Arrange the following in ascending order of magnitude $110_{two}, 31_{eight}, 42_{five}$

D. $110_{two}, 42_{five}, 31_{eight}$

A. $110_{two}, 31_{five}, 42_{eight}$

B. $42_{five}, 110_{two}, 31_{eight}$

C. $42_{five}, 31_{eight}, 110_{two}$

D. $110_{two}, 42_{five}, 31_{eight}$

make x the subject of the relation $y = \frac{ax^3 - b}{3z}$

C. x = $\sqrt[3] \frac{3yz + b}{a}$

A. x = $\sqrt[3] \frac{ax^3 - b}{3z}$

B. x = $\sqrt[3] \frac{3yz - b}{a}$

C. x = $\sqrt[3] \frac{3yz + b}{a}$

D. x = $\sqrt[3] \frac{3yzb}{a}$

For what value of x is $\frac{ x^2 + 2 }{ 10x^2 - 13x - 3}$ is undefined?.

D. $\frac{-1}{5}, \frac{-3}{2}$

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WAEC (WASSCE) 2023

2023 WAEC (WASSCE) Mathematics Past Questions and Answers

The 2023 WAEC (WASSCE) Mathematics past questions with answers: 50 objective questions from the 2023 paper in the DeeepSolve question bank, each with its answer behind "Show answer". Try each question before you check, then practise the paper as a timed WAEC-style CBT.

Practise this paper as a timed WAEC-style CBT All WAEC (WASSCE) Mathematics years

