WAEC Further Mathematics 2023 past questions

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

Ten questions from the 2023 WAEC Further Mathematics paper, with answers and explanations. We hold 44 questions from this paper, and the Skolra app has all of them.

WAEC Further Mathematics 2023 questions and answers

1. 1.A linear transformation on the oxy plane is defined by P: (x,y) -> (2x+y, -2y). Find P².
   1. A
      (4x, -4y)
   2. B
      (4x + 2y, 4y)
   3. C
      (4x, 4y)
      Correct
   4. D
      (2x + y, -2y)
   Show answerHide answer

   Answer: C

   Setting up P as a matrix. The rule P: (x, y) → (2x + y, −2y) can be written as a matrix acting on the column (x, y):

   A = [[2, 1], [0, −2]], so that A(x, y)ᵀ gives (2x + y, −2y)ᵀ.

   Finding P². P² means applying the transformation twice — apply P to the result of P. Let (x′, y′) = P(x, y) = (2x + y, −2y). Then:

   P²(x, y) = P(x′, y′) = (2x′ + y′, −2y′)
   = (2(2x + y) + (−2y), −2(−2y))
   = (4x + 2y − 2y, 4y)
   = (4x, 4y)

   This matches squaring the matrix: A² = [[2,1],[0,−2]]·[[2,1],[0,−2]] = [[4, 2−2],[0, 4]] = [[4,0],[0,4]] = 4I, which sends (x, y) to (4x, 4y). So (4x, 4y) is correct.

2. 2.Solve 6sin2θ tanθ = 4, where 0° < θ < 90°
   1. A
      θ = 19.5°
   2. B
      θ = 41.8°
   3. C
      θ = 54.7°
   4. D
      θ = 35.3°
      Correct
   Show answerHide answer

   Answer: D

   Expand sin2θ: sin2θ = 2 sinθ cosθ, and tanθ = sinθ/cosθ. Substituting into 6 sin2θ tanθ = 4 gives:

   6(2 sinθ cosθ)(sinθ/cosθ) = 4

   The cosθ cancels (it is never zero for 0° < θ < 90°), leaving 12 sin²θ = 4, so sin²θ = 1/3.

   Taking the square root: sinθ = 1/√3 ≈ 0.5774. Since θ is restricted to the first quadrant, sinθ is positive and increasing, so there is exactly one solution: θ = sin⁻¹(0.5774) ≈ 35.3°.

   A common slip is forgetting the "2" inside sin2θ and simplifying 6 sinθcosθ tanθ = 4 as if it were 6 sin²θ = 4. That gives sin²θ = 2/3, sinθ ≈ 0.8165, and θ ≈ 54.7° — the complement of the correct angle, not the answer itself.

3. 3.If (1/9)^(2x-1) = (1/81)^(2-3x), find the value of x
   1. A
      x = 3/4
   2. B
      x = 5/8
      Correct
   3. C
      x = 8/5
   4. D
      x = 3/5
   Show answerHide answer

   Answer: B

   Rewrite both sides with base 3. Since 9 = 3² and 81 = 3⁴, we have 1/9 = 3⁻² and 1/81 = 3⁻⁴.

   So the equation (1/9)^(2x−1) = (1/81)^(2−3x) becomes:

   3^(−2(2x−1)) = 3^(−4(2−3x))

   Since the bases are equal, the exponents must be equal:

   −2(2x−1) = −4(2−3x)
   −4x + 2 = −8 + 12x
   2 + 8 = 12x + 4x
   10 = 16x
   x = 10/16 = 5/8

   Checking: when x = 5/8, 2x−1 = 1/4 and 2−3x = 1/8. The left exponent on 3 is −2 × 1/4 = −1/2, and the right exponent on 3 is −4 × 1/8 = −1/2. They match, so x = 5/8 is correct.

   3/5 is the result of writing 81 as 9² but forgetting that squaring the base doubles the exponent — a student who sets 2x−1 = 2−3x directly (instead of 2x−1 = 2(2−3x)) lands here by mistake.

