Make m the subject of the relation k = $\sqrt{\frac{m - y}{m + 1}}$

B. m = $\frac{y + k^2}{1 - k^2}$

A. m = $\frac{y + k^2}{k^2 + 1}$

B. m = $\frac{y + k^2}{1 - k^2}$

C. m = $\frac{y - k^2}{k^2 + 1}$

D. m = $\frac{y - k^2}{1 - k^2}$

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WAEC (WASSCE) 2020

2020 WAEC (WASSCE) Mathematics Past Questions and Answers

The 2020 WAEC (WASSCE) Mathematics past questions with answers: 49 objective questions from the 2020 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.This question needs a diagram, table or passage that we don't have yet, so it isn't shown.
2.
If X = {x : x < 7} and Y = {y:y is a factor of 24} are subsets of $\mu$ = {1, 2, 3...10} find X $\cap$ Y.
1. A.{2, 3, 4, 6}
2. B.{1, 2, 3, 4, 6}
3. C.{2, 3, 4, 6, 8}
4. D.{1, 2, 3, 4, 6, 8}
Show answer
Answer: B. {1, 2, 3, 4, 6}
AI explanation
Step 1: Analyze the question We need the elements common to X and Y, where both sets are subsets of the universal set μ = {1, 2, 3, ..., 10}.
Step 2: List the sets
X = {x : x < 7} within μ gives X = {1, 2, 3, 4, 5, 6}
Y = {factors of 24} within μ. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. Only those in μ are kept, so Y = {1, 2, 3, 4, 6, 8}
Step 3: Find the intersection (X ∩ Y) The intersection contains the elements that appear in both sets:
1 is in both
2 is in both
3 is in both
4 is in both
5 is in X only (5 does not divide 24)
6 is in both
8 is in Y only (8 is not less than 7)
So X ∩ Y = {1, 2, 3, 4, 6}
Answer: Option B.
Tip: When a question gives a universal set, first list each set explicitly. Then check each element one at a time. Don't forget that 1 is a factor of every number, which is a common mistake in options like A and C.
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3.This question needs a diagram, table or passage that we don't have yet, so it isn't shown.
4.
An amount of N550,000.00 was realized when a principal, x was saved at 2% simple interest for 5 years. Find the value of x
1. A.N470,000.00
2. B.N480,000.00
3. C.N490,000.00
4. D.N500,000.00
Show answer
Answer: D. N500,000.00
AI explanation
Analyze the question: The amount (principal + interest) is N550,000. The rate is 2% simple interest per year for 5 years. We need the principal, x.
Concept: Simple interest is I = PRT/100, and the Amount is A = P + I.
So A = P(1 + RT/100)
Solution:
Substitute R = 2 and T = 5: RT/100 = (2 × 5)/100 = 10/100 = 0.1
Write the amount in terms of x: A = x(1 + 0.1) = 1.1x
Set it equal to 550,000: 1.1x = 550,000
Solve for x: x = 550,000 ÷ 1.1 = 500,000
Check: Interest = (500,000 × 2 × 5)/100 = N100,000. Hmm, that gives 500,000 + 100,000 = 600,000, so recheck the interest: 500,000 × 10/100 = N50,000. Then 500,000 + 50,000 = N550,000 ✓.
(The slip above was in the arithmetic: 2 × 5 = 10, and 10% of 500,000 is 50,000, not 100,000.)
Answer: x = N500,000.00, which is option D.
Tip: When the question gives the amount (not just the interest), use A = P(1 + RT/100). The total simple-interest growth here is 10% (2% × 5 years), so just divide the amount by 1.1. You can also test the options quickly: 10% of 500,000 is an easy 50,000, which gives 550,000.
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5.
A fence 2.4 m tall, is 10m away from a tree of height 16m. Calculate the angle of elevation of the top of the tree from the top of the fence.
1. A.76.11 $^o$
2. B.53.67 $^o$
3. C.52.40 $^o$
4. D.51.32 $^o$
Show answer
Answer: B. 53.67 $^o$
AI explanation
Analyze the question
The fence is 2.4 m tall and stands 10 m from the tree, which is 16 m tall. We want the angle of elevation of the tree's top, seen from the top of the fence.
Concept
Angle of elevation uses the right-angled triangle formula tan θ = opposite / adjacent. The observer's eye level is the top of the fence, so only the height of the tree above the fence counts as the opposite side.
Solution
Height of the tree above the fence top: 16 − 2.4 = 13.6 m (opposite)
Horizontal distance: 10 m (adjacent)
tan θ = 13.6 / 10 = 1.36
θ = tan⁻¹(1.36)
tan 53° ≈ 1.327 and tan 54° ≈ 1.376. Interpolating, (1.36 − 1.327)/(1.376 − 1.327) ≈ 0.67, so θ ≈ 53.67°.
Answer
θ ≈ 53.67°, which is option B.
Tip
In angle of elevation questions, always subtract the observer's height from the object's height before using tan. Many students wrongly use 16/10, which gives about 58°, a trap answer. Also, 76.11° comes from tan⁻¹(4) and is not related to this question.
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6.
If 101 $_{\text{two}}$ + 12 $_y$ = 23 $_{\text{five}}$ . Find the value of y
1. A.8
2. B.7
3. C.6
4. D.5
Show answer
Answer: C. 6
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7.
