2021 WAEC (WASSCE) Mathematics Past Questions and Answers

Questions

1. 1.This question needs a diagram, table or passage that we don't have yet, so it isn't shown.

2. 2.

   From a point T, a man moves 12km due west and then moves 12km due south to another point Q. Calculate the bearing of T from Q.

   1. A.225°
   2. B.315°
   3. C.045°
   4. D.135°

   Show answer

   Answer: C. 045°

   AI explanation

   Analyze the question: The man starts at T, goes 12 km west, then 12 km south to reach Q. We need the bearing of T from Q, which means standing at Q and looking toward T, measured clockwise from North.

   Set up coordinates (East = x, North = y):

   • T = (0, 0)
   • After 12 km west: (−12, 0)
   • After 12 km south: Q = (−12, −12)

   Direction from Q to T: T − Q = (0 − (−12), 0 − (−12)) = (12, 12)

   So T is 12 km east and 12 km north of Q. That is the north-east direction.

   Find the angle: tan θ = 12/12 = 1, so θ = 45° from North toward East.

   Bearing of T from Q = 045°

   Check: The bearing of Q from T would be the back-bearing: 045° + 180° = 225°. That is option A, which is the trap. The question asks for the bearing of T from Q, not Q from T.

   Tip: For "bearing of X from Y," put yourself at Y (the second point named) and look at X. Always draw a small north arrow at that point. Bearings of opposite directions differ by 180°, so you can check your answer with the back-bearing.

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3. 3.

   A trapezium of parellel sides 10cm and 21cm and height 8cm is inscribed in a circle of radius 7cm. calculate the area of the region not covered by the trapezium. π =$\frac{22}{7}$

   1. A.84cm$^2$
   2. B.80cm$^2$
   3. C.30cm$^2$
   4. D.94cm$^2$

   Show answer

   Answer: C. 30cm$^2$

   AI explanation

   Let me work through this carefully. Step 1: Area of the circle Area = πr² = (22/7) × 7 × 7 = 154 cm² Step 2: Area of the trapezium Area = ½(a + b)h = ½(10 + 21)(8) = ½ × 31 × 8 = 124 cm² Step 3: Area not covered 154 − 124 = 30 cm² A check on the geometry: A trapezium inscribed in a circle of radius 7 cm would have a diameter of 14 cm, so a parallel side of 21 cm cannot fit inside the circle, because a chord can never be longer than the diameter. The question as written is therefore geometrically inconsistent. Still, WASSCE expects you to use the given numbers, and the intended calculation is the one above. Answer: C. 30 cm² Tip: For "area not covered" questions, subtract the inscribed shape's area from the circle's area. Compute each area separately, then subtract. With π = 22/7 and r = 7, the 7s cancel, so the circle's area is always easy: 22 × 7 = 154.

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4. 4.

   make u the subject in x =$\frac{2u-3}{3u + 2}$

   1. A.u = $\frac{2x + 3}{3x - 2}$
   2. B.u = $\frac{2x - 3}{3x - 2}$
   3. C.u = $\frac{2x + 3}{2 - 3x}$
   4. D.u = $\frac{2x + 3}{3x + 2}$

   Show answer

   Answer: C. u = $\frac{2x + 3}{2 - 3x}$

   AI explanation

   Analyze the question: We need to rearrange x = (2u − 3)/(3u + 2) so that u stands alone on one side. This is "changing the subject of a formula" where the variable appears in both numerator and denominator.

   Concept: Cross-multiply to clear the fraction, group all u-terms on one side, factorise out u, then divide.

   Solution:

   1. Cross-multiply: x(3u + 2) = 2u − 3

   2. Expand: 3xu + 2x = 2u − 3

   3. Collect u-terms on the left and the other terms on the right: 3xu − 2u = −3 − 2x

   4. Factorise u: u(3x − 2) = −(2x + 3)

   5. Divide: u = −(2x + 3)/(3x − 2)

   6. Multiply the numerator and denominator by −1 to get a positive-looking denominator: u = (2x + 3)/(2 − 3x)

   Check with a number: Let u = 1, then x = (2 − 3)/(3 + 2) = −1/5. Using option C: (2(−1/5) + 3)/(2 − 3(−1/5)) = (2.6)/(2.6) = 1 ✓

   Answer: C

   Tip: In "change of subject" questions where the subject appears twice, always remember: cross-multiply, group, factorise, divide. Watch your signs when moving terms. A quick substitution check with a simple number can confirm your answer in seconds.

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5. 5.

   The sum of the interior angles of a regular polygon with k sides is (3k-10) right angles. Find the size of the exterior angle?

   1. A.60°
   2. B.40°
   3. C.90°
   4. D.120°

   Show answer

   Answer: A. 60°

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6. 6.

   consider the statements: P = All students offering Literature(L) also offer History(H); Q = Students offering History(H) do not offer Geography(G). Which of the Venn diagram correctly illustrate the two statements?

