Practise five Mathematics tasks from a source-labelled 2023 collection
626 reviewed questions with answers and explanations.
Practice selection: full-paper coverage has not been confirmed.
Myschool 2025 collection · Original sitting and question number unconfirmed.
In triangle PQR, S lies on QR with QS=11 cm and SR=9 cm. If PS=13 cm and PS is perpendicular to QR, what is the area of triangle PQR?
C: 130 cm²
The base QR is 11+9=20 cm, and the perpendicular height from P is 13 cm. Area = ½×20×13 = 130 cm².
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
Which set is equivalent to {1,2,3,4,5}, meaning it has exactly five distinct elements?
D: {4,3,1,5,2}
Equivalent sets have the same number of distinct elements. The reference set has five. Option D has the same five elements in a different order; A and C each have four, while B has six.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
Solve x/2−y/5=1 and y−x/3=8 simultaneously.
C: x=6, y=10
Multiply the first equation by 10 to get 5x−2y=10, and the second by 3 to get −x+3y=24. From x=3y−24, substitution gives 13y=130, so y=10 and x=6.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
Solve 2x+3>5x+8.
C: x<−5/3
Subtract 2x and 8 to obtain −5>3x, so x<−5/3. No inequality reversal is needed because the final division is by positive 3.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
Define a*b=(a+b)/(ab)+(a−b). Find 3*2.
B: 11/6
Use the given rule with a=3 and b=2. The fraction is (3+2)/(3×2)=5/6, while the difference is 3−2=1, so their sum is 11/6.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
Simplify 125^(−1/3) × 49^(−1/2) × 10^0.
A: 1/35
The three factors are 1/5, 1/7 and 1 respectively. Their product is 1/(5×7)=1/35.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
Find the midpoint of P(−3,4) and Q(5,6).
D: (1,5)
Average corresponding coordinates: x=(−3+5)/2=1 and y=(4+6)/2=5, so the midpoint is (1,5).
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
If cos A=12/13 and A is acute, find tan A.
A: 5/12
Use a right triangle with adjacent side 12 and hypotenuse 13. Its opposite side is √(13²−12²)=5, so tan A=5/12.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
How many distinct arrangements can be made from the letters of SCHOOL?
B: 360
SCHOOL has six letters, with O repeated twice. Divide all 6! arrangements by 2! for the repeated O: 720/2=360.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
Suppose y=a+bx. Given (x,y)=(3,7) and (5,11), find (a,b) in that order.
B: (1,2)
Subtract the equations 7=a+3b and 11=a+5b to get 4=2b, hence b=2. Then a=7−3×2=1.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
Fifteen iron bars average 1000 kg. Removing the heaviest lowers the remaining fourteen-bar average by 5 kg. Find the removed bar's weight.
C: 1070 kg
The original total is 15×1000=15000 kg. The remaining fourteen average 995 kg and total 14×995=13930 kg. The removed bar weighs 15000−13930=1070 kg.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
Find the sum of all entries of the inverse of [[1,2],[3,5]].
B: −1
The determinant is 1×5−2×3=−1. The inverse is (−1)^−1[[5,−2],[−3,1]]=[[-5,2],[3,−1]]. Its entries sum to −1.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
A varies directly as b². If A=4 when b=1, find A when b=2.
C: 16
Direct-square variation means A=kb² for a constant k. The first pair gives 4=k×1², so k=4. At b=2, A=4×2²=16.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
What is the determinant of a 2×2 identity matrix?
A: 1
The 2×2 identity matrix is [[1,0],[0,1]]. Its determinant is 1×1−0×0=1.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
The determinant of [[−2,3,1],[p,2,1],[1,4,2]] is −5. Find p.
C: 3
Expanding the determinant along the first row gives −2(4−4)−3(2p−1)+(4p−2)=1−2p. Set 1−2p=−5 to obtain p=3.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
The mean of 0, x+2, 3x+6 and 4x+8 is 4. Find x.
C: 0
The four numbers sum to 8x+16. A mean of 4 requires a sum of 16, so 8x+16=16 and x=0.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
Find the equation of the line through (3,15) with slope 16/5.
C: 5y−16x−27=0
Use y−15=(16/5)(x−3). Multiply by 5 and rearrange: 5y−75=16x−48, hence 5y−16x−27=0.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
A dealer pays ₦270,000 for a car and ₦70,000 to refurbish it, then sells it for ₦490,000. What is the profit as a percentage of total cost, to two decimal places?
C: 44.12%
Total cost is ₦270,000+₦70,000=₦340,000. Profit is ₦490,000−₦340,000=₦150,000. Therefore profit percentage is (150,000/340,000)×100≈44.12%.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
Evaluate (3/4 ÷ 2¼) × 1 7/11 × (3⅔ − 15/6).
A: 7/11
The first bracket is (3/4)/(9/4)=1/3. Convert 1 7/11 to 18/11, and the last bracket to 11/3−5/2=7/6. Multiplying gives (1/3)(18/11)(7/6)=7/11.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
A basket contains 6 grapes, 11 bananas and 13 oranges. If one fruit is drawn at random, what is the probability that it is a grape or banana?
A: 17/30
There are 6+11+13=30 fruits. The favourable fruits are the 6 grapes and 11 bananas, so the probability is (6+11)/30=17/30.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
Find one interior angle of a regular pentagon.
C: 108°
The interior angles of a pentagon sum to (5−2)×180°=540°. All five angles are equal, so each is 540°/5=108°.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
A worker spends 1/5 of a salary on shirts, then 1/3 of the remainder on transport. What fraction of the original salary remains?
B: 8/15
After the shirts, 4/5 remains. Transport takes one-third of that remainder, leaving two-thirds of it: (4/5)×(2/3)=8/15 of the original salary.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
What is the probability of rolling an even number on a fair six-sided die?
C: 1/2
A fair die has six equally likely faces. Three outcomes, 2, 4 and 6, are even, so the probability is 3/6=1/2.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
Find an antiderivative of 4x³+2x+cos x.
B: x⁴+x²+sin x+C
Integrate term by term: 4x³ gives x⁴, 2x gives x², and cos x gives sin x. The result is x⁴+x²+sin x+C.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
Evaluate −log₁₀(0.00001).
C: 5
The decimal 0.00001 equals 10⁻⁵. Thus log₁₀(0.00001)=−5, and the leading minus sign gives 5.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
A 30 cm chord lies in a circle of radius 17 cm. Find the shortest distance from the chord to the centre.
C: 8 cm
The perpendicular from the centre bisects the chord, giving a 15 cm half-chord. By Pythagoras, the distance is √(17²−15²)=√64=8 cm.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
Evaluate the limit as x approaches 2 of (x³−2x²+6x−12)/(x−2).
B: 10
Factor the numerator as (x−2)(x²+6). For x different from 2 the quotient is x²+6, whose limit at x=2 is 4+6=10.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
The second term of a geometric progression is 1 and its fifth term is 1/8. Find the common ratio.
C: 1/2
Three ratio steps separate the second and fifth terms, so r³=(1/8)/1=1/8. The real cube root is r=1/2.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
A savings balance reaches ₦600,000 after four years at 5% annual simple interest. Find the principal.
B: ₦500,000
At simple interest the amount is P(1+0.05×4)=1.2P. Hence P=₦600,000/1.2=₦500,000.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
For the arithmetic sequence 3, 5, 7, 9, …, what is the (n−2)th term for n at least 3?
D: 2n−3
The kth term is 3+2(k−1)=2k+1. Put k=n−2 to obtain 2(n−2)+1=2n−3.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
Differentiate y=sin(4x) with respect to x.
D: 4cos(4x)
The chain rule gives the derivative of sin(4x) as cos(4x) multiplied by the derivative of 4x, which is 4. Thus dy/dx=4cos(4x).
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
A circular sector has area A=(θ/360)πr², where θ is in degrees. Make θ the subject.
C: 360A/(πr²)
Multiply A=(θ/360)πr² by 360, then divide by πr². This gives θ=360A/(πr²), for a positive radius.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
For real x, what is the maximum value of y=2−4x−2x²?
A: 4
Complete the square: y=4−2(x+1)². Since (x+1)² is non-negative, y cannot exceed 4, and it reaches 4 at x=−1.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
For each of two independent people, the probability of survival is p and the probability of death is q. What is the probability that exactly one survives?
B: 2pq
There are two disjoint orders: the first survives and the second dies, or the first dies and the second survives. Independence gives probability pq for each order, so the total is pq+qp=2pq.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
Convert 137 in base ten to base five.
A: 1022₅
Take one group of 125, leaving 12. That is zero groups of 25, two groups of 5 and two units, so 137=1022₅.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
Suppose P=a+bQ. If P=32 when Q=16 and P=20 when Q=12, find P when Q=28.
A: 68
Subtract 20=a+12b from 32=a+16b to get 12=4b, so b=3. Then a=20−36=−16, and P=−16+3×28=68.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
How many distinct arrangements can be made from all the letters of HANDIER?
D: 5040
HANDIER has seven different letters. All seven can be arranged in 7!=7×6×5×4×3×2×1=5040 orders.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
A grouped distribution has class intervals 0–9, 10–19, 20–29, 30–39, 40–49 and 50–59, with respective frequencies 1, 5, 6, 12, 8 and 3. Estimate its mode using class boundaries.
C: 35.5
The modal class is 30–39 with frequency 12. Its lower boundary is 29.5, class width 10, and neighbouring frequencies are 6 and 8. The grouped-mode estimate is 29.5+[(12−6)/(2×12−6−8)]×10=35.5.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
Solve (1/2)^(t−1)=64 for t.
B: −5
Write 1/2 as 2⁻¹ and 64 as 2⁶. Then 2^(1−t)=2⁶, so 1−t=6 and t=−5.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
Convert 54 in base ten to a numeral in base four.
B: 312₄
Divide by powers of four: 54=3×16+1×4+2. The digits are therefore 3, 1 and 2, giving 312₄.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
Find an antiderivative of 3x²+2x−5.
A: x³+x²−5x+C
Integrate each term: 3x² becomes x³, 2x becomes x² and −5 becomes −5x. Include the constant C.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
Which expression equals the positive fourth root of 0.16?
A: 2×10^(−1/2)
Write 0.16 as 16×10⁻². Taking its positive fourth root gives 16^(1/4)×10^(−2/4)=2×10^(−1/2).
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
Find all real numbers x satisfying x²+3x−4≤0.
D: −4≤x≤1
Factor x²+3x−4 as (x+4)(x−1). The product is zero at −4 and 1, and the upward-opening quadratic is non-positive between those roots, including both endpoints: −4≤x≤1.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
If √75−√12+√27=y√3, what is the value of y?
D: 6
Simplify each radical: √75=5√3, √12=2√3 and √27=3√3. Thus the left side is (5−2+3)√3=6√3, so y=6.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
Let θ be acute and 0<x<y. If cos θ=x/y, express tan θ in terms of x and y.
A: √(y²−x²)/x
For an acute angle, take adjacent side x and hypotenuse y. Pythagoras gives opposite side √(y²−x²), so tan θ=opposite/adjacent=√(y²−x²)/x.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
Let P be the 2×2 matrix with rows (1, 3) and (2, −5), and Q have rows (3, −7) and (1, 2). What is P+2Q?
A: [[7, −11], [4, −1]]
Double each entry of Q to get rows (6, −14) and (2, 4). Add corresponding entries of P: (1+6, 3−14)=(7, −11) and (2+2, −5+4)=(4, −1).
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
A pie chart represents 16 doctors, 8 engineers, 24 lawyers, 21 information technologists and 27 teachers. What central angle represents the doctors?
A: 60°
The total is 16+8+24+21+27=96 people. Doctors are 16/96=1/6 of the group, so their sector angle is (1/6)×360°=60°.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
In a three-set Venn diagram, only the part of Q outside both P and R is shaded. Which expression describes the shaded region?
D: Q∩(P∪R)'
A point must be in Q, but in neither P nor R. Being in neither set means lying outside their union, (P∪R)'. Intersect that complement with Q.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
Twelve students have weights 54, 56, 58, 60 and 62 kg with respective frequencies x, 1, 2x, 2 and 3. What is the modal weight?
B: 58 kg
The frequencies total x+1+2x+2+3=3x+6=12, so x=2. They are then 2, 1, 4, 2 and 3; the highest frequency, 4, belongs to 58 kg.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
For integers a and b, define a*b=ab+a+b. Find 1*3.
C: 7
Substitute a=1 and b=3 into the definition: 1*3=(1×3)+1+3=7. The expression uses the defined operation, not ordinary multiplication alone.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
A grouped table has classes 1–10, 11–20, 21–30, 31–40 and 41–50, with frequencies 2, 7, 10, 3 and 1 respectively. What is the upper class boundary of the modal class?
B: 30.5
The largest frequency is 10, so the modal class is 21–30. For integer-valued classes, the continuous boundary between 30 and the next class at 31 is halfway: 30.5.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
A bird travels 200 km from P to Q on a bearing of N60°E, then travels from Q to R on a bearing of S30°E. If R is due east of P, how far is Q from R?
D: 200/√3 km
The first leg moves 200cos60°=100 km north. Because R is due east of P, the second leg must move 100 km south. If QR=d, its southerly component is d cos30°=d√3/2, so d=200/√3 km.
AI-reviewed · calculation checkedMyschool 2025 collection · Original sitting and question number unconfirmed.
Triangle XYZ is right-angled at Y. If YZ=11 cm, XZ=13 cm and θ is at X, find cot θ and XY respectively.
B: √48/11 and √48 cm
Pythagoras gives XY=√(13²−11²)=√48 cm. At angle X, adjacent=XY and opposite=YZ=11 cm, so cot θ=XY/YZ=√48/11.
AI-reviewed · calculation checkedMyschool 2024 collection · Original sitting and question number unconfirmed.
Find the determinant of [[3,4,6],[2,1,−1],[−1,3,5]].
C: 30
Expand along the first row: 3(5+3) − 4(10−1) + 6(6+1) = 24 − 36 + 42 = 30.
AI-reviewed · calculation checkedMyschool 2024 collection · Original sitting and question number unconfirmed.
Find the total amount after three years on ₦200,000 at 5% annual simple interest.
D: ₦230,000
Simple interest is ₦200,000 × 0.05 × 3 = ₦30,000. Add the principal to get the total amount of ₦230,000.
AI-reviewed · calculation checkedMyschool 2024 collection · Original sitting and question number unconfirmed.
If y varies inversely with x and y=4 when x=1/2, find x when y=10.
C: 1/5
For inverse variation, xy is constant. The first pair gives xy=(1/2)×4=2, so with y=10, x=2/10=1/5.
AI-reviewed · calculation checkedMyschool 2024 collection · Original sitting and question number unconfirmed.
Using π=22/7, find the central angle of a 33 cm arc in a circle of diameter 28 cm.
D: 135°
The circumference is πd=(22/7)×28=88 cm. The arc is 33/88 of a full circle, so its central angle is (33/88)×360°=135°.
AI-reviewed · calculation checkedMyschool 2024 collection · Original sitting and question number unconfirmed.
Solve 16^x = 0.25.
A: −1/2
Write both sides as powers of 2: 16^x=2^(4x), while 0.25=1/4=2^(−2). Hence 4x=−2 and x=−1/2.
AI-reviewed · calculation checkedMyschool 2024 collection · Original sitting and question number unconfirmed.
A 20-person committee has men and women in the ratio 3:1. How many women must join to make the ratio 3:2?
B: 5
The original committee has 15 men and 5 women. Keeping 15 men, a 3:2 ratio needs 10 women, so 5 more women must join.
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Define p*q=2p+pq+q. Find p if (p*2)−(p*1)=40.
A: 39
The operation gives p*2=4p+2 and p*1=3p+1. Their difference is p+1, so p+1=40 and p=39.
AI-reviewed · calculation checkedMyschool 2024 collection · Original sitting and question number unconfirmed.
Which listed statistic is a measure of central tendency?
D: Mean
The mean describes the centre of a data set. Standard deviation, range and variance describe its spread rather than its central tendency.
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In a circle of radius 9 cm, a perpendicular from the centre meets chord PQ at its midpoint T. If the angle between OT and radius OQ is 30°, how long is chord PQ?
A: 9 cm
In right triangle OTQ, OQ is the 9 cm hypotenuse and TQ is opposite the 30° angle. Thus TQ=9 sin 30°=4.5 cm. The perpendicular from the centre bisects a chord, so PQ=2×TQ=9 cm.
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A Cartesian graph shades the region to the right of the dashed vertical line x=1. Which inequality describes the shaded region?
A: x>1
Every shaded point has an x-coordinate greater than 1 because it lies to the right of x=1. The boundary is dashed, so points with x=1 are excluded. There is no separate restriction on y.