4. 4.If X and Y are two independent events such that P(X)=1/8 and P(X∪Y)=5/8, find P(Y).
   1. A
      4/7
      Correct
   2. B
      2/7
   3. C
      1/2
   4. D
      7/16
   Show answerHide answer

   Answer: A

   Since X and Y are independent, we use P(X∩Y) = P(X) × P(Y) together with the general addition rule:

   P(X∪Y) = P(X) + P(Y) − P(X)P(Y)

   Substituting the given values:
   5/8 = 1/8 + P(Y) − (1/8)P(Y)

   Collect P(Y) terms:
   5/8 − 1/8 = P(Y)(1 − 1/8)
   4/8 = P(Y) × 7/8

   Now divide both sides by 7/8 (which is the same as multiplying by its reciprocal, 8/7):
   P(Y) = (1/2) × (8/7) = 4/7

   You can confirm this: P(X∩Y) = (1/8)(4/7) = 1/14, and P(X) + P(Y) − P(X∩Y) = 7/56 + 32/56 − 4/56 = 35/56 = 5/8, which matches exactly.

5. 5.Given that M is the midpoint of T(2,4) and Q(-8,6), find the length of MQ.
   1. A
      √104 units
   2. B
      2√5 units
   3. C
      √26 units
      Correct
   4. D
      6 units
   Show answerHide answer

   Answer: C

   Step 1: Find M, the midpoint of T(2,4) and Q(-8,6).
   Using the midpoint formula, M = ((2 + (-8))/2, (4 + 6)/2) = (-6/2, 10/2) = (-3, 5).

   Step 2: Find the length MQ using the distance formula.
   MQ = √[(-3 - (-8))² + (5 - 6)²] = √[(5)² + (-1)²] = √(25 + 1) = √26 units.

   This is the correct length, and it checks out neatly: since M is the midpoint of T and Q, MQ should be exactly half of the full distance TQ. TQ = √[(2-(-8))² + (4-6)²] = √[10² + (-2)²] = √104 = 2√26, and indeed half of 2√26 is √26.

   The option √104 units is what you get if you forget that M is only the midpoint and mistakenly compute the full distance from T to Q instead of from M to Q.

6. 6.Find the coefficient of the 6th term in the binomial expansion of (1 - 2x/3)^10 in ascending powers of x.
   1. A
      -1120/27
   2. B
      1120/27
   3. C
      896/27
   4. D
      -896/27
      Correct
   Show answerHide answer

   Answer: D

   Setting up the general term: For (1 − 2x/3)¹⁰, the general term in the expansion (ascending powers of x) is

   T_r+1 = C(10, r) · (1)^10−r · (−2x/3)^r = C(10, r) · (−2/3)^r · x^r

   Since the 1st term corresponds to r = 0 (the constant term, no x), the 6th term corresponds to r = 5, not r = 6 and not r = 4. Counting the term number as if it equalled r directly is a common slip — that is how a candidate lands on 1120/27 or −1120/27 using r = 4 instead of the correct r = 5.

   Computing it:
   C(10, 5) = 252
   (−2/3)⁵ = −32/243 (the power is odd, so the sign stays negative)

   T₆ = 252 × (−32/243) × x⁵ = (−8064/243) x⁵

7. 7.If α and β are the roots of 7x²+12x-4=0, find the value of αβ/(α+β)².
   1. A
      7/36
   2. B
      -7/50
   3. C
      -7/36
      Correct
   4. D
      1/3
   Show answerHide answer

   Answer: C

   For the quadratic 7x² + 12x - 4 = 0, using the sum and product of roots formulas (for ax²+bx+c=0):

   Sum of roots: α + β = -b/a = -12/7
   Product of roots: αβ = c/a = -4/7

   We need αβ/(α+β)². First square the sum:

   (α+β)² = (-12/7)² = 144/49

   Now divide the product by this:

   αβ/(α+β)² = (-4/7) ÷ (144/49) = (-4/7) × (49/144) = -196/1008

   Dividing numerator and denominator by 28 gives -7/36, which is the correct value.