   1. A.A
   2. B.B
   3. C.C
   4. D.D

   Show answer

   Answer: A. A

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7. 7.

   A fair die is tossed twice what is the probability of get a sum of at least 10.

   1. A.$\frac{5}{36}$
   2. B.$\frac{2}{3}$
   3. C.$\frac{5}{18}$
   4. D.$\frac{1}{6}$

   Show answer

   Answer: D. $\frac{1}{6}$

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8. 8.

   Find The quadratic Equation Whose Roots Are -2q And 5q.

   1. A.3x$^2$ + 3qx - 10q$^2$
   2. B.x$^2$ + 3qx + 10q$^2$
   3. C.x$^2$ - 3qx + 10q$^2$
   4. D.x$^2$ - 3qx - 10q$^2$

   Show answer

   Answer: D. x$^2$ - 3qx - 10q$^2$

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9. 9.

   If  $\frac{2}{x-3}$ - $\frac{3}{x-2}$ = $\frac{p}{(x-3)(x -2)}$, find p.

   1. A.5 - x
   2. B.- (x + 5)
   3. C.13 - x
   4. D.- (5x - 13)

   Show answer

   Answer: A. 5 - x

   AI explanation

   Step 1: Analyze the question We need to combine the two fractions on the left into a single fraction and compare the numerator with p.

   Step 2: Concept To subtract fractions, use the common denominator (x − 3)(x − 2).

   Step 3: Solution

   $\frac{2}{x-3}-\frac{3}{x-2}=\frac{2(x-2)-3(x-3)}{(x-3)(x-2)}$

   Expand the numerator:

   • 2(x − 2) = 2x − 4
   • 3(x − 3) = 3x − 9

   Subtract:

   2x − 4 − (3x − 9) = 2x − 4 − 3x + 9 = −x + 5 = 5 − x

   So

   $\frac{2}{x-3}-\frac{3}{x-2}=\frac{5-x}{(x-3)(x-2)}$

   Comparing with the right-hand side, p = 5 − x.

   Step 4: Answer The correct option is A. 5 − x.

   Tip: When subtracting fractions, put brackets around the second numerator after multiplying, such as −(3x − 9). The most common mistake is forgetting to change the sign of the +9. Note that the options include −(x + 5), which comes from that kind of sign error.

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10. 10.

    Mensah is 5 years old and joyce is thrice as old as mensah. In how many years will joyce be twice as old as Mensah?

    1. A.3 years
    2. B.10 years
    3. C.5 years
    4. D.15 years

    Show answer

    Answer: C. 5 years

    AI explanation

    Analyze the question: Mensah is 5. Joyce is three times his age, so Joyce is 15. We need to find how many years from now Joyce's age will be twice Mensah's age.

    Concept: Let the number of years be x. Both people age by the same amount, so add x to each age, then set up an equation.

    Solution:

    • Mensah's age now = 5
    • Joyce's age now = 3 × 5 = 15

    After x years:

    • Mensah = 5 + x
    • Joyce = 15 + x

    Joyce is twice Mensah's age:

    15 + x = 2(5 + x) 15 + x = 10 + 2x 15 − 10 = 2x − x x = 5

    Check: In 5 years, Mensah = 10 and Joyce = 20. Since 20 = 2 × 10, this works.

    Answer: 5 years, which is option C.

    Tip: In age problems, the difference in ages never changes. Here the difference is 10 years. Joyce is twice Mensah's age exactly when Mensah's age equals that difference (10), so Mensah needs to be 10, which is 5 years from now. This shortcut saves time in the exam.

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11. 11.

    What number should be subtracted from the sum of 2 $\frac{1}{6}$ and 2$\frac{7}{12}$ to give 3$\frac{1}{4}$?

    1. A.$\frac{1}{3}$
    2. B.1$\frac{1}{2}$
    3. C.1$\frac{1}{6}$
    4. D.$\frac{1}{2}$

    Show answer

    Answer: B. 1$\frac{1}{2}$

    AI explanation

    Step 1: Analyze the question We add 2 1/6 and 2 7/12, then find what number must be subtracted from that sum to leave 3 1/4.

    Step 2: Add the two mixed numbers Convert to twelfths:

    • 2 1/6 = 2 2/12
    • 2 7/12 = 2 7/12

    Sum = 4 9/12 = 4 3/4

    Step 3: Set up the equation Let the number be x:

    4 3/4 − x = 3 1/4

    x = 4 3/4 − 3 1/4 = 1 2/4 = 1 1/2

    Step 4: Check 4 3/4 − 1 1/2 = 4 3/4 − 1 2/4 = 3 1/4 ✓

    Answer: 1 1/2, which is option B.