Questions

1.
The radius and height of a solid cylinder is 8 cm and 14 cm respectively. Find, correct to two d.p the total surface area. (Take $\pi = \frac{22}{7})$
1. A. $1,106.29cm^2$
2. B. $1,016.29cm^2$
3. C. $1,106.89cm^2$
4. D. $1,206.27cm^2$
Show answer
Answer: A. $1,106.29cm^2$
AI explanation
Analyze the question: We need the total surface area of a closed solid cylinder with r = 8 cm and h = 14 cm, using π = 22/7.
Concept: Total surface area of a cylinder = 2πr(h + r), which is the curved surface (2πrh) plus the two circular ends (2πr²).
Solution:
2πr(h + r) = 2 × (22/7) × 8 × (14 + 8)
= 2 × (22/7) × 8 × 22
= (2 × 22 × 8 × 22) / 7
= 7744 / 7
= 1106.285714...
Correct to 2 d.p.: 1,106.29 cm²
Check using the separate parts:
Curved surface: 2 × (22/7) × 8 × 14 = 704 cm²
Two circular ends: 2 × (22/7) × 64 = 1408/7 ≈ 201.29 cm²
Total: 704 + 201.29 = 905.29 cm²
That doesn't match, so let me recheck. 2πrh = 2 × (22/7) × 8 × 14 = 2 × 22 × 8 × 2 = 704. Correct. The ends: 2πr² = 2 × (22/7) × 64 = 2816/7 ≈ 402.29 cm² (I previously miscalculated 1408/7, which is only one end).
Total: 704 + 402.29 = 1,106.29 cm² ✔
Answer: Option A, 1,106.29 cm²
Tip: Factor first: 2πr(h + r). Here h + r = 22, which cancels neatly with the 7 pattern of π = 22/7 and saves time. Also, watch out for "total" vs "curved" surface area: total includes both circular ends.
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2.
Evaluate, correct to three decimal place $\frac{4.314 × 0.000056}{0.0067}$
1. A.0.037
2. B.0.004
3. C.0.361
4. D.0.036
Show answer
Answer: D. 0.036
AI explanation
Step 1: Analyze the question We need to compute (4.314 × 0.000056) ÷ 0.0067 and round to 3 decimal places.
Step 2: Multiply the numerator 4.314 × 0.000056 = 4.314 × 56 × 10⁻⁶ 4.314 × 56 = 241.584 So the numerator = 241.584 × 10⁻⁶ = 0.000241584
Step 3: Divide by 0.0067 0.000241584 ÷ 0.0067 = 241.584 × 10⁻⁶ ÷ (67 × 10⁻⁴) = (241.584 ÷ 67) × 10⁻²
241.584 ÷ 67:
67 × 3 = 201, remainder 40.584
67 × 0.6 = 40.2, remainder 0.384
67 × 0.005 = 0.335, remainder 0.049
67 × 0.0007 ≈ 0.0469
So 241.584 ÷ 67 ≈ 3.6057
Then 3.6057 × 10⁻² = 0.036057
Step 4: Round to 3 decimal places 0.036057 → 0.036
Answer: D. 0.
Tip: For numbers with many zeros, convert to standard form (powers of 10) first. Handle the digits and the powers of 10 separately, which cuts down on decimal-place errors. Also, estimate to check: 4.3 × 0.000056 ≈ 0.00024, and 0.00024 ÷ 0.0067 ≈ 0.036. This quick check helps you rule out options like 0.361 and 0.004.
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3.
There are 30 students in a class. 15 study woodwork and 13 study metal work. 6 study neither of the 2 subjects. How many student study woodwork but not metal work?
1. A.13
2. B.11
3. C.5
4. D.9
Show answer
Answer: B. 11
AI explanation
Analyze the question: This is a two-set Venn diagram problem. We need the number who study woodwork only.
Concept: For two sets W and M in a total group:
Students in at least one subject = Total − Neither
|W ∪ M| = |W| + |M| − |W ∩ M|
Woodwork only = |W| − |W ∩ M|
Solution:
Students studying at least one subject: 30 − 6 = 24
Find the number studying both (let it be x): 15 + 13 − x = 24 28 − x = 24 x = 4
Woodwork but not metal work: 15 − 4 = 11
Check: Woodwork only = 11, both = 4, metal only = 13 − 4 = 9, neither = 6. 11 + 4 + 9 + 6 = 30 ✓
Tip: In "neither" problems, subtract the "neither" group from the total first to get the union. Then use |W ∪ M| = |W| + |M| − |W ∩ M| to find the overlap. Always check that all regions add up to the total.
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4.This question needs a diagram, table or passage that we don't have yet, so it isn't shown.
5.
A number is chosen at random from 40 and 50 inclusive. Find the probability that the number is prime.
1. A. $\frac{8}{11}$
2. B. $\frac{3}{11}$
3. C. $\frac{4}{11}$
4. D. $\frac{5}{11}$
Show answer
Answer: B. $\frac{3}{11}$
AI explanation
Analyze the question: We choose a number from 40 to 50 inclusive. We need the probability that it is prime.
Concept: Probability = (number of favourable outcomes) ÷ (total number of outcomes). A prime number has exactly two factors: 1 and itself.
Step 1: Total outcomes The numbers are 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50. Total = 50 − 40 + 1 = 11
Step 2: Identify the primes
40: even, not prime
41: not divisible by 2, 3, 5 (and 7² = 49 > 41), so prime
42: even
43: not divisible by 2, 3, 5, so prime
44: even
45: divisible by 5
46: even
47: not divisible by 2, 3, 5, so prime
48: even
49: 7 × 7, not prime
50: even
Primes: 41, 43, 47, which gives 3 primes.
Step 3: Probability P(prime) = 3/11
Tip: To test primality of numbers below 100, you only need to check divisibility by 2, 3, 5 and 7. Also, remember to count inclusively: the number of integers from a to b is b − a + 1. Many students lose marks by forgetting the "+1".
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6.
Mr Manu is 4 times as old as his son, Adu. 7 years ago the sum of their ages was 76. How old is Adu?
1. A.22years
2. B.12years
3. C.18years
4. D.15 years
Show answer
Answer: C. 18years
AI explanation
Analyze the question: Manu's age is 4 times Adu's age now. Seven years ago, the sum of their ages was 76. We need Adu's present age.
Concept: Let Adu's present age be x, so Manu's present age is 4x. Seven years ago, each person was 7 years younger, so subtract 7 from each age.
Solution:
Seven years ago:
Adu: x − 7
Manu: 4x − 7
Sum of their ages: (x − 7) + (4x − 7) = 76 5x − 14 = 76 5x = 90 x = 18
Check: Adu is 18 and Manu is 72. Seven years ago they were 11 and 65, and 11 + 65 = 76. ✓
Tip: In age problems, when the question refers to the past, subtract the number of years from each person. So the total drops by 7 × 2 = 14. You can shortcut this by saying the present sum is 76 + 14 = 90, and since the ages are in ratio 1:4 (5 parts), Adu = 90 ÷ 5 = 18.