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Six distinct people sit around a round table. If rotations count as the same arrangement, how many seatings are possible?
A: 120
Fix one person's seat to remove rotational duplicates, then arrange the remaining five people in 5!=120 ways.
AI-reviewed · calculation checkedMyschool 2024 collection · Original sitting and question number unconfirmed.
A graph shades the first-quadrant region on or above both coordinate axes and below a dashed line 3x+4y=12. Which inequalities describe it?
A: 3x+4y<12, x≥0, y≥0
The shading lies below 3x+4y=12, so the expression is less than 12. The diagonal is dashed and therefore excluded. The solid coordinate axes are included, giving x≥0 and y≥0.
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Let A=[[2,−4,3],[5,1,0]] and B=[[1,4,−2],[−3,3,−1]]. Find A+2B.
A: [[4,4,−1],[−1,7,−2]]
Double each entry of B to obtain [[2,8,−4],[−6,6,−2]], then add corresponding entries of A. The result is [[4,4,−1],[−1,7,−2]].
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An infinite geometric series has sum 100 and common ratio −1/2. Find its first term.
A: 150
Because |−1/2|<1, the infinite-sum formula applies: S=a/(1−r). Rearrange to get a=S(1−r)=100[1−(−1/2)]=150.
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Triangle PQR has PQ=QR=8 cm and angle QPR=60°. What is the area of the triangle?
A: 16√3 cm²
Equal sides PQ and QR give equal opposite angles at R and P. Since angle P is 60°, the triangle is equilateral. Its area is (1/2)×8×8×sin 60°=16√3 cm².
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A 25 m tree is viewed from the top of a 10 m building at 30° elevation. Find their horizontal separation.
A: 15√3 m
The fruit at the tree top is 25−10=15 m above the observer. With tan30°=15/d=1/√3, the horizontal distance is d=15√3 m.
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Oranges cost ₦24 for five and sell for ₦30 for four. What is the profit on four oranges?
C: ₦10.80
Each orange costs ₦24/5=₦4.80. Four cost ₦19.20 and sell for ₦30, giving profit ₦30−₦19.20=₦10.80.
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Differentiate y=(5x+1)^4 with respect to x.
A: 20(5x+1)^3
By the chain rule, differentiate the outer fourth power and multiply by the inner derivative 5: dy/dx=4(5x+1)^3×5=20(5x+1)^3.
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Simplify (5+√7)/(3+√7).
B: 4−√7
Multiply numerator and denominator by 3−√7. The numerator becomes 8−2√7 and the denominator becomes 9−7=2, giving 4−√7.
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Find the exact perimeter of the triangle with vertices (3,2), (4,5) and (6,2).
C: 3+√10+√13
The horizontal base is 3. The other two lengths are √[(4−3)^2+(5−2)^2]=√10 and √[(6−4)^2+(2−5)^2]=√13. Add them.
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Find the seventh term of the geometric sequence −10, 50, −250, …
C: −156250
The common ratio is −5. The seventh term is a₇=a₁r^6=−10(−5)^6=−10×15625=−156250.
AI-reviewed · calculation checkedMyschool 2024 collection · Original sitting and question number unconfirmed.
Find the mode of the scores 4, 3, 3, 2, 1, 2, 5, 7, 8, 3, 5.
B: 3
Count the occurrences: 3 appears three times, while 2 and 5 each appear twice and the other scores once. Thus the mode is 3.
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The midpoint of P(x,4) and Q(10,8) is (9,6). Find x.
C: 8
The midpoint x-coordinate is (x+10)/2=9. Hence x+10=18 and x=8; the y-coordinate (4+8)/2=6 also agrees.
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Solve 16^(2+x)/4 = 64^x.
A: 3
Convert to powers of 2: the left side is 2^[4(2+x)−2]=2^(6+4x), while the right side is 2^(6x). Therefore 6+4x=6x and x=3.
AI-reviewed · calculation checkedMyschool 2024 collection · Original sitting and question number unconfirmed.
Let P={2,3,5,7} and Q={1,3,5,7,9}. Which set contains the elements that belong to exactly one of P and Q?
D: {1,2,9}
The elements only in P form {2}; the elements only in Q form {1,9}. Their union is {1,2,9}.
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Write the binary number 10111.11₂ as a mixed number in base ten.
B: 23¾
The whole-number part is 16+4+2+1=23. The fractional part is 1/2+1/4=3/4, so the result is 23¾.
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Calculate 20045₇ − 14256₇ and give the result in base seven.
B: 2456₇
Convert to base ten: 20045₇=4835 and 14256₇=3912. The difference is 923, which is 2456₇.
AI-reviewed · calculation checkedMyschool 2024 collection · Original sitting and question number unconfirmed.
Four teachers have an average age of 40 years, and eight other teachers average 25 years. What is the average age of all twelve teachers?
D: 30 years
The total age is 4×40+8×25=360 years. Divide by all 12 teachers to get an average of 30 years.
AI-reviewed · calculation checkedMyschool 2024 collection · Original sitting and question number unconfirmed.
How many subsets, including the empty set and the set itself, does a set of five distinct elements have?
C: 32
Each of five elements can either be included or excluded, giving 2⁵=32 possible subsets.
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The mean of 13, 16, x, 18, 21, 2x and 35 is 22. Find x.
B: 17
The seven values must total 7×22=154. Their known values total 103, so 103+3x=154 and x=17.
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A data set has a standard deviation of 12.34. What is its variance to the nearest whole number?
C: 152
Variance is the square of the standard deviation: 12.34²=152.2756, which rounds to 152.
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Find the population standard deviation of the values 5, 4, 6, 7 and 8.
A: √2
The mean is 6. The squared deviations sum to 1+4+0+1+4=10; dividing by five gives variance 2 and standard deviation √2.
AI-reviewed · calculation checkedMyschool 2024 collection · Original sitting and question number unconfirmed.
A class has five girls and seven boys. In how many ways can a committee of two girls and three boys be chosen?
A: 350
Choose the girls in C(5,2)=10 ways and the boys in C(7,3)=35 ways. Multiply to get 350 committees.
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In how many ways can a committee of five be selected from seven men and three women if it must contain exactly one woman?
A: 105
Choose the one woman in C(3,1)=3 ways and the remaining four members from seven men in C(7,4)=35 ways. Multiplying gives 3×35=105 committees.
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A bag has seven red and four black balls. Two independent draws are made with replacement. What is the probability of drawing the same colour twice?
B: 65/121
Both red has probability (7/11)² and both black (4/11)². Add them to get 49/121+16/121=65/121.
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U varies directly with √V. If U=24 when V=9, what is V when U=16?
B: 4
Write U=k√V. The first pair gives k=24/3=8. Then 16=8√V, so √V=2 and V=4.
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B varies inversely with the cube root of C. If B=2 when C=27, find the constant of proportionality.
B: 6
Use B=k/∛C. Since ∛27=3 and B=2, multiply both sides of 2=k/3 by 3 to obtain k=6.
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Find all real roots of x³−19x−30=0.
C: 5, −3, and −2
The polynomial factors as (x−5)(x+3)(x+2): multiplying gives x³−19x−30. Setting each factor to zero gives the three roots 5, −3 and −2.
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If x varies inversely with y and x=9 when y=4, which equation relates x and y?
B: x=36/y
Inverse variation has the form x=k/y, or xy=k. Substituting x=9 and y=4 gives k=36, so the required relation is x=36/y for nonzero y.
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Let P=[[1,2],[2,3]] and I be the 2×2 identity matrix. What is P²−4P−I?
B: [[0,0],[0,0]]
Matrix multiplication gives P²=[[5,8],[8,13]], while 4P=[[4,8],[8,12]]. Their difference is I, and subtracting I leaves the zero matrix [[0,0],[0,0]].
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Let A=[[3,2,1],[4,2,−1]] and B=[[1,4],[0,1],[3,2]]. What is Aᵀ+B?
D: [[4,8],[2,3],[4,1]]
Transposing A gives [[3,4],[2,2],[1,−1]]. Add the corresponding entries of B to obtain [[4,8],[2,3],[4,1]].
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Let θ be an acute angle with tan θ=8/15. Evaluate (sin θ−cos θ)/(sin² θ−sin θ).
B: 119/72
For an acute angle, an 8–15–17 right triangle gives sin θ=8/17 and cos θ=15/17. The numerator is −7/17 and the denominator is 64/289−8/17=−72/289. Dividing gives 119/72.
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An operation on real numbers is defined by x*y=xʸ. Find all real x for which x*2=x.
C: 0 and 1
By the operation rule, x*2=x². Thus x²=x, so x(x−1)=0. The only real solutions are x=0 and x=1; both satisfy the equation.
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For the listed weights 25, 30, 32, 30, 42, 45, 48, 50, 52, 51, 42, 38, 40 and 42 kg, what is the mode?
C: 42 kg
Count the occurrences in the listed data: 42 appears three times, 30 twice, and every other value once. The most frequent weight is therefore 42 kg.
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All 144 students took at least one of Mathematics (M), Chemistry (C) and Physics (P). The Venn regions are: M only 42−x, C only 34, P only 30−x, M∩C only 6+x, M∩P only 4x, C∩P only 3x, and all three 2x. How many took both M and P?
B: 24
Sum all seven disjoint regions: 144=(42−x)+34+(30−x)+(6+x)+4x+3x+2x=112+8x, so x=4. Both M and P includes their pair-only region and the three-way region: 4x+2x=24.
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Given t=√(pq/r−r²q), make q the subject, assuming r≠0 and p≠r³.
A: q=rt²/(p−r³)
Square both sides: t²=pq/r−r²q=q(p−r³)/r. Multiply by r and divide by p−r³ to get q=rt²/(p−r³), with the stated nonzero denominators.
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If (√a+√(8a))²=54+b√2 for positive integers a and b, find a and b.
C: a=6, b=24
Expand the square: a+8a+2√(8a²)=9a+4a√2 for positive a. Matching rational and irrational parts gives 9a=54, so a=6, and then b=4a=24.
AI-reviewed · calculation checkedMyschool 2023 collection · Original sitting and question number unconfirmed.
Count the distinct arrangements of all eight letters in SYLLABUS. Arrangements need not form dictionary words.
C: 8!/(2! × 2!)
There are eight letter positions. S occurs twice and L occurs twice; the other letters occur once. Temporarily distinguish repeated letters: there are 8! arrangements. Each actual arrangement is counted 2! × 2! times. Divide: 40320/4 = 10080.
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Find the value of 16^0.16 × 16^0.04 × 2^0.2.
A: 2
Combine powers with base 16: 16^(0.16 + 0.04) = 16^0.2. Rewrite 16 as 2⁴, so 16^0.2 = 2^0.8. Multiply powers with base 2: 2^(0.8 + 0.2) = 2¹ = 2.
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For a ★ b = a²b and a ◇ b = 2a + b, compute (−4 ★ 2) ◇ (7 ★ −1).
D: 15
Apply ★ inside the left brackets: (−4)² × 2 = 32. Apply ★ inside the right brackets: 7² × (−1) = −49. Apply ◇ to the two results: 2 × 32 + (−49) = 15.
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Calculate the definite integral from 0 to 1 of (4x − 6x^(2/3)) dx.
B: −8/5
For a power xⁿ, increase the exponent by one and divide by that new exponent. An antiderivative is 2x² − (18/5)x^(5/3). Differentiating it returns 4x − 6x^(2/3). At 1 this is 2 − 18/5 = −8/5; at 0 it is zero. Subtract the endpoint values.
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A population changes from 1,230 to 1,040. Find its percentage decrease, rounded to two decimal places.
D: 15.45%
The decrease is 1230 − 1040 = 190. Use the original population as the denominator: 100 × 190/1230 = 15.447154…%. Round only the final percentage to two decimal places: 15.45%.
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A regular polygon has an interior angle five times its exterior angle. Which polygon is it?
A: dodecagon
An interior angle and its adjacent exterior angle add to 180°. If the exterior angle is e, then 5e + e = 180°, so e = 30°. The exterior angles total 360°. Therefore the polygon has 360/30 = 12 sides, making it a dodecagon.
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The area of a circular region increases at 1.5 cm² per second. When its area is 2 cm², how fast is its radius increasing, to three significant figures?
D: 0.299 cm/s
Since A=πr², the radius at this instant is √(2/π) cm. Differentiating with respect to time gives dA/dt=2πr(dr/dt). Therefore dr/dt=1.5/[2π√(2/π)]≈0.2992067 cm/s, which is 0.299 cm/s to three significant figures.
AI-reviewed · calculation checkedMyschool 2023 collection · Original sitting and question number unconfirmed.
Rearrange y = (3x − 9c)/(4x + 5d) to make x the subject, where the required denominators are nonzero.
D: x = −(9c + 5dy)/(4y − 3)
Multiply by 4x + 5d to obtain 4xy + 5dy = 3x − 9c. Collect the terms containing x: 4xy − 3x = −9c − 5dy. Factor to obtain x(4y − 3) = −(9c + 5dy). Divide by 4y − 3: x = −(9c + 5dy)/(4y − 3). The original expression requires 4x + 5d ≠ 0, and this rearrangement requires 4y − 3 ≠ 0.
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Find the values of x satisfying 3(x − 1) ≤ 2(x − 3).
A: x ≤ −3
Expand both brackets: 3x − 3 ≤ 2x − 6. Subtract 2x from both sides, then add 3: x ≤ −3. The inequality direction stays the same because these steps do not multiply or divide by a negative number. At x = −3 both sides equal −12, so the boundary value is included.
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From a point S outside a circle, tangents SP and SR touch it at P and R. If ∠PSR=36° and Q is on the major arc PR, what is ∠PQR?
A: 72°
The radii to P and R meet their tangents at right angles. In quadrilateral formed by the centre, P, S and R, the central angle subtending minor arc PR is 180°−36°=144°. The angle at Q on the major arc subtends that minor arc and is half its central angle: 72°.
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A car covers 225 km in 4.5 hours at steady speed. How many hours does it need for 150 km at that speed?
C: 3 hours
The speed is 225 ÷ 4.5 = 50 km/h. The time for 150 km is distance divided by speed: 150 ÷ 50 = 3 hours.
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Calculate 1101001₂ ÷ 101₂, giving the answer in base two.
D: 10101₂
1101001₂ is 64 + 32 + 8 + 1 = 105 in base ten. 101₂ is 5. Divide 105 by 5 to get 21, then express 21 as 16 + 4 + 1 = 10101₂.
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Write (3 − √3)/(2 + √3) as a + b√3. Find a and b.
A: a = 9, b = −5
Multiply top and bottom by 2 − √3. The denominator becomes 4 − 3 = 1; the numerator expands to 6 − 3√3 − 2√3 + 3 = 9 − 5√3. Thus a = 9 and b = −5.
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Points P(−3, −14) and Q(t, −5) are 9 units apart. Determine t.
C: −3
Their vertical separation is −5 − (−14) = 9. The distance formula gives 9² = (t + 3)² + 9², so (t + 3)² = 0 and t = −3.
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At a fixed simple-interest rate, a deposit becomes three times its principal after 10 years. When will its amount reach five times the principal?
C: 20 years
At 10 years the interest is 2P, where P is the principal. Reaching 5P requires interest of 4P, twice as much. Simple interest grows linearly with time, so the required time is 20 years.
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Evaluate the limit as x approaches 2 of (x² + 4x − 12)/(x² − 2x).
A: 4
Factor the numerator as (x − 2)(x + 6) and the denominator as x(x − 2). For x other than 2, cancel x − 2. The limit is (2 + 6)/2 = 4.
AI-reviewed · calculation checkedMyschool 2023 collection · Original sitting and question number unconfirmed.
Estimate the mean age when the age groups 5–6, 7–8 and 9–10 years have frequencies 29, 40 and 38 respectively.
A: 7.7 years
Use class midpoints 5.5, 7.5 and 9.5. Their weighted total is 29×5.5 + 40×7.5 + 38×9.5 = 820.5. Divide by 107 pupils to get about 7.668 years, or 7.7 to one decimal place.
AI-reviewed · calculation checkedMyschool 2023 collection · Original sitting and question number unconfirmed.
A five-member committee is chosen from six men and four women. How many choices contain more women than men?
C: 66
A female majority means either three women and two men, or four women and one man. Count C(4,3)C(6,2) + C(4,4)C(6,1) = 4×15 + 1×6 = 66.
AI-reviewed · calculation checkedMyschool 2023 collection · Original sitting and question number unconfirmed.
Find the determinant of the matrix with rows (2, −1, 3), (4, 1, 2) and (1, −3, 1).
