WAEC Further Mathematics 2007 past questions

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

Ten questions from the 2007 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 2007 questions and answers

1. 1.Find the area of the circle whose equation is given as $x^{2} + y^{2} - 4x + 8y + 11 = 0$.
   1. A
      97 3/7 sq. units
   2. B
      28 2/7 sq. units
      Correct
   3. C
      62 6/7 sq. units
   4. D
      18 6/7 sq. units
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   Answer: B

   Rewrite the equation in centre-radius form by completing the square.

   x² + y² - 4x + 8y + 11 = 0

   Group the x-terms and y-terms:
   (x² - 4x) + (y² + 8y) + 11 = 0

   Complete the square on each bracket:
   (x - 2)² - 4 + (y + 4)² - 16 + 11 = 0
   (x - 2)² + (y + 4)² - 9 = 0
   (x - 2)² + (y + 4)² = 9

   So r² = 9, giving r = 3.

   Area = πr² = 9π. Using π = 22/7: Area = 9 × 22/7 = 198/7 = 28 2/7 sq. units, which is the correct answer.

   The distractor of 62 6/7 sq. units comes from forgetting to carry the constant 11 into the radius calculation, leaving r² = 4 + 16 = 20 instead of 9.

   The distractor of 97 3/7 sq. units comes from a sign slip that adds the constant instead of subtracting it, giving r² = 4 + 16 + 11 = 31.

2. 2.Three students are working independently on a Mathematics problem. Their respective probabilities of solving the problem are 0.6, 0.7 and 0.8. What is the probability that at least one of them solves the problem?
   1. A
      0.336
   2. B
      0.024
   3. C
      0.664
   4. D
      0.976
      Correct
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   Answer: D

   Since the three students work independently, it is easier to find the probability that none of them solves the problem, then subtract from 1.

   The probability that a student fails to solve is 1 minus their probability of solving: 1 − 0.6 = 0.4, 1 − 0.7 = 0.3, and 1 − 0.8 = 0.2.

   Because the events are independent, the probability that all three fail is found by multiplying these: 0.4 × 0.3 × 0.2 = 0.024.

   So the probability that at least one of them solves the problem is 1 − 0.024 = 0.976.

   0.024 is only the probability that nobody solves the problem — a candidate who stops after the multiplication step and forgets the final "1 minus" gets this by mistake.

3. 3.The polynomial $2x^{3} + 3x^{2} + qx - 1$ has the same reminder when divided by $(x + 2)$ and $(x - 1)$. Find the value of the constant q.
   1. A
      -9
   2. B
      9
   3. C
      3
   4. D
      -3
      Correct
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   Answer: D

   Using the Remainder Theorem: dividing a polynomial p(x) by (x - a) leaves a remainder equal to p(b).

   So the remainder on dividing by (x + 2) is p(-2), and the remainder on dividing by (x - 1) is p(1). We are told these two remainders are equal.

   p(-2) = 2(-2)³ + 3(-2)² + q(-2) - 1 = -16 + 12 - 2q - 1 = -5 - 2q
   p(1) = 2(1)³ + 3(1)² + q(1) - 1 = 2 + 3 + q - 1 = 4 + q

   Setting the two remainders equal: -5 - 2q = 4 + q, which gives -9 = 3q, so q = -3 is correct.

4. 4.The normal to the curve $y = 2x^{2} + x - 3$ at the point $(2, 7)$ meets the x-axis at the point P. Find the coordinates of P.
   1. A
      (25, 0)
   2. B
      (58, 0)
   3. C
      (65, 0)
      Correct
   4. D
      (−61, 0)
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   Answer: C

   Step 1: Confirm the point is on the curve.
   y = 2x² + x − 3. At x = 2: y = 2(4) + 2 − 3 = 7. So (2, 7) checks out.

   Step 2: Find the gradient of the tangent at x = 2.
   dy/dx = 4x + 1. At x = 2, gradient = 4(2) + 1 = 9.

   Step 3: Find the gradient of the normal.
   The normal is perpendicular to the tangent, so its gradient is the negative reciprocal: −1/9.

   Step 4: Write the equation of the normal and find where it crosses the x-axis.
   y − 7 = −1/9(x − 2). At the x-axis, y = 0, so:
   −7 = −1/9(x − 2)
   Multiply both sides by −9: 63 = x − 2
   x = 65

   So P = (65, 0).

5. 5.Solve $x^{\frac{2}{3}} - 5x^{\frac{1}{3}} + 6 = 0$.
   1. A
      x = 2 or x = 3
   2. B
      x = 8 or x = 27
      Correct
   3. C
      x = -2 or x = -3
   4. D
      x = 4 or x = 9
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   Answer: B

   This equation looks tricky because of the fractional powers, but it is really a quadratic in disguise.

   Step 1: Substitute. Let y = x^(1/3). Then x^(2/3) = (x^(1/3))^2 = y^2. The equation becomes:

   y^2 - 5y + 6 = 0

   Step 2: Factorise. This factors as (y - 2)(y - 3) = 0, so y = 2 or y = 3.

   Step 3: Return to x. Since y = x^(1/3), we must cube each value of y to get x (not square it, and not leave it as y).

   x = 2³ = 8, or x = 3³ = 27.

   So the correct solution set is x = 8 or x = 27.