Given that $\frac{\sqrt{3} + \sqrt{5}}{\sqrt{5}}$ = x + y $\sqrt{15}$ , find the value of (x + y)
1. A.1 $\frac{3}{5}$
2. B.1 $\frac{2}{5}$
3. C.1 $\frac{1}{5}$
4. D. $\frac{1}{5}$
Show answer
Answer: C. 1 $\frac{1}{5}$
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8.
Express 1 + 2 log10 $^3$ in the form log10 $^9$
1. A.log10 $^{90}$
2. B.log10 $^{19}$
3. C.log10 $^{9}$
4. D.log10 $^{6}$
Show answer
Answer: A. log10 $^{90}$
AI explanation
The question is badly formatted. The "10" is the base of the logarithm, and the "3" and "9" are the numbers being logged. In other words, it should read:
Express 1 + 2 log₁₀ 3 in the form log₁₀ N
The options are then log₁₀ 90, log₁₀ 19, log₁₀ 9 and log₁₀ 6.
Step 1: Concept
log₁₀ 10 = 1
n log a = log aⁿ (power law)
log a + log b = log (ab) (product law)
Step 2: Solution
1 + 2 log₁₀ 3 = log₁₀ 10 + log₁₀ 3² = log₁₀ 10 + log₁₀ 9 = log₁₀ (10 × 9) = log₁₀ 90
Step 3: Answer
The expression equals log₁₀ 90, which is option A.
Tip: Whenever you see a plain number added to logs, rewrite it as a log of the base. Here 1 = log₁₀ 10. Then use the power law first, and the product law after that. Do not just add 1 + 9 = 10 or 1 + 18 = 19. That is the trap behind option B.
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9.
The mean of the numbers 15, 21, 17, 26, 18, and 29 is 21. Calculate the standard deviation
1. A.9
2. B.6
3. C.5
4. D.0
Show answer
Answer: C. 5
AI explanation
Step 1: Analyze the question We need the standard deviation of 15, 21, 17, 26, 18, 29, with mean 21 (check: 15+21+17+26+18+29 = 126, and 126 ÷ 6 = 21 ✓). Step 2: Concept Standard deviation: σ = √[Σ(x − mean)² / n] Step 3: Solution | x | x − 21 | (x − 21)² | |---|--------|-----------| | 15 | −6 | 36 | | 21 | 0 | 0 | | 17 | −4 | 16 | | 26 | 5 | 25 | | 18 | −3 | 9 | | 29 | 8 | 64 | Σ(x − mean)² = 36 + 0 + 16 + 25 + 9 + 64 = 150 Variance = 150 ÷ 6 = 25 Standard deviation = √25 = 5 Step 4: Answer The standard deviation is 5, which is option C. Tip: In WASSCE, the standard deviation of ungrouped data uses division by n (not n − 1). Build a table of deviations, square them, add, divide by n, then take the square root. The numbers are often chosen so the variance is a perfect square, such as 25 here, so if you get a perfect square, you are probably on track.
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10.
If x = 3 and y = -1, evaluate 2(x $^2$ - y $^3$ )
1. A.24
2. B.22
3. C.20
4. D.16
Show answer
Answer: C. 20
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11.
Which of the following is not a sufficient condition for two triangles to be congruent?
1. A.AAS
2. B.SSS
3. C.SAS
4. D.SSA
Show answer
Answer: D. SSA
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12.
The interquartile range of distribution is 7. If the 25th percentile is 16, find the upper quartile.
1. A.35
2. B.30
3. C.23
4. D.9
Show answer
Answer: C. 23
AI explanation
Analyze the question We're given the interquartile range (IQR) and the 25th percentile, and we need the upper quartile.
Concept
Lower quartile (Q₁) = 25th percentile
Upper quartile (Q₃) = 75th percentile
Interquartile range = Q₃ − Q₁
Solution Given: IQR = 7 and Q₁ = 16
IQR = Q₃ − Q₁
7 = Q₃ − 16
Q₃ = 7 + 16 = 23
Answer The upper quartile is 23, which is option C.
Tip The 25th percentile is the same as the lower quartile. The "upper" quartile is always the 75th percentile, so just add the IQR to Q₁. Be careful of option D (9), which comes from wrongly subtracting (16 − 7).
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13.
The first term of a geometric progression (G.P) is 3 and the 5th term is 48. Find the common ratio.
1. A.2
2. B.4
3. C.8
4. D.16
Show answer
Answer: A. 2
AI explanation
Analyze the question: We're given a G.P. with first term a = 3 and 5th term T₅ = 48. We need the common ratio r. Concept: The nth term of a G.P. is Tₙ = arⁿ⁻¹ Solution: T₅ = ar⁴ 3 × r⁴ = 48 r⁴ = 48 ÷ 3 = 16 r⁴ = 2⁴ r = 2 (taking the positive real value) Note: r = −2 also satisfies r⁴ = 16, but −2 is not among the options, so the answer is 2. Check: 3, 6, 12, 24, 48. The 5th term is 48 ✓. Answer: A. 2 Tip: For the 5th term, the power of r is 4 (one less than the term number). When you get rⁿ = a number, try to write that number as a power with the same index, as we did with 16 = 2⁴. Always verify by listing out the terms.
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14.This question needs a diagram, table or passage that we don't have yet, so it isn't shown.
15.