    Tip: For "what should be subtracted from A to give B" questions, the answer is always A − B. Always convert mixed numbers to a common denominator first, and check by substituting your answer back.

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12. 12.

    A cone has a base radius of 8cm and height 11cm. calculate , correct to 2d.p, the curved surface area

    1. A.341.98cm$^2$
    2. B.276.57cm$^2$
    3. C.201.14cm$^2$
    4. D.477.71cm$^2$

    Show answer

    Answer: A. 341.98cm$^2$

    AI explanation

    Analyze the question: We need the curved surface area of a cone with base radius r = 8 cm and vertical height h = 11 cm. Concept: Curved surface area = πrl, where l is the slant height. Because we are given the vertical height, we must first find l using Pythagoras' theorem: l² = r² + h² Step 1: Find the slant height l² = 8² + 11² = 64 + 121 = 185 l = √185 ≈ 13.6015 cm Step 2: Find the curved surface area CSA = πrl = π × 8 × 13.6015 8 × 13.6015 = 108.812 CSA = 3.14159 × 108.812 ≈ 341.84 cm² With π = 22/7: 22/7 × 108.812 = 341.98 cm². This matches option A exactly, so the examiners used π = 22/7. With π = 3.142 the result would be about 341.9, and with 3.14159 it is 341.84. Only option A is close to these, and it matches the π = 22/7 value exactly. Answer: A (341.98 cm²) Tip: When a cone question gives the vertical height, always find the slant height l = √(r² + h²) first. Do not plug h directly into πrl. Also, if an answer is slightly different from yours, check whether the question expects π = 22/7 or 3.142. Here the options point to 22/7.

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13. 13.

    Simplify 2√7- $\frac{14}{√7} + \frac{7}{√21}$

    1. A.$\frac{√21}{21}$
    2. B.7$\frac{√21}{21}$
    3. C.$\frac{√21}{3}$
    4. D.3√21

    Show answer

    Answer: C. $\frac{√21}{3}$

    AI explanation

    Step 1: Analyze the question

    We need to simplify 2√7 − 14/√7 + 7/√21 by rationalising each denominator and combining like terms.

    Step 2: Simplify each term

    • 14/√7 = 14√7/7 = 2√7
    • 7/√21 = 7√21/21 = √21/3

    Step 3: Combine

    2√7 − 2√7 + √21/3 = √21/3

    The first two terms cancel, leaving only the third term.

    Step 4: Match to the options

    √21/3 is option C.

    Answer: C

    Tip: When you see a fraction like k/√n, multiply the top and bottom by √n to rationalise it. Always check whether terms cancel before doing any heavy computation. Here, 2√7 and 14/√7 are equal, so they cancel.

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14. 14.

    In △LMN, |LM| = 6cm, ∠LNM = x and sin x = sin x = $\frac{3}{5}$. Find the area of △LMN

    1. A.60cm$^2$
    2. B.48cm$^2$
    3. C.24cm$^2$
    4. D.30cm$^2$

    Show answer

    Answer: B. 48cm$^2$

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15. 15.

    A box contains 40 identical balls of which 10 are red and 12 are blue. if a ball is selected at random from the box what is the probability that it is neither red nor blue?

    1. A.$\frac{9}{20}$
    2. B.$\frac{3}{10}$
    3. C.$\frac{1}{4}$
    4. D.$\frac{11}{20}$

    Show answer

    Answer: A. $\frac{9}{20}$

    AI explanation

    Analyze the question: We need the probability that a ball drawn at random is neither red nor blue, so it must be some other colour.

    Concept: Probability = (number of favourable outcomes) ÷ (total number of outcomes).

    Solution:

    • Total balls = 40
    • Red = 10, Blue = 12
    • Red or blue = 10 + 12 = 22
    • Neither red nor blue = 40 − 22 = 18

    P(neither red nor blue) = 18/40 = 9/20

    Check: P(red) = 10/40 = 1/4 and P(blue) = 12/40 = 3/10. Together: 1/4 + 3/10 = 5/20 + 6/20 = 11/20. Then 1 − 11/20 = 9/20. ✔

    Answer: A. 9/20

    Note that D (11/20) is the probability of getting a red or blue ball. It is the trap answer for students who forget to subtract from 1.

    Exam tip: For "neither... nor" questions, first count the outcomes you do not want, then subtract them from the total (or use 1 − P(unwanted)). Always simplify your fraction to match the options.

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16. 16.

    500 tickets were sold for a concert tickets for adults and children were sold at $4.50 and$3.00  respectively if the total receipts for the concerts was $1987.50 how many tickets for adults were sold?

    1. A.325
    2. B.235
    3. C.175
    4. D.400

    Show answer

    Answer: A. 325

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17. 17.

    find the value of (x+y)

    1. A.215°
    2. B.70°
    3. C.135°
    4. D.145°

    Show answer

    Answer: C. 135°

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18. 18.