C: −23
Expand along the first row: 2(1×1 − 2×(−3)) − (−1)(4×1 − 2×1) + 3(4×(−3) − 1×1). This is 14 + 2 − 39 = −23.
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A boat goes 8 km north and then 6 km west. Find the bearing of its final position from its start, to the nearest degree.
B: 323°
The displacement is northwest. Its angle west of north is arctan(6/8) ≈ 36.87°. Bearings increase clockwise from north, so the bearing is 360° − 36.87° ≈ 323°.
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A fair coin is tossed three times. What is the probability of getting at least one head?
A: 7/8
The complement is no heads, meaning three tails. Its probability is (1/2)³ = 1/8. Hence the probability of at least one head is 1 − 1/8 = 7/8.
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Find the line through (2, 3) perpendicular to 3x + 2y + 4 = 0.
D: 3y = 2x + 5
The given line has slope −3/2, so a perpendicular line has slope 2/3. Using y − 3 = (2/3)(x − 2) gives 3y = 2x + 5.
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Differentiate y = x^(2/3)(2x − x²).
B: (10/3)x^(2/3) − (8/3)x^(5/3)
Expand first: y = 2x^(5/3) − x^(8/3). Apply d(xⁿ)/dx = nx^(n−1) term by term to obtain (10/3)x^(2/3) − (8/3)x^(5/3).
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Calculate the mean absolute deviation of the first five prime numbers.
A: 2.72
The primes are 2, 3, 5, 7 and 11; their mean is 5.6. Their absolute deviations from 5.6 total 3.6 + 2.6 + 0.6 + 1.4 + 5.4 = 13.6. Divide by five to get 2.72.
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Find the area enclosed by y = 2x² + 10 and y = 4x + 16.
D: 64/3
The curves meet when 2x² + 10 = 4x + 16, giving x = −1 and x = 3. The straight line is higher between these points. Integrate (4x + 16) − (2x² + 10) = −2x² + 4x + 6 from −1 to 3 to obtain 64/3.
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The base-y numeral 402ᵧ equals 102 in base ten. Find y.
C: 5
402ᵧ has value 4y² + 2. Set 4y² + 2 = 102 to obtain y² = 25. A numeral base is positive and must exceed its largest digit, 4, so y = 5.
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For real y, solve log₁₀(y+8)+log₁₀(y−8)=2log₁₀3+2log₁₀5. Which value satisfies the equation?
B: 17
Both logarithms on the left require y>8. Combine them to obtain log₁₀(y²−64)=log₁₀(225), so y²=289. The algebraic roots are ±17, but −17 is outside the logarithm domain. Hence y=17.
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Factor 16x⁴ − y⁴ completely.
A: (2x − y)(2x + y)(4x² + y²)
Write 16x⁴ − y⁴ as (4x²)² − (y²)² = (4x² − y²)(4x² + y²). Factor the first bracket again: 4x² − y² = (2x − y)(2x + y).
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For A = {1,2,3,4,5,6} and B = {2,4,6,8}, find (A − B) ∪ (B − A).
A: {1,3,5,8}
A − B contains the members only in A: 1, 3 and 5. B − A contains the member only in B: 8. Taking their union gives {1,3,5,8}.
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A bag has eight red balls and some white balls. On one draw, a white ball is half as likely as a red ball. If two balls are drawn without replacement, what is the probability of one red and one white in either order?
C: 16/33
There are four white balls because their count must be half the eight red balls. With twelve balls in total, red then white has probability (8/12)(4/11)=8/33. White then red has the same probability. Adding the two mutually exclusive orders gives 16/33.
AI-reviewed · calculation checkedMyschool 2023 collection · Original sitting and question number unconfirmed.
Estimate the median age when the age groups 5–6, 7–8 and 9–10 years have frequencies 29, 40 and 38 respectively.
A: 7.725 years
There are 107 pupils, so the halfway position is 53.5 and lies in the 7–8 group. Using continuous class boundaries 6.5 to 8.5, estimate the median as 6.5 + ((53.5 − 29)/40)×2 = 7.725 years.
AI-reviewed · calculation checkedMyschool 2023 collection · Original sitting and question number unconfirmed.
Find A if A[[0,1],[2,−1]] = [[2,−1],[1,0]].
B: [[0,1],[1/2,1/2]]
Let A = [[a,b],[c,d]]. The product on the left is [[2b,a−b],[2d,c−d]]. Matching entries with [[2,−1],[1,0]] gives b=1, a=0, d=1/2 and c=1/2.
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A class has 3 students in the score band [0%,25%), 5 in [25%,50%), 3 in [50%,75%), and 8 in [75%,100%]. How many scored at least 25%?
A: 16
The three bands starting at 25% or higher contain 5, 3 and 8 students. Add those frequencies: 5+3+8=16. The first band is below 25% and is not included.
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Find the remainder when −2x³ + 6x² + 17x − 21 is divided by x + 1.
C: −30
The remainder theorem says to substitute x = −1. The result is −2(−1)³ + 6(−1)² + 17(−1) − 21 = 2 + 6 − 17 − 21 = −30.
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For a binary operation * on a set, which condition states that * is commutative?
A: a*b = b*a
Commutativity means swapping the two operands leaves the result unchanged: a*b = b*a. The second identity describes associativity, and the third is a distributive form.
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Solve x² − x − 4 ≤ 2.
B: −2 ≤ x ≤ 3
Move 2 to the left: x² − x − 6 ≤ 0. Factor as (x − 3)(x + 2) ≤ 0. The upward-opening quadratic is nonpositive between its roots, including both endpoints.
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Give the nth term of 3, 8, 13, 18, …, taking n = 1 for the first term.
A: 5n − 2
The sequence has first term 3 and common difference 5. Its nth term is 3 + (n−1)5 = 5n − 2; at n=1 it returns 3.
AI-reviewed · calculation checkedMyschool 2023 collection · Original sitting and question number unconfirmed.
What is the locus of points at equal perpendicular distances from two intersecting straight lines?
D: The two angle bisectors
Every point on either angle bisector of two intersecting lines has equal perpendicular distance from both lines. Together, the two bisectors give the full locus.
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Two quantities are 35% and 80% greater than the same reference quantity. Find their ratio in that order.
B: 3:4
If the reference is 100 units, the quantities are 135 and 180. Their ratio is 135:180, which reduces by 45 to 3:4.
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From the top of a 24 m building, the top of a second building is seen at 30° elevation and its base at 60° depression. How tall is the second building?
D: 32 m
The horizontal gap is 24/tan60° = 8√3 m. The second roof rises (8√3)tan30° = 8 m above the first roof. Its height is 24 + 8 = 32 m.
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In triangle ABC, ∠A=82°, ∠C=56°, and BC=43.2 cm. Find AB to three significant figures.
C: 36.2 cm
The sine rule gives AB/sin56°=BC/sin82°. Hence AB=43.2 sin56°/sin82°≈36.1664 cm, which is 36.2 cm to three significant figures.
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An article sells for ₦230 at a 15% profit. What percentage gain or loss would a selling price of ₦180 produce?
B: 10% loss
A ₦230 selling price at 15% profit means cost = 230/1.15 = ₦200. Selling for ₦180 loses ₦20, which is 20/200 × 100 = 10% of cost.
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A circle has radius 13 cm and a chord 24 cm long. Find the length of the minor arc subtended by this chord, to three significant figures.
B: 30.6 cm
The perpendicular from the centre bisects the chord, leaving a half-chord of 12 cm. Half the central angle is arcsin(12/13), so the full angle in radians is 2arcsin(12/13). Arc length is radius times angle: 13×2arcsin(12/13)≈30.5761 cm, or 30.6 cm.
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A rectangle's sides are reported as 38 m and 52 m, each rounded to the nearest metre. Which interval contains every possible area?
A: 1931.25 m² ≤ A < 2021.25 m²
The true sides lie in [37.5,38.5) and [51.5,52.5). All lengths are positive, so multiply the lower bounds for 1931.25 m² and upper bounds for 2021.25 m². The upper endpoint is excluded.
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One ninth of a shop's items are brand A. Of the remainder, five eighths are brand B; the rest are brand C. If brand C has 81 items, how many more brand B items are there than C items?
C: 54
After brand A, 8/9 of the stock remains. Brand C is 3/8 of that remainder, so C is (3/8)(8/9) = 1/3 of all stock. Total stock is 243; B is (5/8)(8/9)×243 = 135, or 54 more than 81.
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A triangle's sides have ratio 2:3:4 and perimeter 180 cm. Find its area to the nearest square centimetre.
A: 1162 cm²
The nine ratio parts make sides 40, 60 and 80 cm; the semiperimeter is 90 cm. Heron's formula gives √[90(90−40)(90−60)(90−80)] = √1,350,000 ≈ 1161.895, which rounds to 1162 cm².
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Calculate interest only on ₦15,700 invested for two years at 8% per year, compounded annually.
D: ₦2,612.48
The amount is ₦15,700×1.08² = ₦18,312.48. Subtract the principal to isolate compound interest: ₦18,312.48 − ₦15,700 = ₦2,612.48.
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An arithmetic progression has third term 6 and fifth term 12. What is the sum of its first twelve terms?
C: 198
From a+2d=6 and a+4d=12, subtract to obtain d=3 and then a=0. The first twelve terms sum to 12/2 × [2a+11d] = 6×33 = 198.
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Evaluate 16.54×10⁻⁵ − 6.76×10⁻⁸ + 0.23×10⁻⁶. Give the result in standard form to three significant figures.
A: 1.66×10⁻⁴
Write every term with the same power of ten: (1.654−0.000676+0.0023)×10⁻⁴=1.655624×10⁻⁴. The fourth significant digit is 5, so to three significant figures the coefficient rounds from 1.655624 to 1.66.
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A student reads 18 mL from a cylinder, while the accepted reading is 18.4 mL. Find the percentage error relative to the accepted reading.
A: 2.17%
The absolute error is |18 − 18.4| = 0.4 mL. Relative to the accepted 18.4 mL, percentage error is 0.4/18.4 × 100 ≈ 2.17%.
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The exterior angles of regular polygons with n−1 and n+2 sides differ by 6°. Find n.
B: 13
Their exterior angles are 360/(n−1) and 360/(n+2). Their difference is 1080/[(n−1)(n+2)] = 6, so (n−1)(n+2)=180. The positive solution of n²+n−182=0 is n=13.
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Two fair six-sided dice are rolled. What is the probability that their sum is prime?
A: 5/12
Prime sums from 2 to 12 are 2, 3, 5, 7 and 11. Their ordered-outcome counts are 1, 2, 4, 6 and 2, totalling 15. There are 36 equally likely outcomes, so the probability is 15/36 = 5/12.
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An L-shaped solid has a constant width of 6 cm. Its lower beam is 36 cm long and 8 cm high. Over the final 10 cm of its length, the solid rises to a total height of 22 cm. What is its volume?
B: 2568 cm³
Split the solid into a full lower beam and the extra upper block. The lower volume is 36×6×8=1728 cm³. The added block is 10×6×(22−8)=840 cm³. Their sum is 2568 cm³.
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A square has diagonal 20√2 cm. Find its area and perimeter in that order.
D: 400 cm², 80 cm
For a square, diagonal = side×√2. The side is therefore 20 cm. Area = 20² = 400 cm² and perimeter = 4×20 = 80 cm.
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A bicycle costs ₦9,200 and repairs cost ₦1,500. It then sells for ₦13,400. What percentage gain is made, to two decimal places?
A: 25.23%
Total cost is ₦10,700. Gain is ₦13,400 − ₦10,700 = ₦2,700. Divide gain by total cost: 2700/10700 × 100 ≈ 25.23%.
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Evaluate 5/8 − 3/4 ÷ 5/12 × 1/4, applying division and multiplication from left to right.
C: 7/40
Compute the division first, then multiply: (3/4) ÷ (5/12) × (1/4) = (9/5)×(1/4) = 9/20. Now 5/8 − 9/20 = 25/40 − 18/40 = 7/40.
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A rectangle's shorter side is 6 cm less than its longer side. Adding 2 cm to both dimensions raises its area by 68 cm². Find the shorter side.
C: 13 cm
Let the shorter side be x and the longer side x+6. The area increase is (x+2)(x+8) − x(x+6) = 4x+16. Set this equal to 68 to obtain x=13 cm.
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An infinite geometric series has second term −2/3 and sum 3/2. Find its common ratio.
A: −1/3
Write the second term as ar=−2/3 and the sum as a/(1−r)=3/2, where |r|<1. Eliminate a to get 9r²−9r−4=0. The roots are 4/3 and −1/3; only −1/3 meets the convergence condition.
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The line 3y + 6x = 48 contains A(−2,k). Find k.
B: 20
A point on a line must satisfy its equation. Substitute x=−2 and y=k: 3k + 6(−2) = 48. Thus 3k−12=48, so 3k=60 and k=20.
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Adult tickets cost ₦520 and child tickets ₦250. Five times as many adult tickets as child tickets bring in ₦171,000. How many child tickets were sold?
D: 60
Let the number of child tickets be c, so adult tickets number 5c. Revenue is 250c + 520(5c) = 2850c. Solve 2850c = 171000 to get c=60.
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Solve log₂(6−x) = 3 − log₂(x) over its real domain.
A: x = 2 or 4
The domain is 0<x<6. Add log₂(x) to both sides, giving log₂[x(6−x)] = 3. Hence x(6−x)=8 and x²−6x+8=(x−2)(x−4)=0. Both 2 and 4 satisfy the domain.
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A cone has base radius 5 cm and slant height 13 cm. Find its volume.
D: 100π cm³
The radius, height and slant height form a right triangle, so the vertical height is √(13²−5²) = 12 cm. Volume is (1/3)πr²h = (1/3)π×25×12 = 100π cm³.
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A trapezium has area 200 cm² and height 16 cm. Its parallel sides are in the ratio 2:3. Find their lengths.
A: 10 cm and 15 cm
Area = (height/2)×sum of parallel sides, so their sum is 2×200/16 = 25 cm. Split 25 in the ratio 2:3 to get 10 cm and 15 cm.
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A pilot flies to an airport at 120 km/h and returns over the same distance at 150 km/h. Both legs take three hours in total. Find the one-way distance.
B: 200 km
If the one-way distance is d, total time is d/120 + d/150 = 3. Since 1/120 + 1/150 = 3/200, solving gives d = 200 km.
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An isosceles right triangle has perimeter 2 m. What is the length of its longest side?
D: −2 + 2√2 m
Let each equal leg have length s, so the hypotenuse is s√2. Then s(2+√2)=2. Multiply 2√2/(2+√2) by the conjugate to obtain the hypotenuse 2√2−2 m.
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A ship travels from (86°N,56°W) to nearby (86°N,64°W) along the same parallel. Find the distance to the nearest kilometre, using π=3.142 and Earth radius 6370 km.
A: 62 km
The longitude change is 8°. The radius of the small circle at latitude 86° is 6370cos86°. Arc distance is (8/360)×2×3.142×6370cos86° ≈ 62.1 km, or 62 km.
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A circle has radius 13 cm, and a chord is 12 cm from its centre. How long is the chord?
D: 10 cm
The perpendicular from the centre bisects the chord. One half has length √(13²−12²) = √25 = 5 cm, so the full chord is 10 cm.
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A show sells 200 tickets: VIP at ₦1,200 and regular at ₦700. Total sales are ₦180,000. How many of each type sold?
D: 80 VIP, 120 regular
Let v be VIP sales, so regular sales are 200−v. Revenue gives 1200v + 700(200−v) = 180000. Then 500v=40000, so v=80 and regular sales=120.
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Twelve workers at eight hours per day finish a job in four days. At the same rate, how many days would four workers at sixteen hours per day need?
A: 6 days
The job needs 12×8×4 = 384 worker-hours. Four workers at 16 hours each supply 64 worker-hours per day. Divide 384 by 64 to get 6 days.
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For real k, (1/8)^(k+2)=1. Find k.
B: −2
A positive base other than 1 gives a result of 1 only when its exponent is zero. Set k+2=0, so k=−2.
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Cube the binary number 101₂ and express the result in base two.
C: 1111101₂
The binary value 101₂ is 5 in base ten. Cubing gives 125, and 125=64+32+16+8+4+1, so its binary digits are 1111101₂.
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A fair coin is tossed twice. What is the probability of at least one head?
C: 3/4
The equally likely outcomes are HH, HT, TH and TT. Three contain at least one head, so the probability is 3/4; equivalently subtract P(TT)=1/4 from 1.
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Six numbers are 2, 5, x+1, x+2, 7 and 9. Their mean is 6. Find their median.
C: 6.5
A mean of 6 requires a total of 36. The six expressions sum to 26+2x, giving x=5. In order the values are 2, 5, 6, 7, 7, 9; the middle pair averages to 6.5.