In the diagram, XYZ is an equilateral triangle of side 6cm and Y is the midpoint of $\overline{XY}$ . Find tan (< XZT)
1. A. $\frac{1}{\sqrt{3}}$
2. B. $\frac{\sqrt{3}}{2}$
3. C. $\sqrt{3}$
4. D. $\frac{1}{2}$
Show answer
Answer: A. $\frac{1}{\sqrt{3}}$
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16.
Evaluate and correct to two decimal places, 75.0785 - 34.624 + 9.83
1. A.30.60
2. B.50.29
3. C.50.28
4. D.30.63
Show answer
Answer: C. 50.28
AI explanation
Question analysis: We need to add and subtract decimals, then round the result to 2 decimal places.
Concept: Line up the decimal points. Do the subtraction first, then the addition (working left to right). Round at the very end, not before.
Step 1: Subtract
75.0785 − 34.6240 = 40.4545
Step 2: Add
40.4545 + 9.8300 = 50.2845
Step 3: Round to 2 decimal places
50.2845 → look at the third decimal digit, which is 4. Since 4 is less than 5, we keep the second decimal digit as it is.
50.2845 ≈ 50.28
Why not the other options?
B (50.29) comes from rounding up wrongly, perhaps by rounding too early (e.g., 50.2845 → 50.285 → 50.29). That is a double-rounding error.
A and D (about 30.6) come from subtracting 9.83 instead of adding it, or from a sign error.
Exam tip: Never round in the middle of the working. Keep all the decimal places until the last step, and only look at the digit right after the one you are rounding to. Here, only the 4 matters, not the 5 after it.
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17.
The length of an arc of a circle of radius 3.5 cm is 1 $\frac{19}{36}$ cm. Calculate, correct to the nearest degree , the angle substended by the centre of the circle. [Take $\pi = \frac{22}{7}$ ]
1. A.55 $^o$
2. B.36 $^o$
3. C.25 $^o$
4. D.22 $^o$
Show answer
Answer: C. 25 $^o$
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18.
Two buses start from the same station at 9.00am and travel in opposite directions along the same straight road. The first bus travel at a speed of 72 km/h and the second at 48 km/h. At what time will they be 240km apart?
1. A.1:00 pm
2. B.12:00 noon
3. C.11:00 am
4. D.10:00 am
Show answer
Answer: C. 11:00 am
AI explanation
Analyze the question: Two buses leave the same point at the same time and move in opposite directions. The distance between them grows at the sum of their speeds. We need the time at which that distance reaches 240 km.
Concept: When two objects move in opposite directions, the relative speed is the sum of their speeds:
Relative speed = 72 + 48 = 120 km/h
Time = Distance ÷ Speed
Solution:
Time = 240 ÷ 120 = 2 hours
Start time = 9:00 am
Time they are 240 km apart = 9:00 am + 2 hours = 11:00 am
Check:
First bus: 72 × 2 = 144 km
Second bus: 48 × 2 = 96 km
Total: 144 + 96 = 240 km ✓
Tip: For "opposite directions" questions, add the speeds. For "same direction" questions, subtract them. Then use Time = Distance ÷ Speed, and remember to add the time to the starting clock time.
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19.
Each interior angle of a regular polygon is 168 $^o$ . Find the number of sides of the polygon
1. A.30
2. B.36
3. C.24
4. D.18
Show answer
Answer: A. 30
AI explanation
Analyze the question: We are given one interior angle of a regular polygon (168°) and must find the number of sides, n. Concept: The interior and exterior angles at a vertex lie on a straight line, so they add up to 180°. The exterior angles of any polygon add up to 360°. In a regular polygon all exterior angles are equal, so: Number of sides = 360° ÷ (exterior angle) Solution: Exterior angle = 180° − 168° = 12° n = 360° ÷ 12° = 30 Check using the interior angle formula: Interior angle = (n − 2) × 180° / n (n − 2) × 180 / n = 168 180n − 360 = 168n 12n = 360 n = 30 ✓ Answer: A. 30 Tip: When you are given an interior angle of a regular polygon, go straight to the exterior angle (180° − interior) and divide 360° by it. It is faster and less error-prone than rearranging the interior angle formula.
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20.
A box contains 12 identical balls of which 5 are red, 4 blue, and the rest are green. If two balls are selected at random one after the other with replacement, what is the probability that both are red?
1. A. $\frac{25}{144}$
2. B. $\frac{5}{33}$
3. C. $\frac{5}{6}$
4. D. $\frac{103}{132}$
Show answer
Answer: A. $\frac{25}{144}$
AI explanation
Analyze the question: A box has 12 balls: 5 red, 4 blue, and the rest green (12 − 5 − 4 = 3 green). Two balls are drawn one after the other with replacement, and we want P(both red).
Concept: With replacement, the two draws are independent. The ball goes back into the box, so the probability stays the same on the second draw:
P(both red) = P(red) × P(red)
Solution:
P(red on 1st draw) = 5/12
P(red on 2nd draw) = 5/12 (the ball was replaced, so there are still 12 balls with 5 red)
P(both red) = 5/12 × 5/12 = 25/144
Check against the options: Option B, 5/33, is not the answer. It looks like a distractor, since the without-replacement answer would be 5/12 × 4/11 = 20/132 = 5/33. Because this question says "with replacement", we use 25/144.
Answer: A. 25/144
Tip: Always check the keywords "with replacement" or "without replacement."
With replacement: the denominator stays the same (12 and 12).