    If log$_{10}$ 2 = m and log$_{10}$ 3 = n, find log$_{10}$ 24 in terms of m and n.

    1. A.3m + n
    2. B.m + 3n
    3. C.4mn
    4. D.3mn

    Show answer

    Answer: A. 3m + n

    AI explanation

    Analyze the question: We need log₁₀ 24 written using log₁₀ 2 = m and log₁₀ 3 = n.

    Concept: Two log laws are needed:

    • Product law: log(ab) = log a + log b
    • Power law: log(aᵏ) = k log a

    Solution:

    Step 1: Factorise 24 into powers of 2 and 3: 24 = 8 × 3 = 2³ × 3

    Step 2: Take the log of both sides: log₁₀ 24 = log₁₀ (2³ × 3)

    Step 3: Apply the product law: log₁₀ 24 = log₁₀ 2³ + log₁₀ 3

    Step 4: Apply the power law: log₁₀ 24 = 3 log₁₀ 2 + log₁₀ 3

    Step 5: Substitute m and n: log₁₀ 24 = 3m + n

    Answer: Option A (3m + n).

    Tip: For "find log of a number in terms of..." questions, first break the number into prime factors that match the given logs (here 2 and 3). Then use the product law to split and the power law to bring powers down. Options with products like mn (C and D) are usually wrong, since logs of products become sums, not products.

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19. 19.

    if tanθ = $frac{3}{4}$, 180° < θ < 270°, find the value of cosθ.

    1. A.$\frac{4}{5}$
    2. B.$\frac{3}{5}$
    3. C.-$\frac{4}{5}$
    4. D.-$\frac{3}{5}$

    Show answer

    Answer: C. -$\frac{4}{5}$

    AI explanation

    Step 1: Analyze the question We're given tanθ = 3/4 and θ is between 180° and 270° (the third quadrant). We need cosθ.

    Step 2: Concept

    • tanθ = opposite/adjacent, so we can build a right-angled triangle.
    • In the third quadrant, both sine and cosine are negative, and tangent is positive (this matches tanθ = 3/4 > 0).
    • Remember "ASTC": All, Sine, Tangent, Cosine. The third quadrant is the "T" quadrant, where only tan is positive.

    Step 3: Solution Take opposite = 3 and adjacent = 4.

    Hypotenuse = √(3² + 4²) = √(9 + 16) = √25 = 5

    The reference triangle is the 3-4-5 triangle, so:

    • cos (reference) = adjacent/hypotenuse = 4/5
    • sin (reference) = opposite/hypotenuse = 3/5

    Because θ is in the third quadrant, cosine is negative:

    cosθ = -4/5

    Check: sinθ = -3/5, so tanθ = sinθ/cosθ = (-3/5)/(-4/5) = 3/4 ✓

    Step 4: Answer The correct option is C. -4/5.

    Tip: When a trig question gives a quadrant range, first find the magnitude using the 3-4-5 (or other Pythagorean) triangle, then apply the sign using ASTC. Here, 180° to 270° means only tan is positive, so sin and cos are both negative.

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20. 20.

    Find the 5th term of the sequence 2,5,10,17....?

    1. A.22
    2. B.24
    3. C.36
    4. D.26

    Show answer

    Answer: D. 26

    AI explanation

    Analyze the question: We need the 5th term of the sequence 2, 5, 10, 17, ...

    Find the pattern: Look at the differences between consecutive terms:

    • 5 − 2 = 3
    • 10 − 5 = 5
    • 17 − 10 = 7

    The first differences (3, 5, 7) increase by 2 each time. This is a quadratic sequence, so the next difference is 9.

    Solution (Method 1: using differences):

    5th term = 17 + 9 = 26

    Solution (Method 2: using the nth term formula):

    Compare each term with n²:

    • n = 1: 1² + 1 = 2 ✓
    • n = 2: 2² + 1 = 5 ✓
    • n = 3: 3² + 1 = 10 ✓
    • n = 4: 4² + 1 = 17 ✓

    So the nth term is Tₙ = n² + 1.

    T₅ = 5² + 1 = 25 + 1 = 26

    Tip: When the differences between terms are not constant, check the differences of the differences. If those are constant (here, 2), the sequence is quadratic. Also, quickly compare the terms with perfect squares (1, 4, 9, 16, 25...), because many WASSCE sequences are of the form n² + k.

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21. 21.

    If 16 * 2$^{(x + 1)}$ = 4$^x$ * 8$^{(1 - x)}$, find the value of x.

    1. A.-4
    2. B.4
    3. C.1
    4. D.-1

    Show answer

    Answer: D. -1

    AI explanation

    Analyze the question: We have an exponential equation with different bases (16, 4, 8, 2). The trick is to write everything as a power of 2, then equate the indices.