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If sin θ=−3/5 and θ lies in the third quadrant, find cos θ.
D: −4/5
Using sin²θ+cos²θ=1 gives cos²θ=1−9/25=16/25. Both sine and cosine are negative in quadrant III, so cosθ=−4/5.
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Simplify 1/(3−√2) in the form p+q√2.
D: 3/7+√2/7
Multiply numerator and denominator by 3+√2. The denominator becomes 9−2=7, giving 3/7+√2/7.
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Find the length of a chord 3 cm from the centre of a circle of radius 5 cm.
A: 8 cm
The perpendicular from the centre bisects the chord. Half the chord is √(5²−3²)=4 cm, so the full chord is 8 cm.
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Mr Adu spends his annual salary on food (28%), rent (15%), car maintenance (20%), gifts (14%), savings (10%) and miscellaneous expenses (13%). If these amounts are represented on a pie chart, what angle represents food?
B: 100.8°
Food uses 28% of the total. Its sector angle is 28/100×360°=100.8°. The six source percentages sum to 100%.
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The tenth term of an arithmetic progression is 32 and the first term is 3/2. Find its fourth term.
A: 35/3
Use a₁₀=a+9d: 32=3/2+9d, so d=61/18. Then a₄=3/2+3(61/18)=35/3. Check that a+9d returns 32.
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L, N and Z lie on a straight line. Rays NM and NY lie on the same side of that line, in the order NL, NM, NY, NZ. If angle LNM is g degrees and angle YNZ is h degrees, find angle MNY.
A: 180°−g°−h°
Adjacent angles on a straight line add to 180°. Here g + angle MNY + h = 180°, so angle MNY = 180°−g°−h°. Both given angles must be subtracted.
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How many different arrangements use all seven letters of ZOOLOGY?
B: 840
There are seven positions, but the three O letters are indistinguishable. Divide the 7! possible orderings by 3!, giving 5040/6=840 distinct arrangements.
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Which expression is an antiderivative of (2x+1)^3?
A: (2x+1)^4/8+C
Set u=2x+1, so du=2 dx. Then the integral is one half of u^4/4, or (2x+1)^4/8+C. Differentiating this result returns (2x+1)^3.
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Matrix A has rows (2,1), (2,3), (1,2), and matrix B has rows (3,2), (4,2). Find AB.
D: Rows (10,6), (18,10), (11,6)
Multiply each row of A by each column of B: (2×3+1×4,2×2+1×2)=(10,6); (2×3+3×4,2×2+3×2)=(18,10); (1×3+2×4,1×2+2×2)=(11,6). Thus AB has rows (10,6), (18,10), (11,6).
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For h>0, r=√(3V/(πh)). Make V the subject of the formula.
B: πhr²/3
Square both sides to get r²=3V/(πh). Multiply by πh and divide by 3, giving V=πhr²/3.
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Find the determinant of the matrix with rows (2, 3) and (1, 3).
C: 3
For a 2×2 matrix, multiply the main diagonal and subtract the other diagonal: 2×3−3×1=3.
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What is the value of log₂(8√2)?
C: 3.5
Write 8 as 2³ and √2 as 2^(1/2). Their product is 2^(7/2), so its base-two logarithm is 7/2=3.5.
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What is the new value of 96/50 when its numerator is decreased by 37.5% and its denominator by 20%?
A: 1.5
A 37.5% decrease leaves 62.5% of 96, namely 60. A 20% decrease leaves 80% of 50, namely 40. The new fraction is 60/40=1.5.
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Find the limit of (x³+6x−7)/(x−1) as x approaches 1.
C: 9
Factor the numerator as (x−1)(x²+x+7). Cancel x−1 for x≠1, then substitute x=1 into x²+x+7 to obtain 9.
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Find the median of 14, 17, 10, 13, 18 and 10.
C: 13.5
Sort the six values as 10, 10, 13, 14, 17, 18. The median is the mean of the third and fourth values: (13+14)/2=13.5.
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Convert 10110.011₂ to a decimal number.
B: 22.375
The whole-number bits contribute 16+4+2=22. The fractional bits contribute 1/4+1/8=0.375, so the number is 22.375.
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A school's running cost varies directly with student count. If 20 students cost ₦10,000, how many students correspond to ₦1,000,000 at the same rate?
B: 2000
At the stated rate each student accounts for ₦10,000/20=₦500. Divide ₦1,000,000 by ₦500 to get 2,000 students.
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What is the midpoint of the points (−3, 5) and (2, 10)?
B: (−1/2, 15/2)
Average corresponding coordinates: (−3+2)/2=−1/2 and (5+10)/2=15/2. The midpoint is (−1/2, 15/2).
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Evaluate C(n²+1, n+5) when n=3.
C: 45
With n=3, the expression is C(10,8), equal to C(10,2). Choose two from ten in 10×9/2=45 ways.
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A particle has displacement s=3t²+5t−10 metres. What is its velocity at t=0?
B: 5 m/s
Velocity is the derivative ds/dt=6t+5. At t=0 this equals 5 m/s; the constant displacement term has no effect on velocity.
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An operation is defined by a*b=a−b+(ab)². Find 3*5.
A: 223
Apply the definition to a=3,b=5: 3−5+(3×5)²=−2+225=223. Square the entire product ab.
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If y varies inversely as x, and x=3 when y=4, find x when y=12.
B: 1
Inverse variation means xy=k. The initial values give k=3×4=12. When y=12, x=12/12=1.
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The numbers 8,p,q,26 form an arithmetic progression. Find p and q.
D: p=14,q=20
There are three equal steps from 8 to 26, so the common difference is (26−8)/3=6. The terms are 8,14,20,26; hence p=14,q=20.
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Factorize the expression 4a²−9b².
A: (2a−3b)(2a+3b)
Recognize a difference of squares: 4a²−9b²=(2a)²−(3b)²=(2a−3b)(2a+3b). Expanding the factors cancels the cross terms.
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If n(P)=20, n(Q)=30 and n(P∪Q)=40, find n(P∩Q).
A: 10
Use n(P∪Q)=n(P)+n(Q)−n(P∩Q). Thus 40=20+30−n(P∩Q), so the intersection contains 10 elements.
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Three times a number x, minus 2, is less than that number minus 6. Find the possible values of x.
A: x<−2
Translate the statement into 3x−2<x−6. Subtract x and add 2 to obtain 2x<−4. Dividing by positive 2 gives x<−2.
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A dress sells for ₦3,800 after a 20% discount. What was its original price?
D: ₦4,750
After a 20% discount the selling price is 80% of the original. Thus 0.8P=3800 and P=3800/0.8=₦4,750. Check: 20% of ₦4,750 is ₦950; subtracting gives ₦3,800.
AI-reviewed · calculation checkedMyschool 2021 collection · Original sitting and question number unconfirmed.
For real x, determine the maximum value of y=3x²+5x−3.
D: No correct option
The coefficient of x² is positive, so the parabola opens upwards and y grows without bound as |x| increases. It has a minimum, y=3(x+5/6)²−61/12, but no finite maximum; none of the numerical options is a maximum.
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Find the derivative of y=2x²(2x−1) at x=−1.
B: 16
Expand y=2x²(2x−1)=4x³−2x². Then dy/dx=12x²−4x. At x=−1 this is 12+4=16.
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Evaluate 241.34(3×10⁻³)², correct to four significant figures.
D: 0.002172
First square 3×10⁻³ to obtain 9×10⁻⁶. Multiplying by 241.34 gives 0.00217206. To four significant figures, keep 2,1,7,2; the next digit is 0, so the result is 0.002172.
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Three children share a basket of mangoes. The first takes 1/4 of the mangoes, and the second takes 3/4 of the remainder. What fraction of all the mangoes does the third receive?
A: 3/16
After the first child takes 1/4, the remainder is 3/4. The second takes 3/4 of that remainder, leaving 1/4 of it: (1/4)(3/4)=3/16 for the third child.
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Simplify (0.00275×0.00640)/(0.025×0.08) and express the result in standard form.
C: 8.8×10⁻³
The numerator is 0.00275×0.00640=0.0000176 and the denominator is 0.025×0.08=0.002. Dividing gives 0.0088=8.8×10⁻³.
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Three brothers share a profit. The first receives 1/3 of the total, and the second receives 2/3 of the remainder. The third receives the remaining ₦12,000. Find the total profit.
B: ₦54,000
The first brother leaves 2/3 of the profit. The second takes 2/3 of that, leaving 1/3×2/3=2/9 for the third. Thus (2/9)P=12000 and P=₦54,000. The shares ₦18,000,₦24,000,₦12,000 add to ₦54,000.
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Each interior angle of a regular polygon is 140°. How many sides does it have?
A: 9
Each exterior angle is 180°−140°=40°. Exterior angles total 360°, so the number of sides is 360/40=9.
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A metal pipe is 3 cm thick and has an internal radius of 10 cm. Find the volume of metal in a 3 m length of pipe.
D: 20,700π cm³
The outer radius is 10+3=13 cm, and 3 m=300 cm. Subtract inner cylinder volume from outer: π(13²−10²)×300=π×69×300=20,700π cm³.
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The values 4, 16, 30, 20, 10, 14 and 26 are represented on a pie chart. Find the sum of the sector angles for values greater than or equal to 16.
D: 276°
The values total 120. Those at least 16 are 16, 30, 20 and 26, totalling 92. Their share of the full circle is (92/120)×360°=276°.
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The mean of ten positive numbers is 16. After another number is added, the mean becomes 18. Find the eleventh number.
C: 38
The first ten numbers total 10×16=160. Eleven numbers with mean 18 total 11×18=198. The added number is 198−160=38.
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A car travels P km from Calabar to Enugu at an average speed of U km/h, then Q km to Benin at an average speed of W km/h. Find its average speed for the complete journey.
B: UW(P+Q)/(PW+QU)
Average speed is total distance divided by total time. The two journey times are P/U and Q/W hours, so the average is (P+Q)/(P/U+Q/W)=UW(P+Q)/(PW+QU).
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If g(x)=x²+3x, find g(x+1)−g(x).
B: 2(x+2)
Substitute x+1: g(x+1)=(x+1)²+3(x+1)=x²+5x+4. Subtract x²+3x to obtain 2x+4=2(x+2).
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Factorize m³ − m² + 2m − 2.
D: (m² + 2)(m − 1)
Group the first two terms and the last two: m²(m − 1) + 2(m − 1). Both groups contain m − 1, so the result is (m² + 2)(m − 1). Expanding this product gives m³ − m² + 2m − 2.
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Find the set of all integers x satisfying −2 < 2x − 6 < 4.
D: {3, 4}
Add 6 to each part to obtain 4 < 2x < 10. Divide by positive 2 to get 2 < x < 5. The inequalities are strict, so neither endpoint is allowed. The only integers in this interval are 3 and 4.
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X is due east of Y. Z is 6 km due south of Y, and XZ is 12 km. What is the bearing of Z from X?
A: 240°
The east-west and north-south directions form a right angle at Y. Seen from X, Z lies southwest. If θ is the angle south of west, sin θ = 6/12 = 1/2, so θ = 30°. West has bearing 270°, hence the bearing toward Z is 270° − 30° = 240°.
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A group of market sellers each sells at least one of maize, yam and plantain. In the group, 12 sell maize, 10 sell yam and 14 sell plantain. 5 sell both plantain and maize, 4 sell both yam and maize, 2 sell yam and plantain only, and 3 sell all three. How many sellers are in the group?
A: 25
The total who sell both yam and plantain is 2 + 3 = 5, including those who sell all three. Apply inclusion-exclusion: 12 + 10 + 14 − 5 − 4 − 5 + 3 = 25. Subtracting the three pair totals removes the triple group too often, so add its 3 sellers back once.
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Let P(x) = Lx + 2kx² + 24. If x + 2 and x − 1 are factors of P(x), find L and k.
A: L = −12, k = −6
The factors give P(−2) = 0 and P(1) = 0. Thus −2L + 8k = −24 and L + 2k = −24. Double the second equation and add it to the first: 12k = −72, so k = −6 and L = −12. Then P(x) = −12(x + 2)(x − 1), confirming both factors.
AI-reviewed · calculation checkedThree teachers share a packet of chalk. The first receives 2/5 of the packet; the second receives 2/15 of what remains. What fraction of the original packet does the third receive?
C: 13/25
After the first share, 3/5 remains. The second receives (2/15)(3/5) = 2/25 of the original packet. The third receives 1 − 2/5 − 2/25 = 13/25.
AI-reviewed · calculation checkedSolve 6logₓ2 − 3logₓ3 = 3log₅0.2 for a valid logarithm base x.
B: 3/4
Divide by three and use log₅0.2 = −1. This gives logₓ(4/3) = −1, so x⁻¹ = 4/3 and x = 3/4. This is positive and different from one.
AI-reviewed · calculation checkedAt what x values does y = x³ + 2x² − 5x − 6 cross the x-axis?
C: 2, −1 and −3
The polynomial factors as (x − 2)(x + 1)(x + 3). Each factor has a simple root, so the curve crosses the x-axis at x = 2, −1 and −3.
AI-reviewed · calculation checkedFactorise completely ac − 2bc − a² + 4b².
B: (a − 2b)(c − a − 2b)
Group the terms as c(a − 2b) − (a² − 4b²). The difference of squares is (a − 2b)(a + 2b), so factoring gives (a − 2b)(c − a − 2b).
AI-reviewed · calculation checkedPendulum length L varies directly as period T squared. A period of 4 seconds corresponds to length 64 cm. Find the length for a period of 9 seconds.
D: 324 cm
Write L = kT². The given pendulum gives k = 64/4² = 4 cm/s². For T = 9 seconds, L = 4 × 9² = 324 cm.
AI-reviewed · calculation checkedWhich inequality describes the shaded half-plane, excluding its boundary?
Find all integers x satisfying −1 < 3 − 2x ≤ 5.
C: −1, 0, 1
From −1 < 3 − 2x we get x < 2. From 3 − 2x ≤ 5 we get x ≥ −1. Thus −1 ≤ x < 2, whose integers are −1, 0 and 1.
AI-reviewed · calculation checkedFind the inverse of [[2,1],[1,1]].
D: [[1,−1],[−1,2]]
The determinant is 2×1 − 1×1 = 1. Swap the diagonal entries and negate the off-diagonal entries, then divide by one, giving [[1,−1],[−1,2]].
AI-reviewed · calculation checkedThe sum of the interior angles of a pentagon is (6x + 6y)°. Express y in terms of x.
B: y = 90 − x
A pentagon’s interior angles sum to (5 − 2)×180° = 540°. Hence 6x + 6y = 540, so x + y = 90 and y = 90 − x.
AI-reviewed · calculation checkedA regular hexagon PQRSTV of side 7 cm is inscribed in a circle. Find the circle’s circumference, using π = 22/7.
C: 44 cm
The six central triangles are equilateral, so the circle’s radius equals the hexagon side, 7 cm. Circumference is 2πr = 2×(22/7)×7 = 44 cm.
AI-reviewed · calculation checkedIn trapezium PQRU, PQTU and PQRS are parallelograms. U,T,S,R lie in that order on the lower parallel side. If PQ = 4 cm, TS = 6 cm and area PQTU = 32 cm², find area PQRU.
D: 72 cm²
Both parallelograms have height 32/4 = 8 cm. Their opposite sides give UT = SR = 4 cm, so UR = 4 + 6 + 4 = 14 cm. The trapezium area is ½(4 + 14)×8 = 72 cm².
AI-reviewed · calculation checkedInside square PQRS, find the locus equidistant from the adjacent sides PQ and QR.
B: The diagonal QS
Equal distances from two intersecting sides lie on an angle bisector. Inside the square, the bisector of the right angle at Q is the diagonal from Q to S.
AI-reviewed · calculation checkedDescribe the complete locus of points at perpendicular distance 5 cm from the straight line LM.
A: Two lines parallel to LM on opposite sides, each 5 cm away
A fixed perpendicular distance from a line can be reached on either side. Therefore the full locus consists of two parallel lines, each exactly 5 cm from LM.
AI-reviewed · calculation checkedFind the midpoint of the segment joining P(−3,5) and Q(5,−3).
D: (1,1)
The midpoint is obtained by averaging the x coordinates and the y coordinates: ((−3 + 5)/2, (5 − 3)/2) = (1,1).
AI-reviewed · calculation checkedA triangle has a 15 cm side opposite a 60° angle. Its side x is opposite a 45° angle. Find x.
C: 5√6 cm
By the sine rule, x/sin45° = 15/sin60°. Thus x = 15(√2/2)/(√3/2) = 15√2/√3 = 5√6 cm. Each side is paired with its opposite angle.