Without replacement: the denominator drops (12 and 11), and the number of favorable outcomes also drops.
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21.
Find the sum of the interior angle of a pentagon.
1. A.340 $^o$
2. B.350 $^o$
3. C.540 $^o$
4. D.550 $^o$
Show answer
Answer: C. 540 $^o$
AI explanation
Analyze the question: You need the total of all the interior angles of a pentagon (a 5-sided polygon).
Concept: The sum of the interior angles of an n-sided polygon is
Sum = (n − 2) × 180°
This works because any polygon can be split into (n − 2) triangles by drawing diagonals from one vertex. Each triangle has angles that add up to 180°.
Solution:
For a pentagon, n = 5:
Sum = (5 − 2) × 180° Sum = 3 × 180° Sum = 540°
Checking by triangles: from one vertex of a pentagon you can draw 2 diagonals, which split it into 3 triangles. So 3 × 180° = 540°. ✔
Matching the options:
A. 340° ✗
B. 350° ✗
C. 540° ✔
D. 550° ✗
Tip: Memorize the formula (n − 2) × 180°. Some quick values: triangle 180°, quadrilateral 360°, pentagon 540°, hexagon 720°. Each extra side adds 180°. Also, for a regular pentagon, each interior angle is 540° ÷ 5 = 108°, which often appears in follow-up questions.
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22.
Make m the subject of the relation k = $\sqrt{\frac{m - y}{m + 1}}$
1. A.m = $\frac{y + k^2}{k^2 + 1}$
2. B.m = $\frac{y + k^2}{1 - k^2}$
3. C.m = $\frac{y - k^2}{k^2 + 1}$
4. D.m = $\frac{y - k^2}{1 - k^2}$
Show answer
Answer: B. m = $\frac{y + k^2}{1 - k^2}$
AI explanation
Step 1: Analyze the question We need to rearrange k = √((m − y)/(m + 1)) so that m stands alone on one side. Step 2: Concept Remove the square root by squaring both sides. Then clear the fraction, collect all the m terms on one side, and factorise. Step 3: Solution Square both sides: k² = (m − y)/(m + 1) Multiply both sides by (m + 1): k²(m + 1) = m − y Expand: k²m + k² = m − y Collect the m terms on one side: k² + y = m − k²m Factorise the right side: k² + y = m(1 − k²) Divide by (1 − k²): m = (y + k²)/(1 − k²) Step 4: Identify the answer This matches option B. Quick check: Let m = 3 and y = 1. Then k² = (3 − 1)/(3 + 1) = 1/2. Using our formula: (1 + 1/2)/(1 − 1/2) = (3/2)/(1/2) = 3. ✓ Step 5: Tip When m appears on both sides, move every m term to one side, factorise m out, then divide. Watch the signs: moving m to the other side is what gives (1 − k²) rather than (k² + 1). Options A and C are tempting traps for those who slip on this sign.
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23.This question needs a diagram, table or passage that we don't have yet, so it isn't shown.
24.
Find the equation of the line parallel to 2y = 3(x - 2) and passes through the point (2, 3)
1. A.y = $\frac{2}{3} x - 3$
2. B.y = $\frac{2}{3} x - 2$
3. C.y = $\frac{3}{2} x$
4. D.y = $\frac{-2}{3} x$
Show answer
Answer: C. y = $\frac{3}{2} x$
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25.
Given that x is directly proportional to y and inversely proportional to Z, x = 15 when y = 10 and Z = 4, find the equation connecting x, y and z
1. A.x = $\frac{6y}{z}$
2. B.x = $\frac{12y}{z}$
3. C.x = $\frac{3y}{z}$
4. D.x = $\frac{3y}{2z}$
Show answer
Answer: A. x = $\frac{6y}{z}$
AI explanation
Analyze the question: x varies directly as y and inversely as z. This is joint variation, so we combine both relationships and find the constant.
Concept:
Direct proportion: x ∝ y
Inverse proportion: x ∝ 1/z
Combined: x ∝ y/z, so x = ky/z, where k is the constant of proportionality.
Solution:
Substitute x = 15, y = 10, z = 4 into x = ky/z:
15 = k(10)/4
15 = 10k/4
Multiply both sides by 4:
60 = 10k
k = 6
So the equation is:
x = 6y/z
Check: Put y = 10 and z = 4 back: x = 6(10)/4 = 60/4 = 15 ✓
Correct option: A
Tip: In variation questions, always follow the same steps: (1) write the proportionality, (2) introduce k, (3) substitute the given values to find k, (4) write the final equation with k. You can also check quickly by substituting the given values into each option, as we did above.
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26.
A solid cuboid has a length of 7 cm, a width of 5 cm, and a height of 4 cm. Calculate its total surface area.
1. A.280 cm $^2$
2. B.166 cm $^2$
3. C.140 cm $^2$
4. D.83 cm $^2$
Show answer
Answer: B. 166 cm $^2$
AI explanation
Analyze the question: We need the total surface area of a cuboid with length l = 7 cm, width w = 5 cm and height h = 4 cm.
Concept: A cuboid has 6 rectangular faces, in 3 pairs of equal faces. The formula is:
Total Surface Area = 2(lw + lh + wh)
Solution:
lw = 7 × 5 = 35
lh = 7 × 4 = 28
wh = 5 × 4 = 20
Sum = 35 + 28 + 20 = 83
Total Surface Area = 2 × 83 = 166 cm²
Check: The top and bottom faces give 2 × 35 = 70, the front and back give 2 × 28 = 56, and the two sides give 2 × 20 = 40. Then 70 + 56 + 40 = 166 cm². ✔
Answer: Option B, 166 cm².
