WAEC Further Mathematics 2018 past questions

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

Ten questions from the 2018 WAEC Further Mathematics paper, with answers and explanations. We hold 46 questions from this paper, and the Skolra app has all of them.

WAEC Further Mathematics 2018 questions and answers

1. 1.A body of mass 28g, initially at rest is acted upon by a force, F Newtons. If it attains a velocity of $5.4ms^{-1}$ in 18 seconds, find the value of F.
   1. A
      8.4 N
   2. B
      8.4 × 10⁻³ N
      Correct
   3. C
      0.084 N
   4. D
      2.7216 N
   Show answerHide answer

   Answer: B

   Given: mass m = 28 g, initial velocity u = 0 (at rest), final velocity v = 5.4 ms⁻¹, time t = 18 s.

   First convert the mass to kilograms, since force in Newtons requires SI units:
   m = 28 g = 28 ÷ 1000 = 0.028 kg.

   Next find the acceleration using a = (v − u) / t:
   a = (5.4 − 0) / 18 = 0.3 ms⁻².

   Now apply Newton's second law, F = ma:
   F = 0.028 × 0.3 = 0.0084 N, which is the same as 8.4 × 10⁻³ N. That is the correct value of F.

   The option giving 8.4 N comes from forgetting to convert the mass from grams to kilograms before multiplying — a very common slip.

   The option giving 0.084 N comes from converting grams to kilograms wrongly, by dividing by 100 instead of 1000.

2. 2.$P = {1, 3, 5, 7, 9}, Q = {2, 4, 6, 8, 10, 12}, R = {2, 3, 5, 7, 11}$ are subsets of $U = {1, 2, 3, ... , 12}$. Which of the following statements is true?
   1. A
      P ∩ Q = ∅
      Correct
   2. B
      R ⊂ P ∪ Q
   3. C
      Q ∩ R ⊂ P
   4. D
      P ∩ R = Q
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   Answer: A

   P = {1, 3, 5, 7, 9} contains only odd numbers, and Q = {2, 4, 6, 8, 10, 12} contains only even numbers. Since no number can be both odd and even, these two sets share no element at all.

   So P ∩ Q = ∅ — this is the true statement.

   Now see why the others fail:

   R ⊂ P ∪ Q looks tempting because P ∪ Q = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12} covers almost everything in U — but it leaves out 11. Since R = {2, 3, 5, 7, 11} contains 11, R is not a subset of P ∪ Q.

   Q ∩ R ⊂ P fails too: Q ∩ R = {2} (the only even number common to Q and R), but 2 is not in P, so {2} cannot be a subset of P.

3. 3.A car is moving at 120$kmh^{-1}$. Find its speed in $ms^{-1}$.
   1. A
      2 ms⁻¹
   2. B
      33.3 ms⁻¹
      Correct
   3. C
      432 ms⁻¹
   4. D
      120 ms⁻¹
   Show answerHide answer

   Answer: B

   To change km/h to m/s, multiply by 1000/3600, which simplifies to 5/18.

   Speed = 120 × 5/18 = 600/18 = 33.3 ms⁻¹ (to 1 decimal place).

   A candidate who flips the conversion fraction and multiplies by 18/5 instead of 5/18 gets 120 × 18/5 = 432 ms⁻¹, which is far too fast for a car.

   A candidate who forgets to convert at all and just changes the unit label ends up with 120 ms⁻¹, which is also unrealistic — that would be faster than the speed of sound.

   A candidate who mistakenly divides by 60 (treating the hour as 60 seconds instead of 3600 seconds) gets 120 ÷ 60 = 2 ms⁻¹, which is far too slow for a moving car.

   The correct, realistic conversion gives 33.3 ms⁻¹.

4. 4.Find the equation to the circle $x^{2} + y^{2} - 4x - 2y = 0$ at the point (1, 3).
   1. A
      3x − y = 0
   2. B
      2x + y − 5 = 0
   3. C
      x − 2y + 5 = 0
      Correct
   4. D
      x + 2y − 7 = 0
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   Answer: C

   The point (1, 3) lies on the circle, since 1² + 3² − 4(1) − 2(3) = 1 + 9 − 4 − 6 = 0. The question is asking for the tangent to the circle at that point.

   Step 1 — Find the centre. Writing x² + y² − 4x − 2y = 0 in the form x² + y² + 2gx + 2fy + c = 0 gives 2g = −4 and 2f = −2, so g = −2, f = −1. The centre is (−g, −f) = (2, 1).

   Step 2 — Slope of the radius to the point. From the centre (2, 1) to (1, 3): slope = (3 − 1)/(1 − 2) = 2/(−1) = −2.

   Step 3 — Tangent slope. A tangent is perpendicular to the radius drawn to the point of contact, so its slope is the negative reciprocal of −2, which is 1/2.