   Where the wrong options come from:

   Stopping at y = 2 or y = 3 and forgetting to cube back to x gives the tempting but incomplete answer "x = 2 or x = 3".

6. 6.Two vectors m and n are defined by $m = 3i + 4j$ and $n = 2i - j$. Find the angle between m and n.
   1. A
      53.1°
   2. B
      36.9°
   3. C
      79.7°
      Correct
   4. D
      26.6°
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   Answer: C

   Given: m = 3i + 4j and n = 2i - j.

   The angle between two vectors is found from cosθ = (m·n) / (|m||n|).

   Step 1 - Dot product:
   m·n = (3)(2) + (4)(-1) = 6 - 4 = 2.

   Step 2 - Magnitudes:
   |m| = √(3² + 4²) = √25 = 5
   |n| = √(2² + (-1)²) = √5

   Step 3 - Cosine of the angle:
   cosθ = 2 / (5 × √5) = 2 / 11.180 = 0.1789

   θ = cos⁻¹(0.1789) ≈ 79.7°, and that is the correct answer.

   You can check this a second way: the angle m makes with the x-axis is arctan(4/3) ≈ 53.1°, and the angle n makes with the x-axis is arctan(-1/2) ≈ -26.6° (n dips below the x-axis because its j-component is negative). The angle between them is 53.1° - (-26.6°) = 79.7°, which matches.

7. 7.Forces $F_{1} = (10 N, 090°), F_{2} = (20 N, 210°)$ and $F_{3} = (4 N, 330°)$ act on a body at rest on a smooth table. Find, correct to one decimal place, the magnitude of the resultant force.
   1. A
      14.0 N
      Correct
   2. B
      34.0 N
   3. C
      20.9 N
   4. D
      6.0 N
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   Answer: A

   Resolving each force into horizontal (x) and vertical (y) components, using bearings measured clockwise from North (x = F sinθ, y = F cosθ):

   F₁ = 10 N at 090°: x = 10 sin90° = 10, y = 10 cos90° = 0
   F₂ = 20 N at 210°: x = 20 sin210° = −10, y = 20 cos210° = −17.32
   F₃ = 4 N at 330°: x = 4 sin330° = −2, y = 4 cos330° = 3.46

   Summing the components:
   Σx = 10 − 10 − 2 = −2
   Σy = 0 − 17.32 + 3.46 = −13.86

   Magnitude of the resultant:
   R = √((−2)² + (−13.86)²) = √(4 + 192.00) = √196.00 = 14.0 N

8. 8.Given that the straight lines $kx - 5y + 6 = 0$ and $mx + ny - 1 = 0$ are parallel, find a relationship connecting the constants m, n and k.
   1. A
      5m − kn = 0
   2. B
      5n + km = 0
   3. C
      km = 5n
   4. D
      5m + kn = 0
      Correct
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   Answer: D

   Finding the slopes: Write each line in the form y = (slope)x + intercept.

   From kx − 5y + 6 = 0: 5y = kx + 6, so y = (k/5)x + 6/5. The slope is k/5.

   From mx + ny − 1 = 0: ny = −mx + 1, so y = (−m/n)x + 1/n. The slope is −m/n.

   Applying the parallel condition: Two lines are parallel only when their slopes are equal, so:
   k/5 = −m/n

   Cross-multiplying gives kn = −5m, which rearranges to 5m + kn = 0. This is the correct relationship.

   The option 5m − kn = 0 comes from equating k/5 directly to m/n without carrying the negative sign from −m/n — a common slip when the equation isn't rearranged into y = mx + c form first.

9. 9.A bag contains 2 red and 4 green sweets of the same size and shape. Two boys pick a sweet each from the box, one after the other, without replacement. What is the probability that at least a sweet with green wrapper is picked?
   1. A
      8/15
   2. B
      14/15
      Correct
   3. C
      2/3
   4. D
      1/15
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   Answer: B

   The bag has 2 red and 4 green sweets, 6 sweets in all. Two sweets are picked one after another without replacement.

   The easiest way to handle "at least one green" is to work with the opposite event first: no green at all, which means both sweets picked are red.

   P(first sweet red) = 2/6.
   P(second sweet also red, given the first was red) = 1/5 (only 1 red sweet and 5 sweets left).

   So P(both red) = 2/6 x 1/5 = 2/30 = 1/15.

   Since "at least one green" is the complement of "both red":

   P(at least one green) = 1 - 1/15 = 14/15.

   A candidate who stops after finding P(both red) and writes down 1/15 has answered the wrong question — that is the probability of the complement, not of the event asked for.

10. 10.Given that $p = 4i + 3j$, find the unit vector in the direction of p.
    1. A
       4i + 3j
    2. B
       (4/5)i + (3/5)j
       Correct
    3. C
       (3/5)i + (4/5)j
    4. D
       (4/7)i + (3/7)j
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    Answer: B

    To find a unit vector in the direction of a given vector, you divide the vector by its own magnitude.

    Step 1: Find the magnitude of p.
    p = 4i + 3j, so |p| = √(4² + 3²) = √(16 + 9) = √25 = 5.

    Step 2: Divide the vector by its magnitude.
    Unit vector = p / |p| = (4i + 3j) / 5 = (4/5)i + (3/5)j.

    This is the correct answer, and you can check it: (4/5)² + (3/5)² = 16/25 + 9/25 = 25/25 = 1, confirming it truly is a unit vector.

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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