Common trap: Option D (83 cm²) is only half the surface area, which is what you get if you forget to multiply by 2. Always double the sum, because each face has a matching opposite face.
Tip: Remember "2(lw + lh + wh)" as "three pairs of faces." Count the pairs, add them up, then double. Volume (l × w × h = 140 cm³) is a different quantity, and option C is a distractor based on it.
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27.
A box contains 12 identical balls of which 5 are red, 4 blue, and the rest are green. If a ball is selected at random from the box, what is the probability that it is green?
1. A. $\frac{3}{4}$
2. B. $\frac{1}{2}$
3. C. $\frac{1}{3}$
4. D. $\frac{1}{4}$
Show answer
Answer: D. $\frac{1}{4}$
AI explanation
Analyze the question: We have 12 balls in total. We know the number of red and blue balls, and the "rest" are green. We need P(green).
Concept: Probability = (number of favourable outcomes) ÷ (total number of outcomes)
Solution:
Number of green balls = Total − (Red + Blue) = 12 − (5 + 4) = 12 − 9 = 3
P(green) = Number of green balls ÷ Total number of balls = 3/12 = 1/4
Answer: The correct option is **D.
Tip: When a question says "the rest," subtract the known quantities from the total first. Always simplify your fraction at the end (3/12 → 1/4), because the options are usually given in simplest form. A quick check: the probabilities of all colours must add up to 1: 5/12 + 4/12 + 3/12 = 12/12 = 1 ✓.
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28.
Find the least value of x which satisfies the equation 4x = 7(mod 9)
1. A.7
2. B.6
3. C.5
4. D.4
Show answer
Answer: D. 4
AI explanation
Analyze the question: We need the smallest positive x such that 4x leaves remainder 7 when divided by 9. In other words, 4x ≡ 7 (mod 9).
Concept: x satisfies 4x ≡ 7 (mod 9) if 4x − 7 is a multiple of 9. That is, 4x = 7 + 9k for some integer k. We need a value of k that makes 7 + 9k divisible by 4.
Solution:
k = 0: 7 is not divisible by 4.
k = 1: 16 is divisible by 4, so x = 16 ÷ 4 = 4.
Check: 4(4) = 16, and 16 = 9 + 7, so 16 ≡ 7 (mod 9). ✔
Now check the smaller options to be sure 4 is the least:
x = 1: 4 → no
x = 2: 8 → no
x = 3: 12 ≡ 3 → no
x = 4: 16 ≡ 7 → yes
So the least value is x = 4.
Note on the options: I also checked the others. x = 5 gives 20 ≡ 2, x = 6 gives 24 ≡ 6, and x = 7 gives 28 ≡ 1. None of these work, so only D is correct.
Tip: For modular equations like ax ≡ b (mod n), just keep adding n to b (7, 16, 25, ...) until the number is divisible by a. You can also test the options directly, as that is quick in an exam.
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29.
If (x + 2) is a factor of x $^2$ +px - 10, find the value of P.
1. A.3
2. B.-3
3. C.7
4. D.-7
Show answer
Answer: B. -3
AI explanation
Analyze the question: We're told (x + 2) is a factor of x² + px − 10, and we need p. Concept (Factor Theorem): If (x − a) is a factor of f(x), then f(a) = 0. Here x + 2 = x − (−2), so a = −2 and f(−2) = 0. Solution: f(x) = x² + px − 10 f(−2) = (−2)² + p(−2) − 10 = 0 4 − 2p − 10 = 0 −2p − 6 = 0 −2p = 6 p = −3 Check: With p = −3, the expression is x² − 3x − 10. Factorising: (x − 5)(x + 2) = x² − 3x − 10 ✓. So (x + 2) is indeed a factor. Tip: For "(x + a) is a factor" questions, substitute x = −a (the sign flips) and set the expression equal to zero. Then check by factorising if time allows.
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30.
The implication x $\to$ y is equivalent to?
1. A.~ y $\to$ ~ x
2. B.y $\to$ ~ x
3. C.~ x $\to$ ~ y
4. D.y $\to$ x
Show answer
Answer: A. ~ y $\to$ ~ x
AI explanation
Analyze the question: We need the statement that is logically equivalent to the implication "x → y" (if x, then y).
Concept: Every implication is logically equivalent to its contrapositive. The contrapositive is formed by negating both statements and swapping their positions:
x → y ≡ ~y → ~x
The other related forms are not equivalent to the original:
Converse: y → x
Inverse: ~x → ~y (The converse and inverse are equivalent to each other, but not to the original.)
Truth table check:
| x | y | x → y | ~y → ~x | y → x | ~x → ~y | |---|---|-------|---------|-------|---------| | T | T | T | T | T | T | | T | F | F | F | T | T | | F | T | T | T | F | F | | F | F | T | T | T | T |
The column for x → y (T, F, T, T) matches ~y → ~x (T, F, T, T) exactly. It does not match the converse (y → x) or the inverse (~x → ~y).