   A candidate who forgets the negative sign on the product (treating c/a as +4/7 instead of -4/7) lands on 7/36 — a sign slip, not a calculation error.

   A candidate who forgets to square the denominator and simply computes αβ/(α+β) gets (-4/7)/(-12/7) = 1/3, missing the squaring step entirely.

8. 8.Evaluate ∫₀¹ x(x²-2)² dx
   1. A
      13/6
   2. B
      1/6
   3. C
      7/6
      Correct
   4. D
      19/6
   Show answerHide answer

   Answer: C

   Step 1: Expand the bracket.
   (x² - 2)² = x⁴ - 4x² + 4

   Step 2: Multiply by x.
   x(x² - 2)² = x⁵ - 4x³ + 4x

   Step 3: Integrate term by term.
   ∫(x⁵ - 4x³ + 4x) dx = x⁶/6 - x⁴ + 2x²

   Step 4: Apply the limits 0 to 1.
   At x = 1: 1/6 - 1 + 2 = 1/6 + 1 = 7/6
   At x = 0: everything is 0
   So the result is 7/6, which is the correct value.

   Checking by substitution confirms this: letting u = x² - 2 gives x dx = du/2, and the limits change from x = 0 (u = -2) to x = 1 (u = -1). This gives (1/2)∫u² du = u³/6 from -2 to -1, which is (-1 - (-8))/6 = 7/6 — the same answer, so we can be confident in it.

9. 9.The volume of a cube is increasing at the rate of 3.5 cm³s⁻¹. Find the rate of change of the side of the base when its length is 6cm.
   1. A
      0.194 cm s⁻¹
   2. B
      0.0972 cm s⁻¹
   3. C
      0.0324 cm s⁻¹
      Correct
   4. D
      0.0162 cm s⁻¹
   Show answerHide answer

   Answer: C

   The volume of a cube of side x cm is V = x³. Since both V and x change with time, we differentiate with respect to time t using the chain rule:

   dV/dt = 3x² · (dx/dt)

   We are given dV/dt = 3.5 cm³s⁻¹ and we want dx/dt when x = 6 cm. Substituting:

   3.5 = 3(6)² · (dx/dt)
   3.5 = 3(36) · (dx/dt)
   3.5 = 108 · (dx/dt)

   So dx/dt = 3.5/108 = 7/216 ≈ 0.0324 cm s⁻¹. This is the correct rate at which the side is increasing.

   The option 0.194 cm s⁻¹ comes from a common slip: differentiating x³ as if it gives 3x (forgetting to square x), so the student divides 3.5 by 3×6 = 18 instead of 3×6² = 108.

10. 10.Adu's scores in five subjects in an examination are 85, 84, 83, 86 and 87. Calculate the standard deviation.
    1. A
       1.58
    2. B
       1.41
       Correct
    3. C
       3.16
    4. D
       2.00
    Show answerHide answer

    Answer: B

    Step 1: Find the mean.
    Mean = (85 + 84 + 83 + 86 + 87) ÷ 5 = 425 ÷ 5 = 85.

    Step 2: Find each deviation from the mean and square it.
    85 − 85 = 0 → 0²= 0
    84 − 85 = −1 → (−1)² = 1
    83 − 85 = −2 → (−2)² = 4
    86 − 85 = 1 → 1² = 1
    87 − 85 = 2 → 2² = 4
    Sum of squared deviations = 0 + 1 + 4 + 1 + 4 = 10.

    Step 3: Find the variance, then the standard deviation.
    Variance = 10 ÷ 5 = 2.
    Standard deviation = √2 ≈ 1.41, which is the correct answer.

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

Spotted a wrong answer? Tell us from the app or on the contact page and we check it.

Practise all 44 questions we hold from the 2023 paper, on a timed CBT screen

Every question has the answer and an explanation, in the Skolra app.

Download for WindowsDownload for Mac

About the app · Skolra for Windows · Skolra for Mac