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7.
Solve $1 + \sqrt[3]{ x - 3} = 4$
1. A.30
2. B.6
3. C.12
4. D.66
Show answer
Answer: A. 30
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8.This question needs a diagram, table or passage that we don't have yet, so it isn't shown.
9.
An equilateral triangle has a side 2 cm. Calculate the height of the triangle.
1. A.5cm
2. B. $\sqrt5$ cm
3. C.3cm
4. D. $\sqrt{ 3}$ cm
Show answer
Answer: D. $\sqrt{ 3}$ cm
AI explanation
Analyze the question: An equilateral triangle has all sides equal (2 cm) and all angles 60°. The height (altitude) drops from one vertex to the midpoint of the opposite side. This splits the triangle into two congruent right-angled triangles.
Concept: The altitude of an equilateral triangle bisects the base. We then use Pythagoras' theorem in one of the right-angled triangles.
Solution:
Hypotenuse = the slanted side = 2 cm
Base of the right triangle = half of 2 cm = 1 cm
Height = h
Using Pythagoras:
h² + 1² = 2²
h² + 1 = 4
h² = 3
h = √3 cm
Check with the formula: Height of an equilateral triangle = (√3 / 2) × side = (√3 / 2) × 2 = √3 cm ✓
Tip: For any equilateral triangle, remember h = (√3/2) × side. Also, the 30°-60°-90° triangle has sides in the ratio 1 : √3 : 2. Here, the side of 2 is the hypotenuse, 1 is the short side, and √3 is the height. Spotting this ratio will save you time in the exam.
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10.
How many students scored at least 3 marks?
1. A.44
2. B.52
3. C.38
4. D.18
Show answer
Answer: C. 38
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11.
Factorize completely: $x^2 - (y + z)^2$
1. A.(x - y - z)(x - y - z)
2. B.(x + y + z)(x - y - z)
3. C.(x + y + z)(x + y - z)
4. D.(x + y - z)(x - y + z)
Show answer
Answer: B. (x + y + z)(x - y - z)
AI explanation
Analyze the question: The expression x² − (y + z)² is a difference of two squares, so we can factorize it using a standard identity.
Concept: Difference of two squares: a² − b² = (a + b)(a − b)
Solution:
Let a = x and b = (y + z).
x² − (y + z)² = [x + (y + z)] [x − (y + z)]
Remove the inner brackets carefully, remembering the minus sign changes both signs in the second factor:
= (x + y + z)(x − y − z)
Check by expanding (optional): (x + y + z)(x − y − z) = x² − x(y + z) + x(y + z) − (y + z)² = x² − (y + z)² ✓
Matching the options:
A. (x − y − z)(x − y − z): this is a perfect square, not a difference of squares. ✗
B. (x + y + z)(x − y − z): matches our result. ✓
C. (x + y + z)(x + y − z): wrong second factor. ✗
D. (x + y − z)(x − y + z): this expands to x² − (y − z)², which is not the same. ✗
Tip: When you see a square minus a bracket squared, treat the whole bracket as a single term. Write (a + b)(a − b) first, then open the brackets, and watch the signs in the second factor, where the minus sign applies to everything inside the bracket.
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12.This question needs a diagram, table or passage that we don't have yet, so it isn't shown.
13.
John was facing S35°E. If he turned 90° in the anticlockwise direction, find his new direction.
1. A.S55°E.
2. B.S35°W.
3. C.N55°E.
4. D.N35°W.
Show answer
Answer: C. N55°E.
AI explanation
Analyze the question: John faces S35°E, which is 35° east of due south. He turns 90° anticlockwise (as seen from above). We need his new bearing.
Concept: Convert to a three-figure bearing, measured clockwise from North.
S35°E = 180° − 35° = 145°.
Anticlockwise turning subtracts from the bearing; clockwise turning adds.
Solution:
Starting bearing = 145°.
Turn 90° anticlockwise: 145° − 90° = 055°.
A bearing of 055° is 55° from North toward East, which is N55°E.
Check with a picture: Facing south-south-east, with south at the bottom. Turning anticlockwise from facing south moves you toward the east (south → east → north → west). So he swings from S35°E toward East, and after 90° he faces northeast-ish, N55°E. This matches.
Answer: The new direction is N55°E, which is option C.
Tip: Convert compass directions to three-figure bearings first. Then anticlockwise means subtract, clockwise means add. Convert back at the end. Remember that bearing 0° to 90° is N…E, 90° to 180° is S…E, 180° to 270° is S…W, and 270° to 360° is N…W.
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14.
One-third of the sum of two numbers is 12, twice their difference is 12. Find the numbers.
1. A.22 and 14
2. B.20 and 16
3. C.21 and 15
4. D.23 and 13
Show answer
Answer: C. 21 and 15
AI explanation
Analyze the question
We have two unknown numbers, so we form two equations from the two statements.
Let the numbers be x and y (x > y).
"One-third of the sum is 12": (x + y)/3 = 12
"Twice their difference is 12": 2(x − y) = 12
Concept
This is a pair of simultaneous linear equations. Simplify each equation, then add them to eliminate one variable.
Solution
From (x + y)/3 = 12: x + y = 36 ... (i)
From 2(x − y) = 12: x − y = 6 ... (ii)
Add (i) and (ii): 2x = 42, so x = 21
Substitute into (i): 21 + y = 36, so y = 15
Check
Sum = 21 + 15 = 36, and one-third of 36 = 12 ✓
Difference = 21 − 15 = 6, and twice 6 = 12 ✓
The numbers are 21 and 15, which is option C.
Checking the other options (a quick way to confirm in the exam):
A. 22 + 14 = 36 but the difference is 8, so twice is 16 ✗
B. 20 + 16 = 36 but the difference is 4, so twice is 8 ✗
D. 23 + 13 = 36 but the difference is 10, so twice is 20 ✗
Tip: All four options have the same sum (36), so the difference is what separates them. Just solve x − y = 6 and look for the pair with a difference of 6. That takes only a few seconds.
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15.
M varies jointly as the square of n and square root of q. If M = 24 when n = 2 and q = 4, find M when n = 5, q = 9.