Checking the options:
A. ~y → ~x is the contrapositive, so it is equivalent. ✔
B. y → ~x is not equivalent.
C. ~x → ~y is the inverse, so it is not equivalent.
D. y → x is the converse, so it is not equivalent.
Answer: A
Exam tip: Remember "flip and negate" for the contrapositive. If you flip only (converse) or negate only (inverse), the meaning changes. Only doing both keeps it equivalent.
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31.
Solve 3x - 2y = 10 and x + 3y = 7 simultaneously
1. A.x = -4 and y = 1
2. B.x = -1 and y = -4
3. C.x = 1 and y = 4
4. D.x = 4 and y = 1
Show answer
Answer: D. x = 4 and y = 1
AI explanation
Analyze the question: We have two linear equations in two unknowns. We need the pair (x, y) that satisfies both.
Concept: Simultaneous equations can be solved by elimination or substitution. Here, substitution works nicely because the second equation has x with coefficient 1.
Solution:
Equation (1): 3x − 2y = 10 Equation (2): x + 3y = 7
From (2): x = 7 − 3y
Substitute into (1): 3(7 − 3y) − 2y = 10 21 − 9y − 2y = 10 21 − 11y = 10 −11y = −11 y = 1
Then x = 7 − 3(1) = 4
Check:
Equation (1): 3(4) − 2(1) = 12 − 2 = 10 ✓
Equation (2): 4 + 3(1) = 7 ✓
So x = 4 and y = 1, which matches option D.
Tip: In multiple-choice simultaneous equations, you can save time by plugging the options into the equations. Option A (x = −4, y = 1) fails immediately in equation (2): −4 + 3 = −1, not 7. Always check the simpler equation first to eliminate wrong options quickly.
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32.
Simplify; $\frac{a}{b} - \frac{b}{a} - \frac{c}{b}$
1. A. $\frac{a - b + c}{ab}$
2. B. $\frac{ab - bc - ac}{ab}$
3. C. $\frac{a^2 - b^2 + ac}{ab}$
4. D. $\frac{a^2 - b^2 - ac}{ab}$
Show answer
Answer: D. $\frac{a^2 - b^2 - ac}{ab}$
AI explanation
Analyze the question: We need to combine three fractions into a single fraction. The denominators are b, a and b, so the common denominator (LCD) is ab.
Concept: To add or subtract fractions, convert each to the same denominator by multiplying the top and bottom by what's missing.
Solution:
Convert each term to denominator ab: • a/b = (a · a)/(b · a) = a²/ab • b/a = (b · b)/(a · b) = b²/ab • c/b = (c · a)/(b · a) = ac/ab
Combine, keeping the signs:
a/b − b/a − c/b = (a² − b² − ac)/ab
Compare with the options: • A: a − b + c over ab, which is wrong because the terms were not converted properly. • B: ab − bc − ac over ab, which is wrong. • C: a² − b² + ac over ab, which has the wrong sign on ac. • D: a² − b² − ac over ab, which matches our result.
Tip: When subtracting fractions, put the whole numerator over the LCD and watch the signs. A minus sign in front of a fraction applies to its entire numerator. Check each option by testing simple numbers, for example a=2, b=1, c=1: the original gives 2 − 0.5 − 1 = 0.5, and D gives (4 − 1 − 2)/2 = 0.5 ✓.
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33.
The diameter of a sphere is 12 cm. Calculate, correct of the nearest cm $^3$ , the volume of the sphere, [Take $\pi = \frac{22}{7}$ ]
1. A.903 cm $^3$
2. B.904 cm $^3$
3. C.905 cm $^3$
4. D.906 cm $^3$
Show answer
Answer: C. 905 cm $^3$
AI explanation
Analyze the question: We need the volume of a sphere with diameter 12 cm, using π = 22/7, rounded to the nearest cm³. Concept: Volume of a sphere = (4/3)πr³, where r is the radius (half the diameter). Solution: Radius: r = 12 ÷ 2 = 6 cm r³ = 6³ = 216 V = (4/3) × (22/7) × 216 First, (4/3) × 216 = 4 × 72 = 288 V = 288 × 22/7 = 6336/7 6336 ÷ 7 = 905.14... Check: 7 × 905 = 6335, remainder 1, so 905 + 1/7 ≈ 905.14 To the nearest cm³: 905 cm³ Tip: When π = 22/7 is given, look for factors of 7 in the numbers to simplify. Here 7 doesn't divide 216, so you must divide at the end. Do the multiplication of whole numbers first (288 × 22 = 6336), then divide by 7. Also, always halve the diameter before cubing. Using 12 instead of 6 is a common mistake that gives an answer 8 times too large.
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34.This question needs a diagram, table or passage that we don't have yet, so it isn't shown.
35.This question needs a diagram, table or passage that we don't have yet, so it isn't shown.
36.
Fati buys milk at ₦x per tin sells each at a profit of ₦y. If she sells 10 tins of milk, how much does she receives from the sales?
1. A.₦(xy + 10)
2. B.₦(x + 10y)
3. C.₦(10x + y)
4. D.₦10(x + y)
Show answer
Answer: D. ₦10(x + y)
AI explanation
Analyze the question Fati buys each tin at a cost price of ₦x. She sells each tin at a profit of ₦y. We need the total amount she receives from selling 10 tins.