AI-reviewed · calculation checkedFind the rate of change of a hemisphere’s volume V with respect to its radius r when r = 2.
C: 8π
A hemisphere has volume V = 2πr³/3. Differentiating gives dV/dr = 2πr². Substituting r = 2 yields 2π×4 = 8π.
AI-reviewed · calculation checkedEvaluate ∫₁³(x² − 1) dx.
A: 6⅔
An antiderivative is x³/3 − x. At 3 its value is 6, and at 1 its value is −2/3. The definite integral is 6 − (−2/3) = 6⅔.
AI-reviewed · calculation checkedA crop pie chart has angles: millet 150°, maize 90°, others 60°, with the remainder representing beans. If millet weighs 3000 tonnes, find the weight of beans.
D: 1200 tonnes
Beans occupy 360° − 150° − 90° − 60° = 60°. Weights are proportional to the sector angles, so beans weigh (60/150)×3000 = 1200 tonnes.
AI-reviewed · calculation checkedFor a histogram with equal-width class intervals, consider: I. Rectangular bars have equal width. II. Their heights are proportional to class frequencies. III. Adjacent class bars meet without gaps. Which statements apply?
C: I, II and III
With equal class widths, frequency density and frequency are proportional. The bars therefore have equal widths, heights proportional to frequencies, and adjacent boundaries without gaps.
AI-reviewed · calculation checkedUse the cumulative-frequency graph to estimate the percentage of students who scored more than 20 marks.
Find the mean absolute deviation about the mean of 1, 2, 3 and 4.
A: 1.0
The mean is (1 + 2 + 3 + 4)/4 = 2.5. Absolute deviations are 1.5, 0.5, 0.5 and 1.5, totalling 4. Dividing by four gives 1.0.
AI-reviewed · calculation checkedIn how many ways can two students be selected from five for a debating competition?
A: 10 ways
Only the selected pair matters, not its order. The number of combinations is ⁵C₂ = (5×4)/(2×1) = 10. Dividing by two avoids counting each pair twice.
AI-reviewed · calculation checkedMultiply 213₄ by 23₄, giving the result in base four.
D: 12231₄
In decimal, 213₄ = 39 and 23₄ = 11, so their product is 429. Dividing repeatedly by four, or using place values, gives 12231₄.
AI-reviewed · calculation checkedEvaluate log_(√2)(4) + log_(1/2)(16) − log₄(32).
A: −2.5
The three logarithms are 4, −4 and 5/2 respectively: (√2)⁴ = 4, (1/2)⁻⁴ = 16 and 4^(5/2) = 32. Thus the expression equals −2.5.
AI-reviewed · calculation checkedLet U be the even integers strictly between 0 and 30, P the multiples of 6 in U and Q the multiples of 4 in U. Find the complement of P ∪ Q in U.
C: {2,10,14,22,26}
Remove from U every integer divisible by 6 or 4. The remaining even integers are 2, 10, 14, 22 and 26, which form the required complement.
AI-reviewed · calculation checkedEvaluate [(1/0.06) ÷ (1/0.042)]⁻¹, correct to two decimal places.
D: 1.43
Dividing the reciprocals gives 0.042/0.06 = 0.7. The exponent −1 takes the reciprocal, giving 1/0.7 = 1.428571… , or 1.43 to two decimal places.
AI-reviewed · calculation checkedFactorise completely 4abx − 2axy − 12b²x + 6bxy.
D: 2x(a − 3b)(2b − y)
Factor out 2x to obtain 2x[2ab − ay − 6b² + 3by]. Grouping the bracket gives a(2b − y) − 3b(2b − y), yielding the stated factorisation.
AI-reviewed · calculation checkedThe graphs of y = x² + 4 and a straight line PQ are used to solve x² − 3x + 2 = 0. Find the equation of PQ.
A: y = 3x + 2
At intersections with y = 3x + 2, equating the y values gives x² + 4 = 3x + 2. Rearrangement produces exactly x² − 3x + 2 = 0.
AI-reviewed · calculation checkedSolve 3x − 5y + 5 = 0 and 4x − 7y + 8 = 0 for x and y respectively.
C: 5, 4
The equations give 3x − 5y = −5 and 4x − 7y = −8. Multiplying by 4 and 3 respectively and subtracting gives y = 4; substitution then gives x = 5.
AI-reviewed · calculation checkedWhich inequalities describe triangle OPQ, including its boundary?
Three consecutive terms of a geometric progression are n − 2, n and n + 3. Find its common ratio.
B: 3/2
Consecutive geometric terms satisfy n² = (n − 2)(n + 3) = n² + n − 6. Hence n = 6 and the terms are 4, 6 and 9, with ratio 3/2.
AI-reviewed · calculation checkedA point P lies on one chosen side of the line through distinct fixed points X and Y, with angle XPY = 90°. Which option most precisely describes its locus, excluding X and Y?
C: A semicircle
By the angle-in-a-semicircle theorem, P lies on the circle with diameter XY. Restricting P to one side of XY selects one semicircle; its endpoints are excluded because the angle is undefined there.
AI-reviewed · calculation checkedFind the locus of P(x,y) equidistant from Q(0,0) and R(2,1).
C: 4x + 2y = 5
Equating squared distances gives x² + y² = (x − 2)² + (y − 1)². Cancelling x² and y² leaves 4x + 2y = 5.
AI-reviewed · calculation checkedAn arc subtends 30° at a point on the circumference of a circle of radius 21 cm. Find its length, using π = 22/7.
C: 22 cm
The corresponding angle at the centre is twice the circumference angle, giving 60°. Arc length is (60/360) × 2π × 21 = 7π = 22 cm.
AI-reviewed · calculation checkedA circle has radius 7 cm. Angle XZY is 45°. Find the shaded minor segment cut off by chord XY, using π = 22/7.
Differentiate y = 3sin(−4x).
A: −12cos(−4x)
The derivative of sin u is cos u multiplied by du/dx. Here u = −4x, so dy/dx = 3cos(−4x) × (−4) = −12cos(−4x).
AI-reviewed · calculation checkedA pie chart has four sectors, three measuring 45°, 90° and 135°. If the smallest sector represents ₦28, how much does the largest represent?
D: ₦84
The fourth angle is 360° − 45° − 90° − 135° = 90°. The largest sector is 135°, three times the smallest 45° sector, so its value is 3 × ₦28 = ₦84.
AI-reviewed · calculation checkedTwo fair six-sided dice are thrown independently. Find the probability that their sum is divisible by three.
B: 1/3
There are 36 equally likely ordered outcomes. Sums 3, 6, 9 and 12 have 2, 5, 4 and 1 outcomes respectively, giving 12 favourable outcomes and probability 12/36 = 1/3.
AI-reviewed · calculation checkedBy how much is the mean of 30, 56, 31, 55, 43 and 44 less than their median? Give the answer to two decimal places.
D: 0.33
In order, the numbers are 30, 31, 43, 44, 55 and 56, so the median is (43 + 44)/2 = 43.5. The mean is 259/6 = 43.1666… . Their difference rounds to 0.33.
AI-reviewed · calculation checkedSimplify (√0.7 + √70)².
C: 84.7
Since √70 = 10√0.7, the bracket equals 11√0.7. Squaring gives 121 × 0.7 = 84.7. Both radicals are nonnegative real numbers.
AI-reviewed · calculation checkedEvaluate (0.21 × 0.072 × 0.0054)/(0.006 × 1.68 × 0.063), correct to four significant figures.
A: 0.1286
Multiplying the numerator and denominator gives a quotient of 0.128571428… . Its first four significant digits are 1, 2, 8 and 5; the next digit rounds this to 0.1286.
AI-reviewed · calculation checkedSimplify 52.4 − 5.7 − 3.45 − 1.75.
C: 41.5
The quantities being subtracted total 5.7 + 3.45 + 1.75 = 10.9. Subtracting this from 52.4 gives 41.5. Align decimal places when adding or subtracting.
AI-reviewed · calculation checkedTwo concentric circles have radii r and R, with r = 2R/5. Find the area shaded between the circles.
XZ is a diameter of a circle of radius 15/2 cm. Point Y lies on the circle and XY = 12 cm. Find the area of triangle XYZ.
B: 54 cm²
The angle at Y is 90° because it subtends a diameter. XZ = 15 cm, so YZ = √(15² − 12²) = 9 cm. Area = ½ × 12 × 9 = 54 cm².
AI-reviewed · calculation checkedFind the midpoint of the segment joining the x- and y-intercepts of 2y = 4x − 8.
D: (1,−2)
Setting y = 0 gives the x-intercept (2,0). Setting x = 0 gives the y-intercept (0,−4). Averaging the coordinates gives midpoint (1,−2).
AI-reviewed · calculation checkedIf tan θ = 4/3, calculate sin²θ − cos²θ.
A: 7/25
Using tan²θ = 16/9 and sin²θ + cos²θ = 1 gives sin²θ = 16/25 and cos²θ = 9/25. Their difference is 7/25.
AI-reviewed · calculation checkedWhich option precisely identifies the complete locus of points P equidistant from distinct fixed points S and T?
D: The perpendicular bisector of ST
The required points lie on the perpendicular through the midpoint of ST. This is the perpendicular bisector, whose every point has equal distances to S and T.
AI-reviewed · calculation checkedTriangle PQR has angle PQR = 50° and an exterior angle at R of 128°. Classify the triangle.
A: A scalene triangle
The interior angle at R is 180° − 128° = 52°. The third angle is 180° − 50° − 52° = 78°. All three angles differ, so all three sides differ and the triangle is scalene.
AI-reviewed · calculation checkedFind the locus of points equidistant from the parallel lines x = 1 and x = 7.
D: x = 4
The equidistant locus is the parallel line halfway between them. Its x-coordinate is (1 + 7)/2 = 4, so the equation is x = 4.
AI-reviewed · calculation checkedIn a class of 80 students, 30 offer Music, 40 offer History and 20 offer neither. Find the probability that a randomly chosen student offers Music only.
B: 0.25
The union contains 80 − 20 = 60 students. Therefore the overlap is 30 + 40 − 60 = 10. Music only contains 30 − 10 = 20 students, giving 20/80 = 0.25.
AI-reviewed · calculation checkedCalculate the mean absolute deviation about the mean of 7, 3, 14, 9, 7 and 8.
B: 2⅓
The mean is 48/6 = 8. Absolute deviations are 1, 5, 6, 1, 1 and 0, totalling 14. Their average is 14/6 = 7/3 = 2⅓.
AI-reviewed · calculation checkedFind the maximum value of y = 1 − 2x − 3x².
B: 4/3
Completing the square gives y = 4/3 − 3(x + 1/3)². The squared term is nonnegative, so the largest possible value is 4/3, attained at x = −1/3.
AI-reviewed · calculation checkedThe ninth term of an arithmetic progression is five times its fifth term. Find the relationship between first term a and common difference d.
B: a + 3d = 0
The condition is a + 8d = 5(a + 4d). Rearranging gives 4a + 12d = 0; dividing by four gives a + 3d = 0.
AI-reviewed · calculation checkedTime is inversely proportional to the number of workers. If 45 workers complete a job in 5 days, how long will 25 workers take?
B: 9 days
The job requires a constant 45 × 5 = 225 worker-days. With 25 workers, time is 225/25 = 9 days, assuming the stated inverse relationship.
AI-reviewed · calculation checkedThe boundary of px + qy < 4 crosses the axes at (−4,0) and (0,2). Find p and q.
C: p = −1, q = 2
Points on the boundary satisfy px + qy = 4. At (−4,0), −4p = 4, giving p = −1. At (0,2), 2q = 4, giving q = 2.
AI-reviewed · calculation checkedFind the inverse function of f(x) = 3x + 4.
D: (x − 4)/3
Let y = 3x + 4. Solving for x gives x = (y − 4)/3. Interchanging the variable names gives f⁻¹(x) = (x − 4)/3.
AI-reviewed · calculation checkedSolve x³ − 5x² − x + 5 = 0.
D: 1, −1 or 5
Group the terms as x²(x − 5) − (x − 5) = (x − 5)(x² − 1). Factoring again gives (x − 5)(x − 1)(x + 1), so the roots are 5, 1 and −1.
AI-reviewed · calculation checkedFind the sum to infinity of 1 + 1/3 + 1/9 + 1/27 + … .
A: 3/2
This is a geometric series with first term 1 and common ratio 1/3. Since the ratio has magnitude below one, its sum is 1/(1 − 1/3) = 3/2.
AI-reviewed · calculation checkedEvaluate ∫sin(3x) dx.
B: −(1/3)cos(3x) + c
The derivative of cos(3x) is −3sin(3x). Multiplying by −1/3 cancels that factor, so an antiderivative is −(1/3)cos(3x) + c.
AI-reviewed · calculation checkedGiven dy/dx = 2x − 3 and y = 3 when x = 0, find y.
B: x² − 3x + 3
Integrating the derivative gives y = x² − 3x + C. Substitution of x = 0 and y = 3 gives C = 3, hence y = x² − 3x + 3.
AI-reviewed · calculation checkedDifferentiate y = sin²(5x) with respect to x.
C: 10sin(5x)cos(5x)
Apply the chain rule twice: the square contributes 2sin(5x), and differentiating sin(5x) contributes 5cos(5x). Their product is 10sin(5x)cos(5x).
AI-reviewed · calculation checkedEvaluate 21.05347 − 1.6324 × 0.43 to three decimal places.
B: 20.352
Apply multiplication before subtraction: 1.6324 × 0.43 = 0.701932. The difference is 20.351538, which rounds to 20.352 at three decimal places.
AI-reviewed · calculation checkedEvaluate (0.14² × 0.275)/(7 × 0.02), correct to three decimal places.
B: 0.039
The numerator is 0.0196 × 0.275 = 0.00539 and the denominator is 0.14. Their quotient is 0.0385, which rounds up to 0.039 at three decimal places.
AI-reviewed · calculation checkedGiven p = 1 + √2 and q = 1 − √2, evaluate (p² − q²)/(2pq).
C: −2√2
We have p + q = 2, p − q = 2√2 and pq = −1. Hence p² − q² = 4√2, and division by 2pq = −2 gives −2√2.
AI-reviewed · calculation checkedSolve m² + n² = 29 and m + n = 7. Give the ordered pairs (m,n).
D: (2,5) and (5,2)
Squaring the sum gives 49 = 29 + 2mn, so mn = 10. Thus m and n are roots of t² − 7t + 10 = 0, namely 2 and 5, in either order.
AI-reviewed · calculation checkedDivide a^(3x) − 26a^(2x) + 156a^x − 216 by a^(2x) − 24a^x + 108, where the divisor is nonzero and a > 0.
C: a^x − 2
Let t = a^x. The numerator equals (t − 2)(t² − 24t + 108), as expansion confirms. Cancelling the nonzero divisor gives t − 2 = a^x − 2.
AI-reviewed · calculation checkedFind all integer pairs (x,y) satisfying 3y + 5x ≤ 15, with 0 < y < 3 and x > 0.
C: (1,1), (1,2), (2,1)
The possible y values are 1 and 2. For y = 1, x can be 1 or 2; for y = 2, only x = 1 works. These give the three listed pairs.
AI-reviewed · calculation checkedFor K = [[2,1],[3,4]], find K² + K + I, where I is the 2 × 2 identity matrix.
B: [[10,7],[21,24]]
Matrix multiplication gives K² = [[7,6],[18,19]]. Adding K and the identity matrix gives [[10,7],[21,24]]. The identity contributes one only to each diagonal entry.
AI-reviewed · calculation checkedA chord ST is equal in length to the radius r of its circle. Find the length of the minor arc ST.
B: πr/3
The two radii and the chord form an equilateral triangle, so the minor central angle is 60° = π/3 radians. Arc length rθ is therefore πr/3.
AI-reviewed · calculation checkedFind the complete locus of a point whose perpendicular distance from y = 4 is a constant k ≥ 0.
D: y = 4 ± k
Distance from a horizontal line is the absolute difference of y coordinates. Thus |y − 4| = k, giving the two parallel lines y = 4 + k and y = 4 − k.
AI-reviewed · calculation checkedP(−6,1) and Q(6,6) are the endpoints of a circle’s diameter. Calculate its radius.
B: 6.5 units
The diameter length is √[(6 + 6)² + (6 − 1)²] = √169 = 13 units. The radius is half the diameter, giving 6.5 units.
AI-reviewed · calculation checkedDifferentiate (2x + 5)²(x − 4) with respect to x.
A: (2x + 5)(6x − 11)
The product rule gives 4(2x + 5)(x − 4) + (2x + 5)². Factoring out 2x + 5 leaves 4x − 16 + 2x + 5 = 6x − 11.