1. A.288
2. B.400
3. C.300
4. D.225
Show answer
Answer: D. 225
AI explanation
Analyze the question: "Varies jointly" means M is proportional to the product of the given quantities: M ∝ n² × √q. Concept: Joint variation gives M = k·n²·√q, where k is a constant. Step 1: Find k M = 24, n = 2, q = 4 24 = k × (2²) × √4 24 = k × 4 × 2 24 = 8k k = 3 Step 2: Find M when n = 5, q = 9 M = 3 × 5² × √9 M = 3 × 25 × 3 M = 225 Answer: M = 225, which is option D. Tip: For joint variation, always write the full equation first (M = k·n²·√q), find k using the first set of values, then substitute the second set. Be careful to apply the square to n and the square root to q, not the other way around.
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16.
Simplify $3\sqrt{12} + 10\sqrt{3} - \frac{6}{\sqrt{3}}$
1. A.10 $\sqrt{3}$
2. B.18 $\sqrt{3}$
3. C.14 $\sqrt{3}$
4. D.7 $\sqrt{3}$
Show answer
Answer: C. 14 $\sqrt{3}$
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17.
A bag contains 4 white marbles and 3 blue marbles. Another bag contains 5 red and 6 blue marbles. If a marble is picked at random from each bag, find the probability that they are of the same colour.
1. A. $\frac{9}{11}$
2. B. $\frac{18}{77}$
3. C. $\frac{1}{2}$
4. D. $\frac{11}{12}$
Show answer
Answer: B. $\frac{18}{77}$
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18.
Solve: $log_3 x + log_3 (x - 8) = 2$
1. A.8
2. B.6
3. C.9
4. D.7
Show answer
Answer: C. 9
AI explanation
Analyze the question: We have two logarithms with the same base 3 added together, and the sum equals 2. We need to find x.
Concept: Product rule: log_b M + log_b N = log_b (MN). Also, log_b y = k means y = b^k. Remember the domain: the logarithm needs positive arguments, so x > 0 and x - 8 > 0, which means x > 8.
Solution:
Combine the logs: log₃ [x(x - 8)] = 2
Convert to index form: x(x - 8) = 3² = 9
Expand and rearrange: x² - 8x - 9 = 0
Factorise (find two numbers with product -9 and sum -8: -9 and +1): (x - 9)(x + 1) = 0
So x = 9 or x = -1.
Check the domain: x must be greater than 8. x = -1 is rejected because log of a negative number is undefined. So x = 9.
Verification: log₃ 9 + log₃ (9 - 8) = log₃ 9 + log₃ 1 = 2 + 0 = 2 ✓
Answer: C (9)
Tip: Always check your roots in the original log equation. WASSCE often includes an extra root that makes a log argument negative, and options like 8 would make (x - 8) = 0, which is also invalid.
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19.
Find the mean deviation of assets of numbers: 14, 15, 16, 17, 18, and 19.
1. A.2.5
2. B.1.7
3. C.1.5
4. D.3.5
Show answer
Answer: C. 1.5
AI explanation
Analyze the question: We need the mean deviation of the data set 14, 15, 16, 17, 18, 19. Mean deviation is the average of the absolute distances of each value from the mean.
Concept: Mean deviation = Σ|x − x̄| / n
Step 1: Find the mean x̄ = (14 + 15 + 16 + 17 + 18 + 19) / 6 = 99 / 6 = 16.5
Step 2: Find each absolute deviation
|14 − 16.5| = 2.5
|15 − 16.5| = 1.5
|16 − 16.5| = 0.5
|17 − 16.5| = 0.5
|18 − 16.5| = 1.5
|19 − 16.5| = 2.5
Step 3: Sum the deviations 2.5 + 1.5 + 0.5 + 0.5 + 1.5 + 2.5 = 9
Step 4: Divide by n Mean deviation = 9 / 6 = 1.5
Tip: For a list of consecutive numbers, the deviations are symmetric around the mean, so you only need to calculate half of them and double the result. Always use absolute values, because without them the deviations would sum to zero.
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20.This question needs a diagram, table or passage that we don't have yet, so it isn't shown.
21.
Arrange the following in ascending order of magnitude $110_{two}, 31_{eight}, 42_{five}$
1. A. $110_{two}, 31_{five}, 42_{eight}$
2. B. $42_{five}, 110_{two}, 31_{eight}$
3. C. $42_{five}, 31_{eight}, 110_{two}$
4. D. $110_{two}, 42_{five}, 31_{eight}$
Show answer
Answer: D. $110_{two}, 42_{five}, 31_{eight}$
AI explanation
Analyze the question: The three numbers are in different bases. To compare them, convert each to base 10, then arrange from smallest to largest.
Concept: In base b, each digit is multiplied by a power of b, starting from b⁰ on the right.
Solution:
110₂ = 1×2² + 1×2¹ + 0×2⁰ = 4 + 2 + 0 = 6
31₈ = 3×8¹ + 1×8⁰ = 24 + 1 = 25
42₅ = 4×5¹ + 2×5⁰ = 20 + 2 = 22
Ascending order: 6 < 22 < 25, which gives
110₂ < 42₅ < 31₈
Checking the options:
A. Uses 31₅ and 42₈, which are not the given numbers, so it is wrong.
B. 42₅ (22), 110₂ (6), 31₈ (25) is not in order, so it is wrong.
C. 42₅ (22), 31₈ (25), 110₂ (6) is not in order, so it is wrong.
D. 110₂ (6), 42₅ (22), 31₈ (25) is in ascending order, so it is correct.
Tip: In base-comparison questions, always convert everything to base 10 first. Also check that the digits and bases in an option match the original question, as option A tries to trick you by swapping bases.
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22.
A line L passing through the point (6, -13) is parallel to the line which passes through (7, 4) and (-3, 9). Find the equation of the line L.
1. A.y = $\frac{1}{2}x - 10$
2. B.y = $\frac{-1}{2}x + 10$
3. C.y = $\frac{-1}{2}x - 10$
4. D.y = $\frac{1}{2}x +10$
Show answer
Answer: C. y = $\frac{-1}{2}x - 10$
AI explanation
Analyze the question: Line L passes through (6, -13) and is parallel to the line through (7, 4) and (-3, 9). Parallel lines have equal gradients, so we find the gradient of the given line first. Concept: Gradient m = (y₂ − y₁)/(x₂ − x₁), and the equation of a line through (x₁, y₁) is y − y₁ = m(x − x₁). Step 1: Gradient of the given line m = (9 − 4)/(−3 − 7) = 5/(−10) = −1/2 Step 2: Gradient of L Since L is parallel to this line, its gradient is also m = −1/2. Step 3: Equation of L through (6, −13) y − (−13) = −1/2 (x − 6) y + 13 = −1/2 x + 3 y = −1/2 x + 3 − 13 y = −1/2 x − 10 Check: When x = 6, y = −3 − 10 = −13 ✓ Tip: For parallel lines, copy the gradient. For perpendicular lines, the product of gradients is −1. Always substitute the given point back into your final equation to check quickly. Options A and D have the wrong gradient sign, and B has the wrong intercept, so a quick check of the gradient and one point eliminates them.