    Concept: If aᵐ = aⁿ, then m = n. Also, aᵐ × aⁿ = aᵐ⁺ⁿ and (aᵐ)ⁿ = aᵐⁿ.

    Solution:

    Rewrite each number as a power of 2:

    • 16 = 2⁴
    • 4 = 2², so 4ˣ = 2²ˣ
    • 8 = 2³, so 8^(1−x) = 2^(3(1−x)) = 2^(3−3x)

    Left side: 2⁴ × 2^(x+1) = 2^(4 + x + 1) = 2^(x+5)

    Right side: 2^(2x) × 2^(3−3x) = 2^(2x + 3 − 3x) = 2^(3−x)

    Equate the indices: x + 5 = 3 − x 2x = 3 − 5 2x = −2 x = −1

    Check: Left: 16 × 2^(−1+1) = 16 × 1 = 16 Right: 4^(−1) × 8^(2) = (1/4) × 64 = 16 ✓

    Answer: x = −1, which is option D.

    Tip: In any exponential equation with different bases, first convert everything to the same prime base (2, 3, 5...). Then just equate the powers. Always substitute your answer back to confirm it.

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22. 22.

    For what value of x is $\frac{4 - 2x}{x + 1}$ undefined.

    1. A.2
    2. B.-1
    3. C.1
    4. D.-2

    Show answer

    Answer: B. -1

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23. 23.

    The height of an equilateral triangle of side is 10 3√ cm. calculate its perimeter.

    1. A.20cm
    2. B.60cm
    3. C.40cm
    4. D.30cm

    Show answer

    Answer: B. 60cm

    AI explanation

    Analyze the question: The question is a little garbled. "10 3√ cm" most likely means the height is 10√3 cm. We need the perimeter of an equilateral triangle with that height.

    Concept: For an equilateral triangle of side s, drawing the height splits it into two right-angled triangles (30°-60°-90°). By Pythagoras:

    h² = s² − (s/2)² = (3/4)s²

    so h = (√3/2) × s

    Solution:

    1. Set h = 10√3: (√3/2) × s = 10√3

    2. Divide both sides by √3: s/2 = 10

    3. So s = 20 cm

    4. Perimeter = 3 × s = 3 × 20 = 60 cm

    Check: With s = 20, h = √(20² − 10²) = √300 = 10√3 ✓

    Answer: The perimeter is 60 cm, which is option B.

    Tip: Memorize h = (√3/2) × s for equilateral triangles. When the height contains √3, the √3 cancels and the side comes out as a whole number. Don't stop at the side length, because the question asks for the perimeter, and option A (20 cm) is the trap for those who forget to multiply by 3.

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24. 24.

    A trader paid import duty of 38 kobo in the naira on the cost of an engine. If a total of #22,800.00 was paid as import duty, calculate the cost of the engine.

    1. A.#60,000.00
    2. B.#120,000.00
    3. C.#24,000.00
    4. D.#18,000.00

    Show answer

    Answer: A. #60,000.00

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25. 25.

    correct 0.007985 to three significant figures.

    1. A.0.0109
    2. B.0.0800
    3. C.0.00799
    4. D.0.008

    Show answer

    Answer: C. 0.00799

    AI explanation

    Analyze the question: We need to round 0.007985 to three significant figures.

    Concept: Significant figures start counting from the first non-zero digit. Leading zeros (the zeros after the decimal point and before the first non-zero digit) are not significant.

    Solution:

    1. Write the number: 0.007985
    2. Leading zeros: 0.00 are not significant.
    3. The significant digits are 7, 9, 8, 5.
    4. For three significant figures, keep 7, 9, 8 and look at the next digit, which is 5.
    5. Since the next digit is 5, we round up the last kept digit: 8 becomes 9.

    0.007985 ≈ 0.00799

    Check the other options:

    • A. 0.0109 is far too large, so it is wrong.
    • B. 0.0800 is wrong because it changes the size of the number.
    • D. 0.008 is only 1 significant figure, so it does not meet the requirement.

    Tip: Always start counting significant figures from the first non-zero digit. When the digit after your cut-off is 5 or more, round up. Don't forget to keep the zeros in front as place holders (0.00799, not 0.799).

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26. 26.

    Solve: 2$^{\sqrt{2x + 1}}$ = 32

    1. A.13
    2. B.24
    3. C.12
    4. D.11

    Show answer

    Answer: C. 12

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27. 27.

    If 4x+2y=16 and 6x-2y=4 , find the value of (y-x).