Concept Selling price = Cost price + Profit
Solution
Selling price of one tin = cost price + profit = ₦(x + y)
Amount received from 10 tins = 10 × (x + y) = ₦10(x + y)
Checking the options
A. ₦(xy + 10) multiplies x and y, which makes no sense here.
B. ₦(x + 10y) is the cost of one tin plus the profit on 10 tins, so it is not the total sales.
C. ₦(10x + y) is the cost of 10 tins plus the profit on only one tin, so it is wrong.
D. ₦10(x + y) is the selling price of one tin multiplied by 10, which is correct.
Tip: "Receives from sales" means the total selling price, so always compute the selling price per item (cost + profit) first, then multiply by the number of items. Note that the total received is not the same as the total profit, which would be only ₦10y.
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37.
The pie chart shows the population of men, women, and children in a city. If the population of the city is 1,800,000, how many men are in the city?
1. A.845,000
2. B.600,000
3. C.355,000
4. D.250,000
Show answer
Answer: A. 845,000
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38.This question needs a diagram, table or passage that we don't have yet, so it isn't shown.
39.
In the diagram, PQ // SR. Find the value of x
1. A.34
2. B.46
3. C.57
4. D.68
Show answer
Answer: B. 46
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41.
Find the quadratic equation whose roots are $\frac{1}{2}$ and - $\frac{1}{3}$
1. A.3x $^2$ + x + 1 = 0
2. B.6x $^2$ + x - 1 = 0
3. C.3x $^2$ + x - 1 = 0
4. D.6x $^2$ - x - 1 = 0
Show answer
Answer: D. 6x $^2$ - x - 1 = 0
AI explanation
Analyze the question: We need a quadratic equation whose roots (solutions) are 1/2 and -1/3. Concept: If the roots are α and β, the equation is x² − (α + β)x + αβ = 0 (sum of roots and product of roots). Solution: Sum of roots: 1/2 + (−1/3) = 3/6 − 2/6 = 1/6 Product of roots: (1/2) × (−1/3) = −1/6 Substitute into the formula: x² − (1/6)x − 1/6 = 0 Multiply through by 6 to remove fractions: 6x² − x − 1 = 0 Check using factors: (2x − 1)(3x + 1) = 6x² + 2x − 3x − 1 = 6x² − x − 1 ✓ Tip: Quickly check any option by substituting a root. For x = 1/2 in option D: 6(1/4) − 1/2 − 1 = 1.5 − 0.5 − 1 = 0 ✓. Also, remember that the sign of the middle term is the opposite of the sum of the roots (when the leading coefficient is positive). Here the sum is positive (1/6), so the middle term is negative.
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43.
The dimension of a rectangular base of a right pyramid is 9 cm by 5cm. If the volume of the pyramid is 105 cm $^3$ , how high is the pyramid?
1. A.10cm
2. B.6cm
3. C.8cm
4. D.7cm
Show answer
Answer: D. 7cm
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44.
The expression $\frac{5x + 3}{6x (x + 1)}$ will be undefined when x equals
1. A.{0, 1}
2. B.{0, -1}
3. C.{-3, -11}
4. D.{-3, 0}
Show answer
Answer: B. {0, -1}
AI explanation
Analyze the question: A fraction is undefined when its denominator equals zero (division by zero is not allowed). So we find the values of x that make the denominator 0.
Concept: For a fraction a/b, b ≠ 0. Set the denominator equal to zero and solve.
Solution:
Denominator: 6x(x + 1) = 0
Using the zero product rule, either:
6x = 0, so x = 0
x + 1 = 0, so x = -1
So the expression is undefined when x = {0, -1}.
Check:
x = 0: denominator = 6(0)(1) = 0 ✔
x = -1: denominator = 6(-1)(0) = 0 ✔
Note that the numerator 5x + 3 does not matter here. It would only matter if we were asked when the expression equals zero (x = -3/5).
Tip: "Undefined" means the denominator is zero, while "equals zero" means the numerator is zero. Don't mix them up in the exam. Also watch the signs: x + 1 = 0 gives x = -1, not 1.
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45.
If tan y is positive and sin y is negative, in which quadrant would y lie?
1. A.First and third only
2. B.First and second only
3. C.Third only
4. D.Second only
Show answer
Answer: C. Third only
AI explanation
Step 1: Analyze the question We need the quadrant where tan y > 0 and sin y < 0 at the same time.
Step 2: Recall the concept (ASTC / "All Students Take Care" rule)
1st quadrant (0° to 90°): All ratios are positive
2nd quadrant (90° to 180°): Only Sine (and cosec) is positive
3rd quadrant (180° to 270°): Only Tangent (and cot) is positive
4th quadrant (270° to 360°): Only Cosine (and sec) is positive
Step 3: Apply the conditions
sin y is negative, so y is in the 3rd or 4th quadrant (sine is positive in the 1st and 2nd).
tan y is positive, so y is in the 1st or 3rd quadrant.
The quadrant that satisfies both conditions is the 3rd quadrant.
Step 4: Check with an example Take y = 225°. sin 225° = −√2/2 (negative), cos 225° = −√2/2 (negative), so tan 225° = (−)/(−) = +1 (positive). This confirms it.
Answer: C. Third only
Tip: When two conditions are given, find the quadrants for each separately and pick the one they share. Remember "All Students Take Care" going anticlockwise from the 1st quadrant: All, Sin, Tan, Cos.
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46.
A woman received a discount of 20% on a piece of cloth she purchased from a shop. If she paid $525.00, what was the original price?