5. 5.Resolve $\frac{3x - 1}{(x - 2)^{2}}, x \neq 2$ into partial fractions.
   1. A
      $\frac{5}{x-2}+\frac{3}{(x-2)^{2}}$
   2. B
      $\frac{3}{x-2}-\frac{5}{(x-2)^{2}}$
   3. C
      $-\frac{3}{x-2}+\frac{5}{(x-2)^{2}}$
   4. D
      $\frac{3}{x-2}+\frac{5}{(x-2)^{2}}$
      Correct
   Show answerHide answer

   Answer: D

   We want constants A and B such that:

   $\frac{3x-1}{(x-2)^{2}} = \frac{A}{x-2} + \frac{B}{(x-2)^{2}}$

   Multiply both sides by $(x-2)^{2}$:

   $3x - 1 = A(x-2) + B$

   Put x = 2 (this cancels the A-term):

   $3(2) - 1 = B \Rightarrow B = 5$

   Now compare the coefficient of x on both sides: on the right, expanding $A(x-2)$ gives $Ax - 2A$, so the coefficient of x is A. On the left it is 3. So A = 3.

   Check with the constant term: $-2A + B = -2(3) + 5 = -1$, which matches the constant on the left (−1), confirming the values.

   So the correct resolution is $\frac{3}{x-2}+\frac{5}{(x-2)^{2}}$.

6. 6.Given that $y = x(x + 1)^{2}$, calculate the maximum value of y.
   1. A
      0
      Correct
   2. B
      4/27
   3. C
      -1
   4. D
      -4/27
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   Answer: A

   Expand first: y = x(x + 1)² = x(x² + 2x + 1) = x³ + 2x² + x.

   Differentiate and find the turning points:
   dy/dx = 3x² + 4x + 1 = (3x + 1)(x + 1) = 0, so x = -1 or x = -1/3.

   Classify each turning point with the second derivative, d²y/dx² = 6x + 4:
   At x = -1: 6(-1) + 4 = -2, which is negative, so this is the maximum point.
   At x = -1/3: 6(-1/3) + 4 = 2, which is positive, so this is the minimum point.

   Evaluate y at the maximum point (x = -1): y = (-1)(-1 + 1)² = (-1)(0)² = 0. So the maximum value of y is 0.

7. 7.Evaluate $\int_{\frac{1}{2}}^{1} \frac{x^{3} - 4}{x^{3}} \mathrm {d} x$.
   1. A
      -3/2
   2. B
      13/2
   3. C
      -11/2
      Correct
   4. D
      11/2
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   Answer: C

   We need to evaluate ∫ from x = 1/2 to x = 1 of (x³ − 4)/x³ dx.

   First split the fraction: (x³ − 4)/x³ = 1 − 4/x³ = 1 − 4x⁻³.

   Integrate term by term: ∫1 dx = x, and ∫4x⁻³ dx = 4 × (x⁻²/−2) = −2x⁻² = −2/x². So the antiderivative of (1 − 4x⁻³) is x − (−2/x²) = x + 2/x².

   Now apply the limits. At x = 1: 1 + 2/1² = 1 + 2 = 3.
   At x = 1/2: 1/2 + 2/(1/2)² = 1/2 + 2/(1/4) = 1/2 + 8 = 17/2.

   The definite integral is (value at the top limit) − (value at the bottom limit) = 3 − 17/2 = 6/2 − 17/2 = −11/2. That is the correct result.

8. 8.Express $\frac{13}{4}\pi$ radians in degrees.
   1. A
      675°
   2. B
      540°
   3. C
      585°
      Correct
   4. D
      292.5°
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   Answer: C

   To change radians to degrees, we use the fact that π radians = 180°, so we multiply the given radian measure by 180/π.

   Since the angle already has a π in it, the π's cancel neatly:
   (13/4)π × (180/π) = (13/4) × 180 = 13 × 45 = 585°.

   The option 292.5° comes from mistakenly using 90° in place of 180° (perhaps confusing the conversion factor with a right angle), giving (13/4) × 90 = 292.5°, which is exactly half the correct value.

9. 9.What is the probability of obtaining a head and a six when a fair coin and and a die are tossed together?
   1. A
      1/36
   2. B
      1/12
      Correct
   3. C
      2/3
   4. D
      1/6
   Show answerHide answer

   Answer: B

   A fair coin has two equally likely outcomes, so P(head) = 1/2. A fair die has six equally likely outcomes, so P(six) = 1/6.

   Tossing the coin and the die together are independent events — what happens on the coin has no effect on the die. For independent events, we multiply the individual probabilities to get the probability that both happen together:

   P(head and six) = 1/2 × 1/6 = 1/12

10. 10.If $\alpha$ and $\beta$ are the roots of $2x^{2} - 5x + 6 = 0$, find the equation whose roots are $(\alpha + 1)$ and $(\beta + 1)$.
    1. A
       2x² - x + 3 = 0
    2. B
       2x² + 9x + 13 = 0
    3. C
       2x² - 9x + 8 = 0
    4. D
       2x² - 9x + 13 = 0
       Correct
    Show answerHide answer

    Answer: D

    Step 1: Find the sum and product of the original roots.
    For 2x² - 5x + 6 = 0, dividing by 2 gives x² - (5/2)x + 3 = 0, so α + β = 5/2 and αβ = 3.

    Step 2: Find the sum and product of the new roots.
    The new roots are (α + 1) and (β + 1).
    Sum = (α + 1) + (β + 1) = α + β + 2 = 5/2 + 2 = 9/2.
    Product = (α + 1)(β + 1) = αβ + α + β + 1 = 3 + 5/2 + 1 = 13/2.

    Step 3: Form the new equation.
    Using x² - (sum)x + (product) = 0:
    x² - (9/2)x + 13/2 = 0.
    Multiplying through by 2 to clear fractions gives 2x² - 9x + 13 = 0, which is the correct equation.

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