    1. A.8
    2. B.2
    3. C.4
    4. D.6

    Show answer

    Answer: B. 2

    AI explanation

    Analyze the question: We have two linear equations in x and y. We solve them together (simultaneous equations), then compute y − x. Concept: Because the y-terms are +2y and −2y, adding the two equations eliminates y. Solution: Equation 1: 4x + 2y = 16 Equation 2: 6x − 2y = 4 Add them: (4x + 6x) + (2y − 2y) = 16 + 4 10x = 20 x = 2 Substitute x = 2 into Equation 1: 4(2) + 2y = 16 8 + 2y = 16 2y = 8 y = 4 Check in Equation 2: 6(2) − 2(4) = 12 − 8 = 4 ✓ Now find y − x: y − x = 4 − 2 = 2 Answer: Option B (2). Tip: When the coefficients of one variable are equal and opposite (like +2y and −2y), just add the equations. If they are equal and the same sign, subtract instead. Always check your values in the other equation before choosing an option.

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28. 28.

    The diagonal of a rhombus are 12cm and 5cm. calculate its perimeter

    1. A.26cm
    2. B.24cm
    3. C.17cm
    4. D.34cm

    Show answer

    Answer: A. 26cm

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29. 29.

    The pie chart represents the distribution of fruits on display in the shop if there are 60 apples on display how many oranges are there?

    1. A.80
    2. B.270
    3. C.120
    4. D.90

    Show answer

    Answer: B. 270

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31. 31.

    find the first quartile of 7,8,7,9,11,8,7,9,6 and 8.

    1. A.8.5
    2. B.7.0
    3. C.7.5
    4. D.8.0

    Show answer

    Answer: B. 7.0

    AI explanation

    Step 1: Analyze the question We need the first quartile (Q1), the value below which 25% of the ordered data lies.

    Step 2: Arrange the data in order The data: 7, 8, 7, 9, 11, 8, 7, 9, 6, 8 (n = 10)

    Ordered: 6, 7, 7, 7, 8, 8, 8, 9, 9, 11

    Step 3: Find Q1 Using the common method, Q1 is the median of the lower half.

    Since n = 10, the lower half is the first 5 values: 6, 7, 7, 7, 8 The median of these is the 3rd value = 7.

    Check with the position formula: Q1 position = ¼(n + 1) = ¼(11) = 2.75th value.

    • 2nd value = 7, 3rd value = 7
    • Q1 = 7 + 0.75(7 − 7) = 7

    Both methods give 7.0.

    Step 4: Verify with the cumulative frequency approach (often used in WASSCE) | x | 6 | 7 | 8 | 9 | 11 | |---|---|---|---|---|---| | f | 1 | 3 | 3 | 2 | 1 | | cf | 1 | 4 | 7 | 9 | 10 |

    Q1 is at the ¼(10) = 2.5th position, which falls in the group with cf reaching 4, so x = 7. This agrees.

    Answer: B (7.0).

    Tip: Always sort the data first. For Q1, take the median of the lower half. If the 2nd and 3rd values (or the relevant neighbours) are equal, there's no need to interpolate. Also count carefully: here the data has 10 values, and it is easy to miss one.

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32. 32.

    Find, correct to two decimal, the mean of 1$\frac{1}{2}$, 2$\frac{2}{3}$, 3$\frac{3}{4}$, 4$\frac{4}{5}$, and 5$\frac{5}{6}$.

    1. A.3.71
    2. B.3.70
    3. C.3.69
    4. D.3.72

    Show answer

    Answer: A. 3.71

    AI explanation

    Analyze the question: We need the mean of five mixed numbers, given to 2 decimal places.

    Concept: Mean = (sum of values) ÷ (number of values).

    Solution:

    Separate the whole numbers and the fractions.

    Whole numbers: 1 + 2 + 3 + 4 + 5 = 15

    Fractions: 1/2 + 2/3 + 3/4 + 4/5 + 5/6

    The LCM of 2, 3, 4, 5, 6 is 60:

    • 1/2 = 30/60
    • 2/3 = 40/60
    • 3/4 = 45/60
    • 4/5 = 48/60
    • 5/6 = 50/60

    Sum of fractions = (30 + 40 + 45 + 48 + 50)/60 = 213/60 = 3.55

    Total sum = 15 + 3.55 = 18.55

    Mean = 18.55 ÷ 5 = 3.71

    Answer: The mean is 3.71, which is option A.

    Tip: With mixed numbers, add the whole parts and the fractional parts separately. This keeps the working simple and reduces errors. Always divide by the number of items, which is 5 here.

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33. 33.

    if p = {-3<x<1} and Q = {-1<x<3}, where x is a real number, find P n Q.

    1. A.0
    2. B.-3, -2, -1, 0 and 1
    3. C.-2, -1 and 0
    4. D.-1, 0 and 1

    Show answer

    Answer: A. 0

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34. 34.This question needs a diagram, table or passage that we don't have yet, so it isn't shown.

35. 35.This question needs a diagram, table or passage that we don't have yet, so it isn't shown.

36. 36.