AI-reviewed · calculation checkedFind dV/dr for a sphere when its radius r = 1.
A: 4π
A sphere has volume V = 4πr³/3. Differentiating with respect to its radius gives dV/dr = 4πr². At r = 1, the rate is 4π.
AI-reviewed · calculation checkedFind the dimensions of the rectangle with greatest area for a fixed positive perimeter p.
A: A square of side p/4
If one side is x, the other is p/2 − x. The area x(p/2 − x) is a downward-opening quadratic with its maximum at x = p/4. Both sides are therefore p/4.
AI-reviewed · calculation checkedEvaluate ∫2(2x − 3)^(2/3) dx.
D: (3/5)(2x − 3)^(5/3) + k
Let u = 2x − 3, so du = 2 dx. The integral becomes ∫u^(2/3) du = (3/5)u^(5/3) + k. Substituting back gives the stated result.
AI-reviewed · calculation checkedCars counted by colour were: yellow 3, white 4, red 8, green 2, blue 6 and black 2. What fraction of the cars were yellow?
C: 3/25
The counts total 3 + 4 + 8 + 2 + 6 + 2 = 25 cars. Three were yellow, so the fraction is 3/25. Each colour count contributes once to the total.
AI-reviewed · calculation checkedScores 4, 7, 8, 11, 13 and 8 have respective frequencies 3, 5, 2, 7, 2 and 1. Find the mean score.
C: 8.7
Multiply each score by its frequency. The total score is 12 + 35 + 16 + 77 + 26 + 8 = 174 across 20 observations, giving mean 174/20 = 8.7.
AI-reviewed · calculation checkedIf Q₁, Q₂ and Q₃ denote the lower quartile, median and upper quartile of a distribution, which expression gives its interquartile range?
A: Q₃ − Q₁
The interquartile range measures the span of the middle half of the data. Subtract the lower quartile from the upper quartile: IQR = Q₃ − Q₁.
AI-reviewed · calculation checkedA bowl contains 1 blue, 2 black, 4 yellow, 5 white and 3 brown beads. A bead is chosen uniformly at random. Find the probability it is blue or white.
C: 2/5
There are 15 beads in total. The blue and white groups contain 1 + 5 = 6 beads, so their combined probability is 6/15 = 2/5.
AI-reviewed · calculation checkedA town has population 240000 in January 1998 and grows by 2% annually. What is its population in January 2000?
B: 249696
Two annual increases occur between the dates. Compound growth gives 240000×1.02² = 249696, rather than applying the increase just once.
AI-reviewed · calculation checkedEvaluate [(2.813 × 10⁻³) × 1.063]/(5.637 × 10⁻²), first rounding each number to two significant figures, then expressing the answer to two significant figures.
B: 0.055
The rounded inputs are 2.8 × 10⁻³, 1.1 and 5.6 × 10⁻². Their quotient is 0.00308/0.056 = 0.055, already expressed to two significant figures.
AI-reviewed · calculation checkedThe third term of an arithmetic progression is 4x − 2y and the ninth is 10x − 8y. Find its common difference.
C: x − y
There are six common differences between the third and ninth terms. Thus d = [(10x − 8y) − (4x − 2y)]/6 = (6x − 6y)/6 = x − y.
AI-reviewed · calculation checkedFor p ≠ 1, find the inverse of p under p * q = p + q − pq, where the identity is zero.
C: p/(p − 1)
The inverse q satisfies p + q − pq = 0. Rearranging gives q(1 − p) = −p, so q = p/(p − 1). The denominator requires p ≠ 1.
AI-reviewed · calculation checkedIf x − 1, x + 1 and x − 2 are factors of ax³ + bx² + cx − 1, find a, b and c respectively.
A: −1/2, 1, 1/2
The product of the factors is x³ − 2x² − x + 2. Its multiplier must be −1/2 to give constant −1, yielding coefficients a = −1/2, b = 1 and c = 1/2.
AI-reviewed · calculation checkedSolve 2 − x > x².
D: −2 < x < 1
Move all terms to one side to get x² + x − 2 < 0, or (x + 2)(x − 1) < 0. The product is negative strictly between its roots, so −2 < x < 1.
AI-reviewed · calculation checkedIf α and β are roots of 3x² + 5x − 2 = 0, find 1/α + 1/β.
D: 5/2
By the root sum and product formulas, α + β = −5/3 and αβ = −2/3. Therefore 1/α + 1/β = (α + β)/(αβ) = 5/2.
AI-reviewed · calculation checkedAn equilateral triangle of side √3 cm is inscribed in a circle. Find the circle’s radius.
C: 1 cm
For an equilateral triangle of side a, the circumradius is a/√3. This also follows by taking two-thirds of the altitude a√3/2. Here the radius is 1 cm.
AI-reviewed · calculation checkedThe lines 3y = 4x − 1 and Ky = x + 3 are perpendicular. Find K.
A: −4/3
Their slopes are 4/3 and 1/K. Perpendicular nonvertical lines have slope product −1, so 4/(3K) = −1 and K = −4/3.
AI-reviewed · calculation checkedIn a circle centred at O, diameter FH meets chord EG at right angles at N. If NH = 8 cm and EG = 24 cm, find FH.
C: 26 cm
The perpendicular diameter bisects EG, so EN = NG = 12 cm. Intersecting chords give FN × NH = EN × NG = 144. Thus FN = 18 cm and FH = 18 + 8 = 26 cm.
AI-reviewed · calculation checkedFor distinct fixed points P and Q, which option most precisely describes the complete locus of X such that XP = XQ?
D: The perpendicular bisector of PQ
Points equidistant from two fixed points form their perpendicular bisector. Although this is a straight line, the perpendicular-bisector description uniquely identifies which line is required.
AI-reviewed · calculation checkedThe diagram shows y = x². Find the shaded area.
The expression ax² + bx + c equals 5 when x = 1, and its derivative is 2x + 1. Find a, b and c respectively.
D: 1, 1, 3
Comparing 2ax + b with 2x + 1 gives a = 1 and b = 1. Substituting x = 1 into the original expression gives 1 + 1 + c = 5, so c = 3.
AI-reviewed · calculation checkedHow many distinct arrangements can be made using all the letters of MATHEMATICS?
C: 11!/(2! × 2! × 2!)
There are eleven letters. M, A and T each appear twice, while all other letters appear once. Divide 11! by 2! for each repeated pair, giving 11!/(2! × 2! × 2!).
AI-reviewed · calculation checkedAn integer is chosen uniformly at random from 1 through 20. Find the probability that it is not a multiple of 4.
A: 3/4
The multiples of 4 are 4, 8, 12, 16 and 20, making five excluded outcomes. The remaining fifteen out of twenty integers give probability 15/20 = 3/4.
AI-reviewed · calculation checkedFind the population variance of x, 2x, 3x, 4x and 5x.
B: 2x²
The mean is 3x. The squared deviations are 4x², x², 0, x² and 4x². Their sum is 10x², so dividing by five gives population variance 2x².
AI-reviewed · calculation checkedPupils’ ages in years are grouped as 10–12, 13–15, 16–18, 19–20 and 21–23, with frequencies 6, 14, 15, 10 and 5 respectively. What percentage are over 15 but under 21?
C: 50%
The relevant groups are 16–18 and 19–20, containing 15 + 10 = 25 pupils. There are 6 + 14 + 15 + 10 + 5 = 50 pupils altogether, so the percentage is 50%.
AI-reviewed · calculation checkedSolve √2/(x + √2) = 1/(x − √2), excluding zero denominators.
A: 3√2 + 4
Cross-multiplying gives√2x − 2 = x + √2, so x = (2 + √2)/(√2 − 1). Rationalising gives 4 + 3√2.
AI-reviewed · calculation checkedIf log₈10 = x, express log₈5 in terms of x.
C: x − 1/3
Since 5 = 10/2, log₈5 = log₈10 − log₈2. Because 8^(1/3) = 2, the result is x − 1/3. The logarithm of a quotient is the difference of its logarithms.
AI-reviewed · calculation checkedA convergent geometric progression has first term twice its common ratio and infinite sum 8. Find the sum of its first two terms.
C: 72/25
Writing a = 2r gives 2r/(1 − r) = 8, so r = 4/5 and a = 8/5. The first two terms sum to a(1 + r) = 72/25.
AI-reviewed · calculation checkedDivide 4x³ − 3x + 1 by 2x − 1.
D: 2x² + x − 1
The quotient is 2x² + x − 1. Multiplying by 2x − 1 gives 4x³ − 3x + 1, with no remainder. Each coefficient agrees with the original cubic polynomial.
AI-reviewed · calculation checkedTwo real numbers have positive difference p and sum 2p. Find the larger number.
B: 3p/2
If the larger number is L and the smaller is S, then L − S = p and L + S = 2p. Adding gives 2L = 3p, so L = 3p/2.
AI-reviewed · calculation checkedFind the locus of P(x,y) such that PV = PW, where V = (1,1) and W = (3,5).
D: x + 2y = 8
Equating squared distances gives (x − 1)² + (y − 1)² = (x − 3)² + (y − 5)². Cancelling and simplifying gives 4x + 8y = 32, hence x + 2y = 8.
AI-reviewed · calculation checkedFind the tangent of the acute angle between 2x + y = 3 and 3x − 2y = 5.
C: 7/4
The slopes are −2 and 3/2. For the acute angle, tan θ = |(m₂ − m₁)/(1 + m₁m₂)| = |(7/2)/(−2)| = 7/4.
AI-reviewed · calculation checkedFrom P, the bearings of Q and R are N67°W and N23°E. From Q, R is on bearing N68°E. If PQ = 150 m, find PR.
C: 150 m
The angle at P is 67° + 23° = 90°. The reverse bearing QP is 113°, so angle PQR is 113° − 68° = 45°. The remaining angle is also 45°, giving PR = PQ = 150 m.
AI-reviewed · calculation checkedAt what value of x does y = x³ − x have a local minimum?
C: √3/3
Stationary points satisfy 3x² − 1 = 0. The second derivative is 6x, which is positive at x = 1/√3. Thus the local minimum occurs at √3/3.
AI-reviewed · calculation checkedEvaluate [(1/0.03) ÷ (1/0.024)]⁻¹ to two decimal places.
B: 1.25
The ratio inside the brackets is 0.024/0.03 = 0.8. Raising it to−1 takes its reciprocal,1/0.8 = 1.25.
AI-reviewed · calculation checkedThe polynomial pm² + qm + 1 leaves remainder 2 on division by m − 1 and remainder 4 on division by m + 1. Find p and q.
A: 2, −1
The remainder theorem gives p + q + 1 = 2 and p − q + 1 = 4. Thus p + q = 1 and p − q = 3, yielding p = 2 and q = −1.
AI-reviewed · calculation checkedFactorise r² − r(2p + q) + 2pq.
C: (r − q)(r − 2p)
The two terms q and 2p have sum 2p + q and product 2pq. Therefore the expression factors as(r − q)(r − 2p).
AI-reviewed · calculation checkedSolve √x − √(x − 2) − 1 = 0 over the real numbers.
D: 9/4
Rearrange to√x = 1 + √(x − 2). Squaring gives 1 = 2√(x − 2), so x = 9/4. Substitution gives 3/2 − 1/2 − 1 = 0.
AI-reviewed · calculation checkedDivide 2x³ + 11x² + 17x + 6 by 2x + 1.
A: x² + 5x + 6
Polynomial division gives x² + 5x + 6. Multiplication confirms(2x + 1)(x² + 5x + 6) = 2x³ + 11x² + 17x + 6.
AI-reviewed · calculation checkedResolve(11x + 2)/(6x² − x − 1) into partial fractions, excluding denominator zeros.
D: 1/(3x + 1) + 3/(2x − 1)
The denominator is(3x + 1)(2x − 1). Combining 1/(3x + 1) and 3/(2x − 1) gives numerator(2x − 1) + 3(3x + 1) = 11x + 2.
AI-reviewed · calculation checkedFor positive real x, solve 1/(3x) + 1/2 > 1/(4x).
B: x > 0
Subtracting the right side gives 1/(12x) + 1/2 > 0. Both terms are positive whenever x > 0, so every positive x satisfies the inequality.
AI-reviewed · calculation checkedWhich inequalities describe the shaded region, including the boundary lines?
If p + 1,2p − 10 and 1 − 4p² are consecutive arithmetic-progression terms, find p.
C: −11/4, 2
Twice the middle term equals the sum of the outer terms:4p − 20 = p + 2 − 4p². Thus 4p² + 3p − 22 = 0, giving p = 2 or−11/4.
AI-reviewed · calculation checkedThe first three terms of a convergent geometric progression sum to half its nonzero infinite sum. Find its positive common ratio.
D: 1/∛2
For common ratio r, the first-three-term sum is(1 − r³) times the infinite sum. Hence 1 − r³ = 1/2, so r = ∛(1/2) = 1/∛2.
AI-reviewed · calculation checkedPQST is a parallelogram with TS = 10 cm. T,S,R lie in that order on a straight line, SR = 8 cm, and triangle QRS has area 20 cm². Find trapezium PQRT’s area.
C: 70 cm²
Triangle QRS has height 2×20/8 = 5 cm above the baseline. The parallelogram shares that height and has area 10×5 = 50 cm². Adding the triangle gives 70 cm².
AI-reviewed · calculation checkedDistinct points S and T lie on a circle of radius 8 cm centred at N. If ST = SN, find the area of triangle STN.
B: 16√3 cm²
Both SN and TN are radii of length 8 cm, and ST = SN. The triangle is equilateral, so its area is(√3/4)×8² = 16√3 cm².
AI-reviewed · calculation checkedQ,T,R lie in that order on a straight line. PQ = 15, PT = 8, PR = 10 and angle PQT = 30°. Find sin(angle PTR).
D: 15/16
In triangle PQT, the sine rule gives sin(angle PTQ)/15 = sin 30°/8. The angles PTQ and PTR are supplementary, so they have the same sine,15/16.
AI-reviewed · calculation checkedFor 0° ≤ x ≤ 180°, at what x does 6sin(2x − 25)° attain its maximum?
C: 57.5°
Sine reaches 1 when its angle is 90° plus a full turn. In the given range,2x − 25 = 90 gives the only maximum, x = 57.5°.
AI-reviewed · calculation checkedFrom a150 m mast, huts due east and west are seen at depression angles 60° and 45°. Find their separation on level ground.
B: 50(3 + √3) m
The horizontal distances are 150/tan 60° = 50√3 m and 150/tan 45° = 150 m. They lie on opposite sides, so the separation is 50(3 + √3) m.
AI-reviewed · calculation checkedFor y = 243(4x + 5)⁻², find dy/dx at x = 1.
A: −8/3
The chain rule gives dy/dx = −1944(4x + 5)⁻³. At x = 1 this is−1944/9³ = −1944/729 = −8/3. The factor 4 comes from differentiating the inner expression.
AI-reviewed · calculation checkedDifferentiate x/cos x with respect to x, where cos x ≠ 0.
D: sec x + x sec x tan x
Write the function as x sec x. The product rule gives 1×sec x + x×sec x tan x, or sec x + x sec x tan x.
AI-reviewed · calculation checkedFind the curve through(2,5) with derivative 6x − 5.
D: 3x² − 5x + 3
Integrating gives y = 3x² − 5x + C. At(2,5),5 = 12 − 10 + C, so C = 3 and y = 3x² − 5x + 3.
AI-reviewed · calculation checkedClass intervals 5–9,10–14,15–19,20–24 have frequencies 17,32,25,24. Estimate the grouped mode to one decimal place using continuous class boundaries.
C: 12.9
The modal class is 10–14. Its lower boundary is 9.5 and width 5, so mode ≈ 9.5 + [(32 − 17)/(64 − 17 − 25)]×5 = 12.9.
AI-reviewed · calculation checkedMarks 0 through 10 have frequencies 1,3,4,7,10,8,7,9,8,2,1 respectively. If the pass mark is 5, what percentage failed, to one decimal place?
C: 41.7%
The frequencies total 60 pupils. Marks below 5 account for 1 + 3 + 4 + 7 + 10 = 25 pupils, giving(25/60)×100 ≈ 41.7%.
AI-reviewed · calculation checkedOnly one team can win gold. If teams X and Y have winning probabilities 2/5 and 1/3, find the probability that either wins.
C: 11/15
The two winning outcomes are mutually exclusive because there is one gold winner. Add their probabilities:2/5 + 1/3 = 6/15 + 5/15 = 11/15.
AI-reviewed · calculation checkedChoose x and y independently and uniformly from{1,2,3,4}. Find the probability that xy ≤ 6.