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23.
Mrs Kebeh stands at a distance of 110 m away from a building of vertical height 58 m. If Kebeh is 2 m tall, find the angle of elevation of the top of the building from her eye.
1. A. $27^0$
2. B. $28^0$
3. C. $20^0$
4. D. $26^0$
Show answer
Answer: A. $27^0$
AI explanation
Analyze the question: Mrs Kebeh's eye is at a height of about 2 m (we treat her height as the eye level). The building is 58 m tall, so the part of the building above her eye level is 58 − 2 = 56 m. She stands 110 m from the building.
Concept: The angle of elevation θ is found from a right-angled triangle using
tan θ = opposite / adjacent
Solution:
Opposite side = 58 − 2 = 56 m
Adjacent side = 110 m
tan θ = 56/110 = 0.5091
θ = tan⁻¹(0.5091)
Checking known values: tan 27° ≈ 0.5095, so θ ≈ 27°.
(If you had wrongly used 58 m, tan θ = 58/110 = 0.527, giving about 28°. That is the trap in option B, because the 2 m height must be subtracted.)
Answer: The angle of elevation is about 27°, which is option A.
Tip: Always subtract the observer's height from the object's height when the angle is measured from the eye. Also remember tan 27° ≈ 0.51, which is handy for questions like this.
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24.
A student measured the height of a pole as 5.98 m which is less than the actual height. If the percentage error is 5%, find correct to two d.p the actual height of the pole.
1. A.6.29m
2. B.7.67m
3. C.7.18m
4. D.6.65m
Show answer
Answer: A. 6.29m
AI explanation
Analyze the question: The measured height (5.98 m) is less than the actual height. The percentage error is 5%. We need the actual height. Concept: Percentage error = (error ÷ actual value) × 100% Error = actual − measured Solution: Let the actual height = H. (H − 5.98)/H × 100 = 5 H − 5.98 = 0.05H 0.95H = 5.98 H = 5.98 ÷ 0.95 = 6.2947... H ≈ 6.29 m Check: 5% of 6.29 = 0.3145, and 6.29 − 0.3145 = 5.9755 ≈ 5.98 ✓. (Note: if you wrongly took the error as 5% of the measured value, you would get 5.98 × 1.05 = 6.28, which isn't an option. The correct approach uses the actual value as the base, giving 6.29.) Answer: Option A, 6.29 m. Tip: Percentage error is always based on the actual (true) value, not the measured one. Set up the equation as (error/actual) × 100 and solve for the actual value.
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25.
m:n = $2\frac{1}{3} : 1\frac{1}{5}$ and n : q = $1\frac{1}{2} : 1\frac{1}{3}$ , find q : m.
1. A.35 : 18
2. B.16 : 35
3. C.18 : 35
4. D.35 : 16
Show answer
Answer: B. 16 : 35
AI explanation
Step 1: Convert to improper fractions
m : n = 7/3 : 6/5
n : q = 3/2 : 4/3
Step 2: Simplify each ratio to whole numbers
m : n = 7/3 : 6/5. Multiply both sides by 15: 35 : 18.
n : q = 3/2 : 4/3. Multiply both sides by 6: 9 : 8.
Step 3: Make n the same in both ratios
n is 18 in the first ratio and 9 in the second. Multiply the second ratio by 2: n : q = 18 : 16.
So m : n : q = 35 : 18 : 16.
Step 4: Find q : m
q : m = 16 : 35
Check: m/n = 35/18 and n/q = 9/8. Then m/q = (35/18)(9/8) = 315/144 = 35/16, so q/m = 16/35. ✓
Answer: B (16 : 35)
Tip: When chaining ratios, clear the fractions first. Then scale the shared term (here n) so it matches in both ratios, and join them into one ratio such as m : n : q. Read off the pair you need, and watch the order, since the question asks for q : m, not m : q.
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26.
Write the name of a triangle with the vertices (1, -3), (6, 2) and (0,4)?
1. A.Scalene triangle
2. B.Isosceles triangle
3. C.Right -angle triangle
4. D.Equilateral triangle
Show answer
Answer: B. Isosceles triangle
AI explanation
Step 1: Analyze the question We need to classify the triangle by finding the lengths of its sides using the distance formula: d = √[(x₂ − x₁)² + (y₂ − y₁)²]
Step 2: Label the points A(1, −3), B(6, 2), C(0, 4)
Step 3: Compute the side lengths
AB² = (6 − 1)² + (2 − (−3))² = 25 + 25 = 50, so AB = √50 ≈ 7.07
BC² = (0 − 6)² + (4 − 2)² = 36 + 4 = 40, so BC = √40 ≈ 6.32
AC² = (0 − 1)² + (4 − (−3))² = 1 + 49 = 50, so AC = √50 ≈ 7.07
Step 4: Compare AB = AC = √50, while BC = √40 is different. So exactly two sides are equal, which makes the triangle isosceles.
Check for a right angle: Does the sum of the squares of the two smaller sides equal the square of the largest? 40 + 50 = 90 ≠ 50, so no. In fact, no combination works (50 + 50 = 100 ≠ 40). So it is not right-angled.
It is also not equilateral, since the sides are not all equal.
Note on options: Strictly, a triangle with two equal sides is not scalene (scalene means all sides different), so A is out.
Exam tip: For "name the triangle" questions with coordinates, compute the squared distances (skip the square roots). Compare them: all equal gives equilateral, two equal gives isosceles, and if a² + b² = c² it is right-angled.
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27.This question needs a diagram, table or passage that we don't have yet, so it isn't shown.
28.
The bar chart represents the distribution of marks scored by students in an economics examination. Use the bar chart to answer questions 30 to 32 If the failed mark was 4, what is the probability that a student selected at random passed?
1. A.0.36
2. B.0.74
3. C.0.52
4. D.0.64
Show answer
Answer: A. 0.36
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29.
Consider the statements: p: Siah is from Foya. q: Foya is in Lofa. Write in symbolic for the statement: "If Siah is from Foya, then Foya is in Lofa"
