WAEC Further Mathematics 2009 past questions

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

Ten questions from the 2009 WAEC Further Mathematics paper, with answers and explanations. We hold 41 questions from this paper, and the Skolra app has all of them.

WAEC Further Mathematics 2009 questions and answers

1. 1.The probabilities of Rotey obtaining the highest mark in Mathematics, Physics and Biology tests are 0.9, 0.75 and 0.8 respectively. Calculate the probability of getting the highest marks in at least two of the subjects.
   1. A
      0.995
   2. B
      0.54
   3. C
      0.375
   4. D
      0.915
      Correct
   Show answerHide answer

   Answer: D

   Getting the highest mark in at least two subjects means Rotey tops exactly two of the three tests, or tops all three. We add those two cases together.

   Let the probabilities of topping be: Mathematics = 0.9, Physics = 0.75, Biology = 0.8. So the probabilities of not topping are 0.1, 0.25 and 0.2 respectively.

   Exactly two subjects:
   Maths & Physics, not Biology: 0.9 × 0.75 × 0.2 = 0.135
   Maths & Biology, not Physics: 0.9 × 0.25 × 0.8 = 0.18
   Physics & Biology, not Maths: 0.1 × 0.75 × 0.8 = 0.06
   Total for exactly two = 0.135 + 0.18 + 0.06 = 0.375

   All three subjects: 0.9 × 0.75 × 0.8 = 0.54

   At least two = (exactly two) + (all three) = 0.375 + 0.54 = 0.915.

2. 2.The first term of a geometric progression is 350. If the sum to infinity is 250, find the common ratio.
   1. A
      7/5
   2. B
      2/5
   3. C
      5/7
   4. D
      -2/5
      Correct
   Show answerHide answer

   Answer: D

   For a geometric progression with first term a and common ratio r (where |r| < 1), the sum to infinity is:

   S∞ = a / (1 − r)

   Here a = 350 and S∞ = 250, so:

   250 = 350 / (1 − r)
   1 − r = 350/250 = 7/5
   r = 1 − 7/5 = −2/5

   Since |−2/5| < 1, the formula applies validly, so r = −2/5 is correct. As a check: 350 ÷ (1 − (−2/5)) = 350 ÷ (7/5) = 350 × 5/7 = 250, which matches the given sum to infinity.

3. 3.A body is acted upon by forces F1 = (10 N, 090°) and F2 = (6 N, 180°). Find the magnitude of the resultant force.
   1. A
      4.00 N
   2. B
      16.00 N
   3. C
      8.00 N
   4. D
      11.66 N
      Correct
   Show answerHide answer

   Answer: D

   Bearings are measured clockwise from North. A bearing of 090° points due East, and a bearing of 180° points due South. These two directions are at right angles to each other, so F1 and F2 act perpendicular to one another.

   When two forces are perpendicular, the magnitude of their resultant is found with Pythagoras’ theorem, not by simple addition or subtraction:

   R = √(F1² + F2²) = √(10² + 6²) = √(100 + 36) = √136 ≈ 11.66 N

   So the resultant force has a magnitude of approximately 11.66 N.

4. 4.If (x+3) is a factor of the polynomial x³ + 3x² + nx - 12, where n is a constant, find the value of n.
   1. A
      n = 4
   2. B
      n = -4
      Correct
   3. C
      n = -14
   4. D
      n = -22
   Show answerHide answer

   Answer: B

   Using the Factor Theorem: if (x + 3) is a factor of f(x) = x³ + 3x² + nx − 12, then x = −3 must make f(x) equal to zero, since x + 3 = 0 gives x = −3.

   Substitute x = −3 into the polynomial:
   f(−3) = (−3)³ + 3(−3)² + n(−3) − 12
   = −27 + 3(9) − 3n − 12
   = −27 + 27 − 3n − 12
   = −3n − 12

   Since (x + 3) is a factor, f(−3) = 0:
   −3n − 12 = 0
   −3n = 12
   n = −4

   This is the value that makes the substitution balance to zero.

5. 5.Given that tan 2A = 2 tan A / (1 - tan²A), evaluate tan 15°, leaving your answer in surd form.
   1. A
      √3 − 1
   2. B
      2 + √3
   3. C
      √3 − 2
   4. D
      2 − √3
      Correct
   Show answerHide answer

   Answer: D

   Let A = 15°, so 2A = 30°. Substitute into the given identity:

   tan 30° = 2 tan 15° / (1 − tan² 15°)

   We know tan 30° = 1/√3. Let t = tan 15°:

   1/√3 = 2t / (1 − t²)  ⟹  1 − t² = 2√3·t  ⟹  t² + 2√3·t − 1 = 0

   Solving this quadratic with the formula:
   t = (−2√3 ± √(12 + 4)) / 2 = (−2√3 ± 4) / 2 = −√3 ± 2

   This gives two roots: t = 2 − √3 or t = −2 − √3. Since 15° lies in the first quadrant, its tangent must be positive, so we reject the negative root and keep tan 15° = 2 − √3.