    The equation of a line is given as 3 x - 5y = 7. Find its gradient (slope)

    1. A.$\frac{5}{3}$.
    2. B.$\frac{3}{5}$.
    3. C.$\frac{-3}{5}$.
    4. D.$\frac{-5}{3}$.

    Show answer

    Answer: B. $\frac{3}{5}$.

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37. 37.

    Simplify: (11$_{two}$)$^2$

    1. A.1001$_2$
    2. B.1101$_2$
    3. C.101$_2$
    4. D.10001$_2$

    Show answer

    Answer: A. 1001$_2$

    AI explanation

    Analyze the question: We need to square the binary number 11₂ and give the answer in base two.

    Concept: Convert to base ten, square it, then convert back to binary. You can also multiply directly in binary.

    Solution:

    Step 1: Convert 11₂ to base ten. 11₂ = (1 × 2¹) + (1 × 2⁰) = 2 + 1 = 3

    Step 2: Square it. 3² = 9

    Step 3: Convert 9 to base two.

    • 9 ÷ 2 = 4 remainder 1
    • 4 ÷ 2 = 2 remainder 0
    • 2 ÷ 2 = 1 remainder 0
    • 1 ÷ 2 = 0 remainder 1

    Reading the remainders from bottom to top: 9 = 1001₂

    Check by direct binary multiplication:

        11
      × 11
      ----
        11
       110
      ----
      1001

    This gives 1001₂, which matches.

    Answer: Option A, 1001₂.

    Tip: For binary questions, convert to base ten, do the arithmetic, and convert back. Then check with the options. Also remember that 11₂ × 11₂ is like (2+1)² = 4 + 2·2 + 1 = 9, which is 8 + 1 = 1001₂.

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38. 38.

    Height(cm) 160 161 162 163 164 165 No. of players 4 6 3 7 8 9 the table shows the height of 37 players of a basketball team calculates correct to one decimal place the mean height of the players.

    1. A.163.0
    2. B.162.0
    3. C.160.0
    4. D.165.0

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    Answer: A. 163.0

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39. 39.

    A man will be (x+10)years old in 8years time. If 2years ago he was 63 years., find the value of x

    1. A.55
    2. B.63
    3. C.57
    4. D.67

    Show answer

    Answer: B. 63

    AI explanation

    Analyze the question

    We're told two things about the man's age:

    • In 8 years' time he will be (x + 10) years old.
    • 2 years ago he was 63 years old.

    We need to find x.

    Concept

    Work out his present age first. Then use "age in 8 years' time" to form an equation.

    Solution

    Step 1: Present age 2 years ago he was 63, so now he is 63 + 2 = 65 years.

    Step 2: Age in 8 years' time 65 + 8 = 73 years.

    Step 3: Form the equation x + 10 = 73 x = 73 − 10 = 63

    Check: If x = 63, then in 8 years he is 63 + 10 = 73. Going back 8 years gives 65 now, and 2 years ago he was 63. ✓

    Answer: x = 63, which is option B.

    Tip: For age problems, always find the present age first, then move forward or backward in time. Writing the timeline (2 years ago → now → 8 years later) helps avoid slips.

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40. 40.

    The distance d between two villages east more than 18 KM but not more than 23KM. which of these inequalities represents the statements?

    1. A.18 ≤ d ≤ 23
    2. B.18 < d < 23
    3. C.18 ≤ d < 23
    4. D.18 < d ≤ 23

    Show answer

    Answer: D. 18 < d ≤ 23

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41. 41.

    Factorize 6pq-3rs-3ps+6qr

    1. A.3(r -p)(2q + s)
    2. B.3(p + r)( 2q - 2q - s)
    3. C.3(2q - s)(p + r)
    4. D.3(r - p)(s - 2q)

    Show answer

    Answer: C. 3(2q - s)(p + r)

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42. 42.

    A cyclist moved at a speed of Xkm/h for 2 hours. He then increased his speed by 2 km/h for the next 3 hours. If the total distance covered is 36 km, calculate his initials speed.

    1. A.12km/h
    2. B.3km/h
    3. C.4km/h
    4. D.6km/h

    Show answer

    Answer: D. 6km/h

    AI explanation

    Analyze the question

    A cyclist rides at X km/h for 2 hours, then at (X + 2) km/h for 3 hours. The total distance is 36 km. We need X, the initial speed.

    Concept

    Distance = Speed × Time. Add the distances for the two stages and set the total equal to 36 km.

    Solution

    Stage 1: distance = 2 × X = 2X

    Stage 2: the speed is X + 2, so distance = 3(X + 2) = 3X + 6

    Total distance:

    2X + 3X + 6 = 36 5X + 6 = 36 5X = 30 X = 6

    Check:

    • Stage 1: 6 × 2 = 12 km
    • Stage 2: (6 + 2) × 3 = 24 km
    • Total: 12 + 24 = 36 km ✓

    The initial speed is 6 km/h, which is option D.