1. A.~q ⇔ p
2. B.q ⇒ p
3. C.p ⇒ q
4. D.p ⇔ q
Show answer
Answer: C. p ⇒ q
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30.
The interior angle of a regular polygon is 6 times its exterior angle find the number of sides of the polygon.
1. A.12
2. B.15
3. C.10
4. D.14
Show answer
Answer: D. 14
AI explanation
Analyze the question: We have a regular polygon whose interior angle is 6 times its exterior angle. We need the number of sides, n.
Concept:
Interior angle + exterior angle = 180° (they lie on a straight line)
Each exterior angle of a regular polygon = 360°/n
Solution:
Let the exterior angle be x. Then the interior angle is 6x.
x + 6x = 180° 7x = 180° x = 180°/7 ≈ 25.71°
Number of sides:
n = 360° / x = 360 × 7 / 180 = 14
Check: Exterior angle = 360/14 = 180/7°. Interior angle = 180 − 180/7 = 1080/7°, and 6 × 180/7 = 1080/7°. ✓
Tip: For any "interior is k times exterior" question, the exterior angle is 180°/(k+1), so n = 360 ÷ (180/(k+1)) = 2(k+1). Here k = 6, so n = 2(7) = 14. This shortcut saves time in the exam.
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31.
The diagonals of a rhombus are 16 cm and 12 cm find the length of the side.
1. A.20cm
2. B.8cm
3. C.14cm
4. D.10cm
Show answer
Answer: D. 10cm
AI explanation
Analyze the question: We are given a rhombus with diagonals 16 cm and 12 cm, and we need the length of its side.
Concept: The diagonals of a rhombus bisect each other at right angles (90°). This splits the rhombus into four congruent right-angled triangles. Each side of the rhombus is the hypotenuse of one of these triangles, so we can use Pythagoras' theorem.
Solution:
Half of each diagonal: • 16 ÷ 2 = 8 cm • 12 ÷ 2 = 6 cm
Apply Pythagoras' theorem, with side s as the hypotenuse:
s² = 8² + 6² s² = 64 + 36 s² = 100 s = √100 = 10 cm
Answer: The side of the rhombus is 10 cm, which is option D.
Tip: When you see a rhombus with its diagonals given, always halve them first and think "right-angled triangle." Here 6, 8, 10 is a multiple of the 3-4-5 Pythagorean triple (×2). Spotting this triple saves time in the exam.
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32.
An empty cylindrical tank is 140 cm in diameter. If 200 litres of water was poured into the tank. Calculate, correct to the nearest centimeter, the height of the water in the tank. ( $Take \pi = \frac{22}{7})$
1. A.91cm
2. B.7cm
3. C.13cm
4. D.57cm
Show answer
Answer: C. 13cm
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33.
If 2x - 3y = -11 and 3x + 2y = 3, evaluate $(y - x)^2$
1. A.16
2. B.25
3. C.9
4. D.4
Show answer
Answer: A. 16
AI explanation
Step 1: Analyze the question We have two simultaneous equations and need to find (y − x)², so we first solve for x and y. Step 2: Solve the equations (elimination method) Equation 1: 2x − 3y = −11 Equation 2: 3x + 2y = 3 Multiply Equation 1 by 2: 4x − 6y = −22 Multiply Equation 2 by 3: 9x + 6y = 9 Add them: 13x = −13, so x = −1 Substitute into Equation 2: 3(−1) + 2y = 3 −3 + 2y = 3 2y = 6, so y = 3 Check in Equation 1: 2(−1) − 3(3) = −2 − 9 = −11 ✓ Step 3: Evaluate (y − x)² y − x = 3 − (−1) = 4 (y − x)² = 4² = 16 Step 4: Identify the answer The answer is 16, which is option A. Tip: Watch the signs when subtracting a negative: 3 − (−1) = 3 + 1 = 4. Many students wrongly get 2 and pick D. Always check your x and y values in both original equations before finishing.
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34.
The length of the diagonal of a square is 12 cm. Calculate the area of the square.
1. A. $36 cm^2$
2. B. $48 cm^2$
3. C. $72 cm^2$
4. D. $18 cm^2$
Show answer
Answer: C. $72 cm^2$
AI explanation
Analyze the question: We're given the diagonal of a square (12 cm) and asked to find its area.
Concept: For a square with side s, the diagonal d forms the hypotenuse of a right-angled triangle with two sides of length s. By Pythagoras:
d² = s² + s² = 2s²
Since the area is A = s², we get:
A = d²/2
Solution:
Method 1 (using Pythagoras):
12² = 2s²
144 = 2s²
s² = 72
Area = s² = 72 cm²
Method 2 (using the formula A = d²/2):
A = 12²/2 = 144/2 = 72 cm²
(Equivalently, the side is s = 12/√2 = 6√2 cm, so the area is (6√2)² = 36 × 2 = 72 cm².)
Answer: The area is 72 cm², which is option C.
Tip: For any square, remember Area = (diagonal)² ÷ 2. This is also the rhombus formula (½ × d₁ × d₂) with both diagonals equal: ½ × 12 × 12 = 72. Option A (36) is the trap for students who forget to divide by 2 after squaring and instead compute something like 12 × 12 ÷ 4.
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35.
make x the subject of the relation $y = \frac{ax^3 - b}{3z}$
1. A.x = $\sqrt[3] \frac{ax^3 - b}{3z}$
2. B.x = $\sqrt[3] \frac{3yz - b}{a}$
3. C.x = $\sqrt[3] \frac{3yz + b}{a}$
4. D.x = $\sqrt[3] \frac{3yzb}{a}$
Show answer
Answer: C. x = $\sqrt[3] \frac{3yz + b}{a}$
AI explanation
Analyze the question: We need to rearrange y = (ax³ − b)/(3z) so that x stands alone on one side.
Concept: To change the subject, undo the operations around x in reverse order: first the division, then the subtraction, then the multiplication by a, and finally the cube.
Step-by-step solution:
Start with: y = (ax³ − b)/(3z)
Multiply both sides by 3z: 3yz = ax³ − b
Add b to both sides: 3yz + b = ax³
Divide both sides by a: x³ = (3yz + b)/a
Take the cube root of both sides: x = ∛[(3yz + b)/a]
Checking the options:
A: Still has x on the right-hand side, so it is not a valid answer.
B: Has −b, but we added b, so it is wrong.
C: x = ∛[(3yz + b)/a], which matches our result.
D: Multiplies by b instead of adding, so it is wrong.
Tip: Whenever you move a term across the equals sign, reverse its operation: subtraction becomes addition, and multiplication becomes division. Always do the addition or subtraction before dividing by the coefficient. Also, a quick check is that the answer should never contain the subject (x) on both sides.