A: 5/8
There are 16 equally likely ordered pairs. For x = 1,2,3,4, the numbers of allowed y values are 4,3,2,1. Thus 10 pairs work, giving 10/16 = 5/8.
AI-reviewed · calculation checkedExpress (0.006)³ + (0.004)³ in standard form.
C: 2.8 × 10⁻⁷
The cubes are 2.16×10⁻⁷ and0.64×10⁻⁷. Their sum is2.80×10⁻⁷, which is2.8×10⁻⁷ in standard form.
AI-reviewed · calculation checkedGiven logₐ2 = 0.693 and logₐ3 = 1.097, find logₐ13.5.
C: 2.598
Since13.5 = 27/2 = 3³/2, logarithm rules give logₐ13.5 = 3logₐ3 − logₐ2 = 3(1.097) − 0.693 = 2.598.
AI-reviewed · calculation checkedSimplify log₂96 − 2log₂6.
B: 3 − log₂3
Combining the logarithms gives log₂(96/36) = log₂(8/3). This equals log₂8 − log₂3 = 3 − log₂3.
AI-reviewed · calculation checkedIf 8^(x/2) = 2^(3/8) × 4^(3/4), find x.
D: 5/4
Writing everything in base2 gives 2^(3x/2) = 2^(3/8 + 3/2) = 2^(15/8). Thus3x/2 = 15/8 and x = 5/4.
AI-reviewed · calculation checkedAt what simple interest rate per year does ₦1000 grow to ₦1240 in 3 years?
B: 8%
The interest earned is ₦240. Using I = P rt/100 gives r = 100×240/(1000×3) = 8% per year. Simple interest uses the original principal throughout.
AI-reviewed · calculation checkedSolve 2/x − 3/y = 2 and 4/x + 3/y = 10.
B: x = 1/2, y = 3/2
Adding the equations gives 6/x = 12, so x = 1/2. Substituting back gives 4 − 3/y = 2, hence y = 3/2.
AI-reviewed · calculation checkedFind the minimum of x² − 3x + 2 over the real numbers.
A: −1/4
Completing the square gives(x − 3/2)² − 1/4. The squared term is nonnegative, so the minimum is−1/4 at x = 3/2.
AI-reviewed · calculation checkedMake f the subject of t = √[v/(1/f + 1/g)], where the denominators are nonzero.
B: gt²/(gv − t²)
Squaring gives t² = vfg/(g + f). Rearranging yields gt² = f(gv − t²), so f = gt²/(gv − t²).
AI-reviewed · calculation checkedLet f(x) = 2x + 4 and g(x) = 6x + 7 > 0. Solve f(x)/g(x) < 1.
C: x > −3/4
The positive denominator permits multiplication without reversing the sign:2x + 4 < 6x + 7. Thus x > −3/4, which also satisfies 6x + 7 > 0.
AI-reviewed · calculation checkedSolve 12x² < x + 1.
A: −1/4 < x < 1/3
Rearrange to 12x² − x − 1 < 0 and factor as(4x + 1)(3x − 1) < 0. The product is negative between−1/4 and 1/3.
AI-reviewed · calculation checkedIf Sₙ = n² − 1 is the sum of the first n terms, find the nth term for n ≥ 2.
D: 2n − 1
For n ≥ 2, subtract successive partial sums: aₙ = Sₙ − Sₙ₋₁ = n² − 1 − [(n − 1)² − 1] = 2n − 1. The first term is separately S₁ = 0.
AI-reviewed · calculation checkedThe nth term is 3^(1−n), starting at n = 1. Find the sum of the first three terms.
A: 13/9
The first three terms are 1,1/3 and 1/9. Expressing them over denominator 9 gives 9/9 + 3/9 + 1/9 = 13/9.
AI-reviewed · calculation checkedA sector has radius 6 cm and central angle 70°. Find its area using π = 22/7.
A: 22 cm²
The area is the fraction 70/360 of the full circle: (70/360)×(22/7)×6² = 22 cm². The central angle determines the fraction of the circle included.
AI-reviewed · calculation checkedA right prism is 11 cm long. Its trapezoidal cross-section has parallel sides 8 cm and 10 cm separated perpendicularly by 5 cm. Find its volume.
D: 495 cm³
The trapezoid area is½(8 + 10)×5 = 45 cm². Multiplying by the prism length gives 45×11 = 495 cm³.
AI-reviewed · calculation checkedA line makes an angle 135° with the positive x-axis and passes through(2,3). Find its equation.
C: x + y = 5
The gradient is tan 135° = −1. Using the point(2,3), y − 3 = −(x − 2), which rearranges to x + y = 5.
AI-reviewed · calculation checkedIn triangle XYZ, angle XYZ = 60°, XY = 3 cm and YZ = 4 cm. Find XZ.
B: √13 cm
The cosine rule gives XZ² = 3² + 4² − 2×3×4×cos 60° = 25 − 12 = 13. Therefore XZ = √13 cm. The positive square root is required because a side length is positive.
AI-reviewed · calculation checkedIn triangle XYZ, XY = 5 cm and XZ = 2 cm. Extending XZ through Z to E gives exterior angle YZE = 150°. If angle XYZ = θ, find sinθ.
D: 1/5
The interior angle XZY is 180° − 150° = 30°. By the sine rule, sinθ/2 = sin 30°/5, so sinθ = (2/5)×(1/2) = 1/5.
AI-reviewed · calculation checkedDifferentiate (6x³ − 5x² + 1)/(3x²), for x ≠ 0.
C: 2 − 2/(3x³)
Divide each numerator term by 3x² to get 2x − 5/3 + (1/3)x⁻². Differentiating gives 2 − (2/3)x⁻³.
AI-reviewed · calculation checkedFind the gradient of y = 2√x − 1/x at x = 1.
C: 2
Differentiating gives dy/dx = 1/√x + 1/x². At x = 1, the gradient is 1 + 1 = 2. The derivative of−x⁻¹ contributes the positive term x⁻².
AI-reviewed · calculation checkedIntegrate 1/x + cos x with respect to x, for x > 0.
B: ln x + sin x + k
For x > 0, the integral of 1/x is ln x and the integral of cos x is sin x. Adding the constant gives ln x + sin x + k.
AI-reviewed · calculation checkedA₦6000 monthly income is divided in a pie chart: Housing 69°, Transport 60°, Meal 61°, Others 50° and Basic the remainder. Find the Basic amount.
A: ₦2000
The listed sectors total 240°, leaving 120° for Basic. This is 120/360 = 1/3 of the income, so the amount is ₦6000/3 = ₦2000.
AI-reviewed · calculation checkedA basket contains 6 grapes,11 bananas and 13 oranges. One fruit is selected uniformly at random. Find the probability of a grape or banana.
A: 17/30
There are 30 fruits altogether and 17 are grapes or bananas. The favourable outcomes are disjoint, so the probability is 17/30.
AI-reviewed · calculation checkedCalculate 3310₅ − 1442₅.
A: 1313₅
In base ten, 3310₅ = 455 and 1442₅ = 247. Their difference is 208, which is 1×125 + 3×25 + 1×5 + 3 = 1313₅.
AI-reviewed · calculation checkedRound 3.1415926 to five decimal places.
B: 3.14159
The first five decimal digits are 14159. The sixth digit is 2, so the fifth digit stays unchanged and the rounded value is 3.14159.
AI-reviewed · calculation checkedA notebook is 15 cm long but is measured as 16.8 cm. Find the percentage error to two significant figures.
A: 12%
The absolute error is 16.8 − 15 = 1.8 cm. Divide by the true length and multiply by 100: (1.8 ÷ 15)×100 = 12%.
AI-reviewed · calculation checkedExpress 0.0014 × 0.011 in standard form.
D: 1.54 × 10⁻⁵
Multiplying gives 0.0000154. Moving the decimal point five places gives 1.54 × 10⁻⁵, with the coefficient between 1 and 10.
AI-reviewed · calculation checkedEvaluate (81^(3/4) − 27^(1/3))/(3 × 2³).
B: 1
Since 81 = 3⁴ and 27 = 3³, the numerator is 27 − 3 = 24. The denominator is 3×8 = 24, so the quotient is 1.
AI-reviewed · calculation checkedEvaluate 16^(3/2) + log₁₀(0.0001) + log₂(32).
D: 65.00
The three terms are 64, −4 and 5 respectively. Adding them gives 64 − 4 + 5 = 65, matching the value 65.00.
AI-reviewed · calculation checkedSimplify (√12 − √3)/(√12 + √3).
A: 1/3
Since √12 = 2√3, the numerator is √3 and the denominator is 3√3. Cancelling the common nonzero factor gives 1/3.
AI-reviewed · calculation checkedFour players belong to both an 11-player cricket team and a 14-player rugby team. How many belong to at least one team?
B: 21
Adding the team sizes counts each shared player twice. Subtract the four overlaps once: 11 + 14 − 4 = 21 players.
AI-reviewed · calculation checkedIf S = {x : x² = 9 and x > 4}, what is S?
C: ∅
The real solutions of x² = 9 are −3 and 3. Neither is greater than 4, so there are no elements satisfying both conditions.
AI-reviewed · calculation checkedAt which points does y = x² − 5x + 6 cross the x-axis?
D: (2, 0), (3, 0)
Set y = 0 and factor x² − 5x + 6 as (x − 2)(x − 3). Hence x = 2 or x = 3, giving the two intercepts (2, 0) and (3, 0).
AI-reviewed · calculation checkedFactorise abx² + 6y − 3ax − 2byx completely.
A: (ax − 2y)(bx − 3)
Group the terms as ax(bx − 3) − 2y(bx − 3). Factoring the common binomial gives (ax − 2y)(bx − 3).
AI-reviewed · calculation checkedSolve (x − 3)(x − 4) ≤ 0.
A: 3 ≤ x ≤ 4
The roots are 3 and 4. Between them the two factors have opposite signs, and at the roots their product is zero. Therefore 3 ≤ x ≤ 4.
AI-reviewed · calculation checkedSimplify (x² − 1)/(x³ + 2x² − x − 2), for x ≠ −2, −1, 1.
A: 1/(x + 2)
The denominator factors as (x + 2)(x² − 1). Cancelling x² − 1 gives 1/(x + 2), while retaining the original excluded values.
AI-reviewed · calculation checkedWhich binary rule is commutative for all integers a and b?
B: a*b = a + b − ab
Interchanging a and b leaves a + b − ab unchanged. The other listed expressions do not generally give the same value after that interchange.
AI-reviewed · calculation checkedA closed circular cylinder has diameter 14 cm and height 12 cm. Find its total surface area using π = 22/7.
A: 836 cm²
The radius is 7 cm. Including both circular ends, the area is 2πrh + 2πr² = 2×(22/7)×7×(12 + 7) = 836 cm².
AI-reviewed · calculation checkedTriangle PQR has PR = 10 cm, angle QPR = 30° and area 35 cm². Find PQ.
C: 14 cm
Using area = ½×PQ×PR×sin(angle P),35 = ½×PQ×10×½. Thus 35 = 2.5PQ and PQ = 14 cm. The 30° angle is included between PQ and PR.
AI-reviewed · calculation checkedIn triangle QRS, QS = 12 m, angle RQS = 30° and angle QRS = 45°. Find RS.
C: 6√2 m
By the sine rule, RS/sin 30° = 12/sin 45°. Hence RS = 12×(1/2)/(√2/2) = 6√2 m. Each side is paired with its opposite angle.
AI-reviewed · calculation checkedDifferentiate cosec x with respect to x, with angles in radians.
B: −cot x cosec x
Since cosec x = 1/sin x, differentiation gives −cos x/sin²x. This is −(cos x/sin x)(1/sin x) = −cot x cosec x.
AI-reviewed · calculation checkedGiven dy/dx = 4x − 3 and y = 5 at x = 2, find y.
B: 2x² − 3x + 3
Integrating gives y = 2x² − 3x + C. Substituting x = 2 and y = 5 gives 5 = 8 − 6 + C, so C = 3.
AI-reviewed · calculation checkedAges 13,14,15,16,17 have frequencies 3,10,30,42,15. What pie-chart angle represents the 15-year-olds?
D: 108°
The frequencies total 100 students. The 30 students aged 15 represent 30/100 of the circle, giving(30/100)×360° = 108°.
AI-reviewed · calculation checkedIn a class pie chart, Economics occupies 150°, History 90° and French 90°; C.R.K. occupies the remainder. If30 students take French, how many take C.R.K.?
C: 10
C.R.K. occupies 360° − 150° − 90° − 90° = 30°. This is one-third of the French sector, so the number is 30÷3 = 10.
AI-reviewed · calculation checkedFind the population variance of the scores 1,2,3,4,5.
C: 2.0
The mean is 3. The squared deviations are 4,1,0,1,4, totalling 10. Dividing by the five scores gives population variance 2.0.
AI-reviewed · calculation checkedGiven √2 ≈ 1.414, estimate 1/√2 to three decimal places.
D: 0.707
Rationalising gives 1/√2 = √2/2. Using the supplied approximation, 1.414 ÷ 2 = 0.707, to three decimal places.
AI-reviewed · calculation checkedFor A ⊆ B in a universal set E, simplify A ∩ (A ∩ B)′.
B: ∅
Because A is a subset of B, A ∩ B = A. Therefore the expression is A ∩ A′, which has no elements and equals the empty set.
AI-reviewed · calculation checkedFactorise a²x − b²y − b²x + a²y completely.
D: (x + y)(a − b)(a + b)
Grouping gives a²(x + y) − b²(x + y) = (x + y)(a² − b²). Factoring the difference of squares gives (x + y)(a − b)(a + b).
AI-reviewed · calculation checkedFind p and q if (x − 1) and (x − 3) are factors of px³ + qx² + 11x − 6.
B: 1, −6
Substituting x = 1 gives p + q = −5. Substituting x = 3 gives 27p + 9q = −27. Solving gives p = 1 and q = −6.
AI-reviewed · calculation checkedWhich equation represents the curve shown?
Solve 1/x > 2 over the real numbers.
C: 0 < x < 1/2
For negative x, 1/x is negative and cannot exceed 2. For positive x, multiplying by x gives 1 > 2x, so 0 < x < 1/2.
AI-reviewed · calculation checkedWhich inequality represents the shaded region, including its solid boundary?
Find the nth term of 3, 6, 10, 15, 21, …, starting with n = 1.
C: (n + 1)(n + 2)/2
These are triangular numbers beginning with the second triangular number. Replacing the usual index by n + 1 gives (n + 1)(n + 2)/2.
AI-reviewed · calculation checkedThree angles of a quadrilateral are (3y − x − z)°, 3x° and (2z − 2y − x)°. Find the fourth.
A: (360 − x − y − z)°
The three given angles sum to x + y + z degrees. A quadrilateral has an angle sum of 360°, leaving (360 − x − y − z)°.
AI-reviewed · calculation checkedAn open wooden box has uniform thickness 2 cm and internal dimensions 50 cm × 36 cm × 20 cm. Find the volume of wood.
A: 11520 cm³
The outside dimensions are 54 cm × 40 cm × 22 cm because the top is open. Subtract the inner space: 54×40×22 − 50×36×20 = 11520 cm³.
AI-reviewed · calculation checkedFind the perpendicular height h of the triangle shown.
A conical frustum has height h and its top diameter is twice its bottom diameter. Find the height of the complete cone formed by extending its sides to a vertex below.
A: 2h
Similar cones have heights in the ratio of their diameters. If the full height is H, the removed small cone has height H/2, so h = H − H/2 and H = 2h.
AI-reviewed · calculation checkedM(4, q) is the midpoint of L(p, −2) and N(q, p). Find p and q.
D: p = 6, q = 2
The midpoint formula gives p + q = 8 and p − 2 = 2q. Substituting p = 2q + 2 into the first equation gives q = 2 and p = 6.
AI-reviewed · calculation checkedA boat on level water is viewed at a depression angle of 30° from a 10 m high cliff. Find its horizontal distance from the cliff foot.
C: 10√3 m
The depression angle equals the elevation angle from the boat. Thus tan30° = 10/d, giving d = 10/tan30° = 10√3 m.
AI-reviewed · calculation checkedEvaluate sin(−690°).
D: 1/2
Adding two complete turns gives −690° + 720° = 30°. Sine has period360°, so sin(−690°) = sin30° = 1/2.
AI-reviewed · calculation checkedAt which point does y = 2x² − 2x + 3 have gradient2?
A: (1, 3)
Differentiate to obtain dy/dx = 4x − 2. Setting this equal to2 gives x = 1. Substituting into the curve gives y = 3.
AI-reviewed · calculation checkedIntegrate (1 − x)/x³ with respect to x, for x ≠ 0.