6. 6.The coordinates of the centre of a circle is (-2, 3). If its area is 25π cm², find its equation.
   1. A
      (x - 2)² + (y + 3)² = 25
   2. B
      (x + 2)² + (y - 3)² = 25
      Correct
   3. C
      (x + 2)² + (y - 3)² = 625
   4. D
      (x - 2)² + (y - 3)² = 25
   Show answerHide answer

   Answer: B

   Step 1: Find the radius.
   Area of circle = πr² = 25π cm², so r² = 25 (this gives r = 5 cm, but it is r² we actually need for the equation).

   Step 2: Apply the centre-radius form.
   For a circle with centre (h, k) and radius r, the equation is (x − h)² + (y − k)² = r².
   Here the centre is (−2, 3), so h = −2 and k = 3. Substituting carefully:
   (x − (−2))² + (y − 3)² = 25, which simplifies to (x + 2)² + (y − 3)² = 25.

   This is the correct equation. Notice the sign flips: because h = −2 is negative, "x − h" becomes "x + 2".

7. 7.Three men, P, Q and R aim at a target, the probabilities that P, Q and R hit the target are 1/2, 1/3 and 3/4 respectively. Find the probability that exactly 2 of them hit the target.
   1. A
      5/12
      Correct
   2. B
      13/24
   3. C
      1/8
   4. D
      1/4
   Show answerHide answer

   Answer: A

   Let the probabilities of hitting be P(P) = 1/2, P(Q) = 1/3, P(R) = 3/4. Then the probabilities of missing are P(P') = 1/2, P(Q') = 2/3, P(R') = 1/4.

   "Exactly 2 hit" means exactly one man misses. There are three separate ways this can happen, and since they cannot occur together, we add their probabilities:

   P and Q hit, R misses: 1/2 × 1/3 × 1/4 = 1/24
   P and R hit, Q misses: 1/2 × 2/3 × 3/4 = 6/24
   Q and R hit, P misses: 1/2 × 1/3 × 3/4 = 3/24

   Adding these: 1/24 + 6/24 + 3/24 = 10/24 = 5/12. That is the correct probability.

8. 8.p and q are statements such that p ⟹ q. Which of the following is a valid conclusion from the implication?
   1. A
      If p is false, then q is false
   2. B
      If p is true, then q is false
   3. C
      If q is false, then p is false
      Correct
   4. D
      If q is true, then p is true
   Show answerHide answer

   Answer: C

   This is a question on the logic of implication. Given p ⇒ q ("if p then q"), we need the statement that is logically equivalent to it — that is, always true whenever the original implication is true.

   The valid conclusion is the contrapositive: “if q is false, then p is false” (symbolically ¬q ⇒ ¬p). An implication and its contrapositive always have the same truth value, so this is the only one of the four statements that can be safely deduced from p ⇒ q.

9. 9.Two balls are drawn, from a bag containing 3 red, 4 white and 5 black identical balls. Find the probability that they are all of the same colour.
   1. A
      3/22
   2. B
      25/72
   3. C
      19/66
      Correct
   4. D
      47/66
   Show answerHide answer

   Answer: C

   There are 3 + 4 + 5 = 12 balls altogether, and two balls are drawn together (without replacement), so the total number of ways to pick any 2 balls from the bag is ¹²C₂ = (12 × 11) ÷ 2 = 66.

   For the two balls to be the same colour, they must both be red, both white, or both black. We count each case separately and add:

   Both red: ³C₂ = 3
   Both white: ⁴C₂ = 6
   Both black: ⁵C₂ = 10
   Total favourable ways = 3 + 6 + 10 = 19

   So the probability that both balls are the same colour is 19/66.

   25/72 comes from wrongly treating the draws as independent with replacement, i.e. (3/12)² + (4/12)² + (5/12)² — but the question draws two balls together, not one after replacing it.

10. 10.Express the force F = (8 N, 150°) in the form (ai + bj) where a and b are constants.
    1. A
       F = (-4√3i + 4j) N
       Correct
    2. B
       F = (-4i + 4√3j) N
    3. C
       F = (-4√3i - 4j) N
    4. D
       F = (4√3i + 4j) N
    Show answerHide answer

    Answer: A

    Resolving the force: F = (8 N, 150°) means a force of magnitude 8 N acting at an angle of 150° from the positive x-axis. To write it as ai + bj, we find the horizontal and vertical components using a = F cos θ and b = F sin θ.

    Since 150° lies in the second quadrant, cosine is negative and sine is positive there. Using the related acute angle of 30°:
    cos 150° = -cos 30° = -√3/2
    sin 150° = sin 30° = 1/2

    So:
    a = 8 × (-√3/2) = -4√3
    b = 8 × (1/2) = 4

    This gives F = (-4√3i + 4j) N, which is the correct resolution.

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 41 questions we hold from the 2009 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