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36.
The angle of elevation of the top of a building from a point Z on the ground is 50°. If the height of the building is 124 m, find the distance from Z to the foot of the building.
1. A.147.78m
2. B.104.05m
3. C.161.87m
4. D.192.91m
Show answer
Answer: B. 104.05m
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37.
For what value of x is $\frac{ x^2 + 2 }{ 10x^2 - 13x - 3}$ is undefined?.
1. A. $\frac{1}{5}, \frac{3}{2}$
2. B. $\frac{-1}{5}, \frac{3}{2}$
3. C. $\frac{1}{5}, \frac{-3}{2}$
4. D. $\frac{-1}{5}, \frac{-3}{2}$
Show answer
Answer: B. $\frac{-1}{5}, \frac{3}{2}$
AI explanation
Analyze the question: A fraction is undefined when its denominator equals zero. So we solve 10x² − 13x − 3 = 0.
Concept: Factorise the quadratic by finding two numbers whose product is (10)(−3) = −30 and whose sum is −13. Those numbers are −15 and +2.
Solution:
10x² − 13x − 3 = 0
Split the middle term: 10x² − 15x + 2x − 3 = 0
Group: 5x(2x − 3) + 1(2x − 3) = 0
(2x − 3)(5x + 1) = 0
So:
2x − 3 = 0 → x = 3/2
5x + 1 = 0 → x = −1/5
Check: For x = 3/2: 10(9/4) − 13(3/2) − 3 = 22.5 − 19.5 − 3 = 0 ✓. For x = −1/5: 10(1/25) + 13/5 − 3 = 0.4 + 2.6 − 3 = 0 ✓.
The fraction is undefined at x = −1/5 and x = 3/2, which is option B.
Tip: "Undefined" for a fraction always means the denominator is zero, so ignore the numerator (x² + 2 doesn't matter here). Always check your roots by substituting them back into the denominator.
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38.
Solve $2^{5x} \div 2^x = \sqrt[5]{2^{10}}$
1. A. $\frac{3}{2}$
2. B. $\frac{1}{2}$
3. C. $\frac{1}{3}$
4. D. $\frac{5}{3}$
Show answer
Answer: B. $\frac{1}{2}$
AI explanation
Step 1: Analyze the question We have an exponential equation with the same base (2) on both sides. We simplify each side using the laws of indices, then equate the powers.
Step 2: Concept (laws of indices)
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
ⁿ√(aᵐ) = a^(m/n)
Step 3: Solution
Left side: 2^(5x) ÷ 2^x = 2^(5x − x) = 2^(4x)
Right side: ⁵√(2¹⁰) = 2^(10/5) = 2²
Equate the powers: 4x = 2 x = 2/4 = 1/2
Check: 2^(5/2) ÷ 2^(1/2) = 2^(4/2) = 2² = 4, and ⁵√(2¹⁰) = 2² = 4. ✓
Step 4: Answer x = 1/2, which is option B.
Tip: When the bases match, just "drop the base" and equate the exponents. Remember that dividing means subtracting powers, and a root means dividing the power by the root's index.
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39.
Express $413_7$ to base 5
1. A. $2311_5$
2. B. $1131_5$
3. C. $1311_5$
4. D. $2132_5$
Show answer
Answer: C. $1311_5$
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40.
Find the roots of the equations: $3m^2 - 2m - 65 = 0$
1. A. $( -5, \frac{-13}{3})$
2. B. $( 5, \frac{-13}{3})$
3. C. $( 5, \frac{13}{3})$
4. D. $( -5, \frac{13}{3})$
Show answer
Answer: B. $( 5, \frac{-13}{3})$
AI explanation
Analyze the question: We need the values of m that satisfy the quadratic 3m² − 2m − 65 = 0.
Concept: Factorise by splitting the middle term. Find two numbers whose product is a × c = 3 × (−65) = −195 and whose sum is −2.
Solution:
Factors of 195 that differ by 2: 13 and 15. So we use −15 and +13 (product −195, sum −2).
3m² − 15m + 13m − 65 = 0
3m(m − 5) + 13(m − 5) = 0
(m − 5)(3m + 13) = 0
So:
m − 5 = 0 → m = 5
3m + 13 = 0 → m = −13/3
Check:
m = 5: 3(25) − 10 − 65 = 75 − 75 = 0 ✓
m = −13/3: 3(169/9) + 26/3 − 65 = 169/3 + 26/3 − 195/3 = 0 ✓
The roots are m = 5 and m = −13/3, which matches option B.
Tip: For quick checking, the sum of the roots should equal −b/a = 2/3. Here 5 + (−13/3) = 2/3 ✓. The product should equal c/a = −65/3. Here 5 × (−13/3) = −65/3 ✓. You can use this to eliminate wrong options fast in the exam.
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41.
The price of a shoe was decreased by 22%. If the new price is $27.3. what is the original price.
1. A.$62.30
2. B.$42.30
3. C.$72.00
4. D.$35.00
Show answer
Answer: D. $35.00
AI explanation
Analyze the question: The price dropped by 22%, so the new price is 78% of the original price. We need to find the original price. Concept: If the original price is x, then after a percentage decrease: New price = (100 − 22)% of x = 78% of x Solution: 0.78 × x = 27.3 x = 27.3 ÷ 0.78 x = 2730 ÷ 78 = 35 Check: 22% of 35 = 7.70, and 35 − 7.70 = 27.30 ✓ The original price is **$35. **Tip:** For "decreased by r%" questions where you're given the new price, always divide the new price by (100 − r)/100 to get the original. Don't take 22% of the new price and add it back, because that is a common trap. Also, you can test the options quickly: only $35 gives 78% = 27.30.
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42.
A notebook of length 15 cm was measured to be 16.8 cm, calculate, correct to two d.p, the percentage error in the measurement.
1. A.12.00%
2. B.11.71%
3. C.10.71%
4. D.11.21%
Show answer
Answer: A. 12.00%
AI explanation
Analyze the question: The true length is 15 cm and the measured length is 16.8 cm. We need the percentage error, correct to 2 d.p. Concept: Error = |Measured value − True value| Percentage error = (Error ÷ True value) × 100% The usual WASSCE convention is to divide by the true (actual) value. Solution: Error = 16.8 − 15 = 1.8 cm Percentage error = (1.8 ÷ 15) × 100% = 0.12 × 100% = 12.00% Checking the other options: If you divided by the measured value instead, you would get (1.8 ÷ 16.8) × 100 = 10.71%. That is option C, a trap for students who divide by the wrong value. Options B and D do not come from any standard method. 12.00%. **Tip: Percentage error always uses the true/actual value as the denominator, unless the question says otherwise. Ask yourself: "What is the real value?" and divide by that.
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43.
