
   2. B
      1/4
   3. C
      11/36
   4. D
      1/6
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   Answer: A

   When a fair die is thrown twice, there are 36 equally likely outcomes in total (6 × 6).

   We need the sum to be a factor of 12: that is, the sum must be one of 1, 2, 3, 4, 6, or 12. Since the smallest possible sum of two dice is 2, we can drop 1 and check the rest.

   Counting the outcomes for each qualifying sum:
   Sum = 2: (1,1) → 1 way
   Sum = 3: (1,2),(2,1) → 2 ways
   Sum = 4: (1,3),(3,1),(2,2) → 3 ways
   Sum = 6: (1,5),(5,1),(2,4),(4,2),(3,3) → 5 ways
   Sum = 12: (6,6) → 1 way

   Adding these: 1 + 2 + 3 + 5 + 1 = 12 favourable outcomes out of 36.

   So the probability = 12/36 = 1/3, which is the correct answer.

8. 8.If r = i + 2j and n = -i + 3j, find |2n - r|.
   1. A
      5
      Correct
   2. B
      √65
   3. C
      √10
   4. D
      2√5
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   Answer: A

   Given: r = i + 2j and n = -i + 3j. We need |2n - r|.

   Step 1 — Find 2n:
   2n = 2(-i + 3j) = -2i + 6j

   Step 2 — Subtract r:
   2n - r = (-2i + 6j) - (i + 2j) = (-2-1)i + (6-2)j = -3i + 4j

   Step 3 — Find the magnitude:
   |2n - r| = √((-3)² + 4²) = √(9 + 16) = √25 = 5

   So the value 5 is correct.

   The value 2√5 comes from a common slip: doubling the whole difference (n - r) instead of doubling n first before subtracting r.

   The value √65 comes from mistakenly adding the vectors (2n + r) instead of subtracting.

   The value √10 comes from mixing up the order and scaling, working out 2r - n instead of 2n - r.

   Always do the scalar multiplication first, then the vector subtraction, then take the magnitude last.

9. 9.A body of mass 80 kg moving with a velocity of 25 ) ms$^{-1}$ collides with another moving in the opposite direction at 10 ms$^{-1}$. After collision, both bodies moved with a common velocity of 12.8 ms$^{-1}$. Calculate, correct to the nearest whole number, the mass of the second body.
   1. A
      133 kg
   2. B
      43 kg
      Correct
   3. C
      42 kg
   4. D
      76 kg
   Show answerHide answer

   Answer: B

   Momentum is conserved in a collision: the total momentum before collision equals the total momentum after.

   Take the direction of the 80 kg body's motion as positive. Since the second body moves in the opposite direction, its velocity is -10 ms⁻¹, not +10 ms⁻¹.

   Before collision: 80(25) + m(-10)
   After collision (bodies move together): (80 + m)(12.8)

   Setting these equal:
   2000 - 10m = 1024 + 12.8m
   2000 - 1024 = 12.8m + 10m
   976 = 22.8m
   m = 976 ÷ 22.8 = 42.8 kg, which rounds to 43 kg to the nearest whole number.

   The answer of 42 kg comes from cutting off the decimal (42.8) instead of rounding it up.

10. 10.In triangle XYZ, |XY| = 10cm, |YZ| = 9 cm and |XZ| = 7 cm. If XZY = $\alpha$, find the value of cos $\alpha$.
    1. A
       11/15
    2. B
       5/21
       Correct
    3. C
       17/35
    4. D
       10/21
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    Answer: B

    Using the cosine rule, the angle XZY = α is the angle at vertex Z, so the side facing it is XY (the side that does NOT touch Z).

    |XY|² = |XZ|² + |YZ|² − 2·|XZ|·|YZ|·cos α

    10² = 7² + 9² − 2(7)(9) cos α
    100 = 49 + 81 − 126 cos α
    100 = 130 − 126 cos α
    126 cos α = 30
    cos α = 30/126 = 5/21

    The value 10/21 comes from forgetting to double the product of the two sides in the denominator (using 7×9 = 63 instead of 2×7×9 = 126). The values 17/35 and 11/15 come from applying the cosine rule correctly but to the wrong vertex — 17/35 is cos X (using YZ as the opposite side) and 11/15 is cos Y (using XZ as the opposite side), not the angle at Z that the question asks for.

The questions come from past JAMB, WAEC and NECO papers. We use software to help write the answer options and explanations, and every answer is checked twice before it goes on the site. How we check

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