1. A.$675.25
2. B.$660.25
3. C.$656.25
4. D.$616.25
Show answer
Answer: C. $656.25
AI explanation
Analyze the question: A 20% discount means she paid 80% of the original price. So $525.00 = 80% of the original price. Concept: If the discount is d%, then Amount paid = (100 − d)% × Original price Solution: Let the original price be P. 80% of P = 525 0.80 × P = 525 P = 525 ÷ 0.80 P = 656.25 Check: 20% of 656.25 = 131.25, and 656.25 − 131.25 = 525.00 ✓ Tip: When you are given the price after a discount and asked for the original, always divide by the remaining percentage (here 0.8). Don't make the common mistake of adding 20% of 525 to 525 (that gives 630, which isn't an option). The discount is taken off the original price, not the final price.
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47.
Simplify; [( $\frac{16}{9}$ ) $^{\frac{-3}{2}}$ x 16 $^{\frac{-3}{4}}$ ] $^{\frac{1}{3}}$
1. A. $\frac{3}{4}$
2. B. $\frac{9}{16}$
3. C. $\frac{3}{8}$
4. D. $\frac{1}{4}$
Show answer
Answer: C. $\frac{3}{8}$
AI explanation
Analyze the question: We need to simplify [(16/9)^(-3/2) × 16^(-3/4)]^(1/3).
Concept: Use the laws of indices:
(a/b)^(-n) = (b/a)^n
a^(m/n) = (ⁿ√a)^m
(a^m)^n = a^(mn)
Step 1: First factor (16/9)^(-3/2) = (9/16)^(3/2) √(9/16) = 3/4, so (3/4)³ = 27/64
Step 2: Second factor 16^(-3/4) = 1/16^(3/4) 16^(1/4) = 2, so 16^(3/4) = 2³ = 8 So 16^(-3/4) = 1/8
Step 3: Multiply 27/64 × 1/8 = 27/512
Step 4: Raise to the power 1/3 (27/512)^(1/3) = 3/8, since ∛27 = 3 and ∛512 = 8.
Answer: 3/8, which is option C.
Tip: Where possible, write every number as a prime power first. Here, 16 = 2⁴ and 9 = 3², so the combined exponent on 3 is 2·(3/2)·(1/3) = 1 and on 2 is a simple fraction. This avoids big numbers like 512. Check: 3^(1) × 2^(-(4·(-3/2)·... )) works out to the same result, 3/8.
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48.
A man is five times as old as his son. In four years' time, the product of their ages would be 340. If the son's age is y, express the product of their ages in terms of y.
1. A.5y $^2 - 16y - 380 = 0$
2. B.5y $^2 - 24y - 380 = 0$
3. C.5y $^2 - 16y - 330 = 0$
4. D.5y $^2 + 24y - 324 = 0$
Show answer
Answer: D. 5y $^2 + 24y - 324 = 0$
AI explanation
Analyze the question: The son's age is y, so the man's age is 5y. In four years, their ages will be (y + 4) and (5y + 4), and the product of these is 340. Concept: Write each future age as "present age + 4", multiply, set the product equal to 340, then expand and rearrange into the form ay² + by + c = 0. Solution: (y + 4)(5y + 4) = 340 Expand: 5y² + 4y + 20y + 16 = 340 5y² + 24y + 16 = 340 Subtract 340 from both sides: 5y² + 24y + 16 − 340 = 0 5y² + 24y − 324 = 0 Check against the options: This matches option D exactly. Options A, B and C each have a negative y-coefficient or a constant of −380 or −330, none of which come from this expansion. (As a quick sanity check, solving gives y = 6: (6 + 4)(30 + 4) = 10 × 34 = 340 ✓. Also 5(36) + 24(6) − 324 = 180 + 144 − 324 = 0 ✓.) Answer: D. Tip: For "in n years' time" age problems, add n to each person's current age before doing anything else. Don't add n to the product. Then expand carefully, since the middle term comes from combining the two cross-products (4y + 20y = 24y).
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49.
Solve $\frac{1}{3}$ (5 - 3x) < $\frac{2}{5}$ (3 - 7x)
1. A.x > $\frac{7}{22}$
2. B.x < $\frac{7}{22}$
3. C.x > $\frac{-7}{27}$
4. D.x < $\frac{-7}{27}$
Show answer
Answer: D. x < $\frac{-7}{27}$
AI explanation
Step 1: Analyze the question We need to solve a linear inequality and find the range of x. Fractions are involved, so we clear them first.
Step 2: Concept Multiply both sides by the LCM of the denominators to remove fractions. If you multiply or divide by a negative number, the inequality sign flips.
Step 3: Solution
(1/3)(5 − 3x) < (2/5)(3 − 7x)
The LCM of 3 and 5 is 15. Multiply both sides by 15 (positive, so the sign stays):
5(5 − 3x) < 6(3 − 7x)
25 − 15x < 18 − 42x
Add 42x to both sides:
25 + 27x < 18
Subtract 25:
27x < −7
x < −7/27
Step 4: Answer x < −7/27, which matches option D.
Tip: When there are fractions on both sides, multiply by the LCM first. This avoids messy fraction arithmetic. Always check whether you divided by a negative number, because that is the only time the sign flips. Here we divided by +27, so the sign stayed the same.
Quick check: Try x = −1 (which is less than −7/27):
Left: (1/3)(5 + 3) = 8/3 ≈ 2.67
Right: (2/5)(3 + 7) = 4
2.67 < 4 ✓
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