    Tip: In "speed × time" word problems, write each stage's distance as its own expression, then add them and set the sum equal to the total. Always plug your answer back in to confirm it. It takes a few seconds and guards against mistakes in the exam hall.

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43. 43.

    $\overline{XY}$ is a line segments with the coordinates X (- 8,- 12) and Y(p,q). if the midpoint of $\overline{XY}$  is (-4,-2) find the coordinates of Y.

    1. A.(-6,-2)
    2. B.(0,8)
    3. C.(4,10)
    4. D.(0,4)

    Show answer

    Answer: B. (0,8)

    AI explanation

    Analyze the question: We are given one endpoint X(-8, -12) and the midpoint M(-4, -2) of line XY. We need to find the other endpoint Y(p, q). Concept: The midpoint formula is M = ((x₁ + x₂)/2, (y₁ + y₂)/2) Solution: For the x-coordinate: (-8 + p)/2 = -4 -8 + p = -8 p = 0 For the y-coordinate: (-12 + q)/2 = -2 -12 + q = -4 q = 8 So Y = (0, 8). Check: Midpoint of (-8, -12) and (0, 8) = (-8/2, -4/2) = (-4, -2). ✓ Answer: Option B (0, 8). Tip: A quick shortcut is Y = 2M − X. Double the midpoint, then subtract the known endpoint: 2(-4) − (-8) = 0 and 2(-2) − (-12) = 8. Always verify by plugging your answer back into the midpoint formula.

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44. 44.This question needs a diagram, table or passage that we don't have yet, so it isn't shown.

45. 45.This question needs a diagram, table or passage that we don't have yet, so it isn't shown.

46. 46.

    A solid brass cube is melted and recast as a solid cone of height h and base radius r. If the height of the cube is h, find r in terms of h.

    1. A.r = h
    2. B.r = √$\frac{3h^2}{π}$
    3. C.r = πh
    4. D.r = h √$\frac{3}{h}$

    Show answer

    Answer: B. r = √$\frac{3h^2}{π}$

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47. 47.

    Given that sin x = $\frac{3}{5}$, 0 ≤ x ≤ 90, evaluate (tan x + 2cosx)

    1. A.2$\frac{11}{20}$
    2. B.2$\frac{7}{20}$
    3. C.$\frac{11}{20}$
    4. D.$\frac{1}{20}$

    Show answer

    Answer: B. 2$\frac{7}{20}$

    AI explanation

    Step 1: Analyze the question We're given sin x = 3/5 with x between 0° and 90°, so x is in the first quadrant and all ratios are positive. We need tan x + 2cos x.

    Step 2: Concept Use a right-angled triangle: sin x = opposite/hypotenuse. Find the adjacent side with Pythagoras' theorem.

    Step 3: Solution

    • Opposite = 3, hypotenuse = 5
    • Adjacent = √(5² − 3²) = √(25 − 9) = √16 = 4

    So:

    • cos x = 4/5
    • tan x = 3/4

    Now evaluate: tan x + 2cos x = 3/4 + 2(4/5) = 3/4 + 8/5

    Common denominator 20: = 15/20 + 32/20 = 47/20 = 2 7/20

    Step 4: Answer

    Tip: Whenever you see 3/5 (or 4/5), think of the 3-4-5 triangle. It appears very often in WASSCE, JAMB and NECO. Also, when converting an improper fraction to a mixed number, divide: 47 ÷ 20 = 2 remainder 7, so 2 7/20.

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48. 48.

    Which of the following is not an exterior angle of a regular polygon?

    1. A.66°
    2. B.72°
    3. C.24°
    4. D.15°

    Show answer

    Answer: A. 66°

    AI explanation

    Analyze the question: A regular polygon with n sides has equal exterior angles, each equal to 360°/n. So an angle can be an exterior angle of a regular polygon only if 360° divided by that angle gives a whole number (the number of sides).

    Concept: n = 360° ÷ exterior angle, and n must be a whole number (n ≥ 3).

    Check each option:

    • A. 66°: 360 ÷ 66 = 5.4545... → not a whole number ✗
    • B. 72°: 360 ÷ 72 = 5 → a regular pentagon ✓
    • C. 24°: 360 ÷ 24 = 15 → a regular 15-sided polygon ✓
    • D. 15°: 360 ÷ 15 = 24 → a regular 24-sided polygon ✓

    Answer: Only 66° fails to give a whole number of sides, so it is not an exterior angle of a regular polygon.

    Tip: For "which cannot be an exterior angle" questions, divide 360 by each option. If the result is not a whole number, that is your answer. The angle must be a factor of 360.

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49. 49.

    The circumference of a circular track is 9km. A cyclist rides round it a number of times and stops after covering a distance of 302km. How far is the cyclist from the starting point?

    1. A.5km
    2. B.6km
    3. C.7km
    4. D.3km

    Show answer

    Answer: A. 5km

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