Find the gradient of the line passing through the points $(\frac{1}{2}, \frac{- 1}{3}) and ( 3 , \frac{2}{3})$
1. A. $\frac{2}{5}$
2. B. $\frac{5}{2}$
3. C. $\frac{2}{7}$
4. D. $\frac{7}{2}$
Show answer
Answer: A. $\frac{2}{5}$
AI explanation
Analyze the question: We need the gradient (slope) of the line through two given points. Concept: Gradient m = (y₂ − y₁) / (x₂ − x₁) Solution: Points: (x₁, y₁) = (1/2, −1/3) and (x₂, y₂) = (3, 2/3) Change in y: 2/3 − (−1/3) = 2/3 + 1/3 = 1 Change in x: 3 − 1/2 = 6/2 − 1/2 = 5/2 Gradient: m = 1 ÷ (5/2) = 1 × 2/5 = 2/5 My working agrees. Answer: A. 2/5 Tip: With fractions, find the rise and the run separately first, then divide. Dividing by a fraction means multiplying by its reciprocal. Also, keep the points in the same order in both the numerator and denominator. Swapping the order in only one of them gives the wrong sign.
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44.
The radius of a sphere is 3 cm. Find, in terms of π, its volume.
1. A. $30\pi cm^3$
2. B. $108\pi cm^3$
3. C. $27\pi cm^3$
4. D. $36\pi cm^3$
Show answer
Answer: D. $36\pi cm^3$
AI explanation
Analyze the question: We need the volume of a sphere with radius r = 3 cm, written in terms of π.
Concept: The volume of a sphere is
V = (4/3)πr³
Solution:
Substitute r = 3: V = (4/3) × π × 3³
Compute the cube: 3³ = 27
Multiply: V = (4/3) × 27 × π V = 4 × 9 × π V = 36π cm³
Answer: The volume is 36π cm³, which is option D.
Check on the other options:
B (108π) is what you get if you forget to divide by 3: 4 × 27 = 108.
C (27π) is just r³ × π, which leaves out the 4/3.
A (30π) doesn't come from the formula.
Exam tip: Do the division first. Since r³ = 27 is a multiple of 3, the 3 in the denominator cancels right away (27 ÷ 3 = 9), so you only need 4 × 9. Remember the formula as "four-thirds pi r cubed", and don't mix it up with the surface area, 4πr².
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45.
If $log_a 3$ = m and $log_a 5$ = p, find $log_a 75$
1. A. $m^2 + p$
2. B.2m + p
3. C.m + 2p
4. D. $m + p^2$
Show answer
Answer: C. m + 2p
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46.
The angle of a sector of a circle of radius 3.4 cm is 115°. Find the area of the sector. $(Take \pi = \frac{22}{7})$
1. A. $11.6cm^2$
2. B. $12.7cm^2$
3. C. $10.2cm^2$
4. D. $9.4cm^2$
Show answer
Answer: A. $11.6cm^2$
AI explanation
Analyze the question: We need the area of a sector with radius r = 3.4 cm and angle θ = 115°, using π = 22/7.
Concept: Area of a sector = (θ/360) × πr²
Solution:
Compute r²: 3.4² = 11.56
Compute πr²: (22/7) × 11.56 = 254.32/7 ≈ 36.331
Multiply by the fraction of the circle: 115/360 = 23/72 ≈ 0.31944
Area = 0.31944 × 36.331 ≈ 11.606 cm²
Rounded to 1 decimal place, the area is about 11.6 cm².
Checking the other options: 12.7, 10.2 and 9.4 do not match 11.606, so they are ruled out.
Tip: In sector questions, simplify the angle fraction first (115/360 = 23/72) and estimate before calculating. Here the full circle is about 36.3 cm², and 115° is a little less than one third of 360°, so the answer should be a little under 12. That quickly points to 11.6 and away from the other options.
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47.
Find the value of a in the equation: cos (a + 14)° = sin (4a + 6)°
1. A.15
2. B.17
3. C.14
4. D.21
Show answer
Answer: C. 14
AI explanation
Analyze the question: We have cos of one angle equal to sin of another. We need the value of a.
Concept: The complementary angle identity says cos θ = sin(90° − θ). So if cos(a + 14)° = sin(4a + 6)°, then the two angles must add up to 90° (for acute angles).
Solution:
cos(a + 14)° = sin(90 − (a + 14))° = sin(76 − a)°
Now set the angles equal:
76 − a = 4a + 6 76 − 6 = 4a + a 70 = 5a a = 14
Equivalently: (a + 14) + (4a + 6) = 90, so 5a + 20 = 90, so a = 14.
Check:
cos(14 + 14)° = cos 28°
sin(4(14) + 6)° = sin 62°
sin 62° = cos(90 − 62)° = cos 28° ✓
Checking the other options:
a = 15: cos 29° vs sin 66° = cos 24°, not equal.
a = 17: cos 31° vs sin 74° = cos 16°, not equal.
a = 21: cos 35° vs sin 90° = 1, not equal.
Answer: a = 14, which is option C.
Tip: Whenever you see sin and cos equated, think "complementary angles": add the two angles and set the sum equal to 90°. This turns the problem into a simple linear equation.
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48.
The truth set of $8 + 2x - x^2$ = 0 is {p, q}. Evaluate p + q.
1. A.4
2. B.2
3. C.-6
4. D.-2
Show answer
Answer: B. 2
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49.
What percentage of the students scored at most 5 marks?
1. A.58.5%
2. B.63.2%
3. C.38.3%
4. D.41.5%
Show answer
Answer: C. 38.3%
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50.
Find the quadratic equation whose roots are $\frac{2}{3} and \frac{- 3}{4}$
1. A. $12y^2 - y - 6 = 0$
2. B. $12y^2 - y + 6 = 0$
3. C. $12y^2 + y - 6 = 0$
4. D. $y^2 + y - 6 = 0$
Show answer
Answer: C. $12y^2 + y - 6 = 0$
AI explanation
Step 1: Analyze the question We need a quadratic equation whose roots are 2/3 and -3/4. Step 2: The concept If the roots are α and β, the equation is: y² - (α + β)y + αβ = 0 Step 3: Solution Sum of roots: α + β = 2/3 + (-3/4) = 8/12 - 9/12 = -1/12 Product of roots: αβ = (2/3)(-3/4) = -6/12 = -1/2 Substitute into the formula: y² - (-1/12)y + (-1/2) = 0 y² + (1/12)y - 1/2 = 0 Multiply through by 12 to clear the fractions: 12y² + y - 6 = 0 Check using factors: (3y - 2)(4y + 3) = 12y² + 9y - 8y - 6 = 12y² + y - 6 ✓ Step 4: Answer The equation is 12y² + y - 6 = 0, which is option C. Tip: Always use "y² − (sum)y + product = 0". Watch the signs. A negative sum becomes a positive middle term. Also, you can quickly eliminate options: option D has no 12 coefficient, so it can't give fractional roots like these, and the product of roots must be negative, which rules out B.
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