C: 1/x − 1/(2x²) + k
Write the integrand as x⁻³ − x⁻². Integrating term by term gives −1/(2x²) + 1/x + k, where k is a constant.
AI-reviewed · calculation checkedA grade pie chart has sectors A₁ = 144°, A₂ = 72°, A₃ = 64.8° and C₄ = 43.2°, with F taking the remainder. What percentage received C₄?
D: 12.0%
A full pie chart is360°. The C₄ sector represents 43.2/360 of the total, so its percentage is (43.2/360)×100 = 12.0%.
AI-reviewed · calculation checkedClass intervals 1–5, 6–10, 11–15, 16–20 and 21–25 have frequencies6,15,20,7,2. Using continuous class boundaries, estimate the median.
B: 11.5
There are50 observations and21 precede the median class11–15. Using lower boundary10.5 and width5 gives 10.5 + ((25 − 21)/20)×5 = 11.5.
AI-reviewed · calculation checkedA class has6 pupils aged10,27 aged11 and7 aged12. What is the probability that a randomly chosen pupil is at least11?
B: 17/20
There are6 + 27 + 7 = 40 pupils. The27 pupils aged11 and7 aged12 meet the condition, giving34/40 = 17/20.
AI-reviewed · calculation checkedWhich expression describes the shaded region in the Venn diagram?
Make x the subject of (1 + ax)/(1 − ax) = p/q.
B: (p−q)/[a(p+q)]
Cross multiplication gives q+aqx=p−apx. Hence a(p+q)x=p−q, so x=(p−q)/[a(p+q)], where the original relation and denominators are defined.
AI-reviewed · calculation checkedWhich of the following is a factor of 15 + 7x − 2x²?
C: x − 5
By the factor theorem, x − a is a factor when substituting x = a gives zero. At x = 5, 15 + 35 − 50 = 0. Equivalently, 15 + 7x − 2x² = (5 − x)(2x + 3) = −(x − 5)(2x + 3), so x − 5 is a factor.
AI-reviewed · calculation checkedSolve x² + y − 5 = 0 and y − 7x + 3 = 0 for x.
D: 1, −8
The second equation gives y = 7x − 3. Substitution into the first gives x² + 7x − 8 = 0, or (x − 1)(x + 8) = 0. Hence x = 1 or x = −8. Their corresponding y-values are 4 and −59, and both pairs satisfy the original equations.
AI-reviewed · calculation checkedDivide x³ + 7x² − x − 7 by x² − 1.
D: x + 7
Group the terms: x³ + 7x² − x − 7 = x²(x + 7) − (x + 7) = (x² − 1)(x + 7). The polynomial quotient is x + 7 and the remainder is zero. If treated as a fraction, the original expression is undefined at x = ±1.
AI-reviewed · calculation checkedSolve the inequality y² − 3y > 18.
B: y < −3 or y > 6
Move 18 to the left and factor: y² − 3y − 18 = (y − 6)(y + 3). The product is positive when both factors have the same sign: y < −3 or y > 6. The boundary values −3 and 6 give equality, so neither is included.
AI-reviewed · calculation checkedIf k + 1, 2k − 1 and 3k + 1 are consecutive terms of a geometric progression, find the possible common ratios.
B: −1, 5/3
For three consecutive geometric terms, (2k − 1)² = (k + 1)(3k + 1). Expanding and simplifying gives k² − 8k = 0, so k = 0 or 8. At k = 0 the terms are 1, −1, 1 and the ratio is −1. At k = 8 they are 9, 15, 25 and the ratio is 15/9 = 5/3.
AI-reviewed · calculation checkedIn triangle PQR, PS bisects angle QPR, PR = 9 cm, PQ = 4 cm and QS = 3 cm. Find SR.
ST is the common internal tangent shown. O and Q are the circle centres, OS = 5 cm, QT = 2 cm and OQ = 14 cm. Find ST.
Square PQRS has area100 cm². The smaller squares TUYS and XQVU have areas in the ratio1:16. Find YR.
Find the radius of a sphere whose surface area is 154 cm². Use π = 22/7.
B: 3.50 cm
The surface area of a sphere is 4πr². Therefore r² = 154 ÷ (4 × 22/7) = 12.25. Taking the positive square root gives r = 3.50 cm. A radius cannot be negative.
AI-reviewed · calculation checkedFind the area of a sector of a circle of radius 3 m and angle 60°, correct to one decimal place.
C: 4.7 m²
A sector occupies θ/360 of its circle. Its area is (60/360) × π × 3² = 3π/2 m², approximately 4.7124 m². To one decimal place this is 4.7 m².
AI-reviewed · calculation checkedIf sin θ = cos θ, find θ for 0° < θ < 360°.
A: 45°, 225°
At a solution cos θ cannot be zero, because sin θ would then be ±1. Divide by cos θ to obtain tan θ = 1. Tangent is positive in the first and third quadrants, giving θ = 45° and 225° in the stated interval.
AI-reviewed · calculation checkedTwo angles of a triangle are 30° each and its longest side is 10 cm. Find the length of each of the other sides.
D: 10√3/3 cm
The third angle is 180° − 30° − 30° = 120°. The two remaining sides are equal. By the sine rule, a/sin 30° = 10/sin 120°, so a = 10 × (1/2)/(√3/2) = 10/√3 = 10√3/3 cm. This is approximately 5.77 cm.
AI-reviewed · calculation checkedWeight of coconuts (g) | Frequency 0–10 | 10 10–20 | 27 20–30 | 19 30–40 | 6 40–50 | 2
Estimate the mode of the grouped frequency distribution.
C: 16.8 g
The modal class is 10–20 g, with frequency 27. Its preceding and following frequencies are 10 and 19. The grouped-mode estimate is L + [(f₁ − f₀)/(2f₁ − f₀ − f₂)]h = 10 + [17/(54 − 10 − 19)] × 10 = 16.8 g. This interpolates within the modal class; it is not simply the class midpoint.
AI-reviewed · calculation checkedIndependent events X, Y and Z have probabilities 1/2, 2/3 and 1/4 respectively. Find the probability that Y and Z occur but X does not.
C: 1/12
Only Y and Z occurring means Y and Z occur while X does not. Independence lets us multiply the three probabilities: (1 − 1/2) × (2/3) × (1/4) = 1/12. Including X would describe a different event.
AI-reviewed · calculation checkedSimplify 5√18 − 3√72 + 4√50.
C: 17√2
Since √18 = 3√2, √72 = 6√2 and √50 = 5√2, the expression is 15√2 − 18√2 + 20√2 = 17√2.
AI-reviewed · calculation checkedMake t the subject of the formula s = ut + ½at².
D: (−u ± √(u² + 2as))/a
Rearrange to at² + 2ut − 2s = 0. For a ≠ 0, the quadratic formula gives t = (−2u ± √(4u² + 8as))/(2a) = (−u ± √(u² + 2as))/a. Real solutions require u² + 2as ≥ 0; a physical time must also satisfy the conditions of the motion.
AI-reviewed · calculation checkedSolve the equation y − 11√y + 24 = 0.
B: 64, 9
Put z = √y, so z ≥ 0. Then z² − 11z + 24 = (z − 8)(z − 3) = 0. Thus √y is 8 or 3, giving y = 64 or 9. Both values satisfy the original equation.
AI-reviewed · calculation checkedResolve 3/(x² + x − 2) into partial fractions.
A: 1/(x−1) − 1/(x+2)
Factor the denominator as (x−1)(x+2). Writing 3=A(x+2)+B(x−1), x=1 gives A=1 and x=−2 gives B=−1. Thus the partial fractions are 1/(x−1)−1/(x+2), with x≠1,−2.
AI-reviewed · calculation checkedWhat is the nth term of the sequence 2, 6, 12, 20, …?
C: n² + n
The terms are 1×2, 2×3, 3×4 and 4×5. The pattern is n(n + 1) = n² + n, starting at n = 1. It gives 2, 6, 12 and 20, matching all four terms.
AI-reviewed · calculation checkedFor real numbers m and n, the binary operation * is defined by m*n = mn + m + n. Find its identity element.
D: e = 0
An identity e must satisfy m*e = m for every m. Thus me + m + e = m, so e(m + 1) = 0 for every m. Taking m = 0 gives e = 0. Substitution confirms m*0 = m and 0*m = m.
AI-reviewed · calculation checkedP and R are the endpoints of a diameter of a circle, and Q is a point on the semicircle. PQ and QR are chords. S lies on PR, with QS perpendicular to PR. Which expression gives the length QS?
B: QS = √(PS × SR)
The angle PQR is 90° because PR is a diameter. The altitude QS divides the right triangle into two similar triangles, PQS and QRS. Their corresponding sides give PS/QS = QS/SR, so QS² = PS × SR. Taking the positive square root gives QS = √(PS × SR).
AI-reviewed · calculation checkedTowns P and Q both lie on latitude 60°N. Their longitudes are 20°E and 25°W respectively. Calculate the distance between them along this latitude, taking Earth’s radius as 6400 km.
C: 800π km
The longitude difference is 20° + 25° = 45°. At latitude 60°N, the parallel has radius 6400 cos 60° = 3200 km. The required arc is 45/360 of its circumference: (45/360) × 2π × 3200 = 800π km.
AI-reviewed · calculation checkedTwo perpendicular radii of a circle with centre O each have length 6 cm. A chord joins their endpoints. Find the area of the minor segment between this chord and the quarter-circle arc.
B: 9(π − 2) cm²
The perpendicular radii form a 90° sector. Its area is (90/360) × π × 6² = 9π cm². The right triangle formed by the radii has area ½ × 6 × 6 = 18 cm². The segment area is therefore 9π − 18 = 9(π − 2) cm².
AI-reviewed · calculation checkedThe locus of a point which is equidistant from two given fixed points is the
A: perpendicular bisector of the straight line joining them
Place the fixed points at (−a, 0) and (a, 0). Equal distances from (x, y) require (x + a)² + y² = (x − a)² + y². For two distinct fixed points, a ≠ 0, so x = 0. This is the perpendicular bisector of the segment joining the points.
AI-reviewed · calculation checkedFind the equation of the line through (5, 7) parallel to the line 7x + 5y = 12.
B: 7x + 5y = 70
A parallel line has the same coefficients of x and y, so write it as 7x + 5y = c. Substituting (5, 7) gives c = 7×5 + 5×7 = 70. Hence the required equation is 7x + 5y = 70.
AI-reviewed · calculation checkedA frequency distribution has values 2, 4, 6 and 8, with respective frequencies 4, y, 6 and 5. If its mean is 5.2, find y.
C: 5.0
The weighted total is 2×4 + 4y + 6×6 + 8×5 = 84 + 4y, and the total frequency is 15 + y. Thus (84 + 4y)/(15 + y) = 5.2. Multiplying out gives 84 + 4y = 78 + 5.2y, so 6 = 1.2y and y = 5.
AI-reviewed · calculation checkedThe numbers of children in families are 0, 1, 2, 3, 4, 5 and 6, with respective frequencies 7, 11, 6, 7, 7, 5 and 3. Find the mode and median, respectively.
B: 1,2
The highest frequency is 11, corresponding to 1 child, so the mode is 1. There are 46 families. The median uses the 23rd and 24th observations. Cumulative frequencies are 7, 18 and 24 through 0, 1 and 2 children, so both middle observations are 2. The mode and median are therefore 1 and 2.
AI-reviewed · calculation checkedTwo fair six-sided dice are thrown together. Determine the probability of obtaining a total score of 8.
B: 5/36
The dice have 6 × 6 = 36 equally likely ordered outcomes. A total of 8 occurs for (2, 6), (3, 5), (4, 4), (5, 3) and (6, 2): five outcomes. The probability is 5/36.
AI-reviewed · calculation checkedEvents P and Q have probabilities 3/4 and 1/6, respectively. If the probability that both occur is 1/12, what is the probability that at least one occurs?
C: 5/6
Add the probabilities and subtract the overlap counted twice: Pr(P ∪ Q) = Pr(P) + Pr(Q) − Pr(P ∩ Q) = 3/4 + 1/6 − 1/12 = 9/12 + 2/12 − 1/12 = 10/12 = 5/6.
AI-reviewed · calculation checkedCalculate (1/0.05) ÷ (1/5.005) − (0.05 × 2.05), correct to three decimal places.
A: 99.998
Dividing20 by1/5.005 gives20×5.005=100.1. Subtracting0.05×2.05=0.1025 gives99.9975, which rounds to99.998.
AI-reviewed · calculation checkedExpress62/3 as a decimal correct to three significant figures.
D: 20.7
62÷3=20.666… . The first three significant figures are2,0 and6. The next digit is6, so rounding gives20.7.
AI-reviewed · calculation checkedMusa borrows ₦10.00 at2% interest per month and repays ₦8.00 after four months. How much does he still owe?
C: ₦2.80
The monthly interest is2% of₦10, or₦0.20. Four months of simple interest total₦0.80, making₦10.80 due. After paying₦8.00, Musa still owes₦2.80.
AI-reviewed · calculation checkedThree gallons of spirit containing20% water are mixed with five gallons containing15% water. What percentage of the mixture is water?
B: 16⅞%
The water volumes are3×0.20=0.60 gallon and5×0.15=0.75 gallon. Thus1.35 gallons of the8-gallon mixture is water: (1.35/8)×100=16.875%=16⅞%.
AI-reviewed · calculation checkedSimplify 2 log₁₀(2/5) − log₁₀(72/125) + log₁₀9.
D: 1 − 2 log₁₀2
Combining logarithms gives log₁₀[(2/5)²×9÷(72/125)]=log₁₀(5/2). Since 5/2=10/4, this equals 1−2log₁₀2.
AI-reviewed · calculation checkedIf w varies inversely as uv/(u + v), and w = 8 when u = 2 and v = 6, find the relationship between u, v and w.
C: uvw = 12(u + v)
Inverse variation gives w = k(u + v)/(uv). Substituting the given values, 8 = k(2 + 6)/(2 × 6), so k = 12. Therefore uvw = 12(u + v). The relation applies where u, v and u + v are nonzero.
AI-reviewed · calculation checkedWhich of the following is a factor of rs + tr − pt − ps?
C: r − p
Group the terms: rs + tr − pt − ps = r(s + t) − p(t + s) = (r − p)(s + t). Therefore r − p is a factor.
AI-reviewed · calculation checkedFind the two values of y that satisfy the simultaneous equations 3x + y = 8 and x² + xy = 6.
A: −1 and 5
From the first equation, y = 8 − 3x. Substitute into the second: x² + x(8 − 3x) = 6, so x² − 4x + 3 = 0. Factor to obtain (x − 1)(x − 3) = 0. When x = 1, y = 5; when x = 3, y = −1. Thus the two values of y are −1 and 5.
AI-reviewed · calculation checkedFind the range of x that satisfies x/2 + x/3 + x/4 < 1.
A: x < 12/13
Use the common denominator 12: x/2 + x/3 + x/4 = (6x + 4x + 3x)/12 = 13x/12. Multiplying the inequality by the positive number 12/13 gives x < 12/13. The boundary value is excluded because the inequality is strict.
AI-reviewed · calculation checkedSolve the equation (x − 2)(x − 3) = 12.
C: −1 and 6
Expand and collect terms: x² − 5x + 6 = 12, so x² − 5x − 6 = 0. Factor to obtain (x − 6)(x + 1) = 0. Hence x = 6 or x = −1. Substituting either value into the original equation gives 12.
AI-reviewed · calculation checkedSimplify (√(1 + x) + √x)/(√(1 + x) − √x).
B: 1 + 2x + 2√(x(1 + x))
Multiply the numerator and denominator by √(1 + x) + √x. The denominator becomes (1 + x) − x = 1. Expanding the numerator gives 1 + x + x + 2√(x(1 + x)) = 1 + 2x + 2√(x(1 + x)). For real square roots, x ≥ 0.
AI-reviewed · calculation checkedWhat is the nth term of the progression 27, 9, 3, …?
A: 27(1/3)^(n − 1)
This is a geometric progression with first term 27 and common ratio 9/27 = 1/3. Its nth term is ar^(n − 1), so the required term is 27(1/3)^(n − 1), for n = 1, 2, 3, …. It gives 27, 9 and 3 for the first three terms.
AI-reviewed · calculation checkedThe angles of a quadrilateral are (5x − 30)°, (4x + 60)°, (60 − x)° and (3x + 61)°. Which is the smallest angle?
C: (60 − x)°
The interior angles of a quadrilateral sum to 360°. Thus 11x + 151 = 360, giving x = 19. The four angles are 65°, 136°, 41° and 118°. The smallest is 41°, represented by (60 − x)°.
AI-reviewed · calculation checkedThe area of a square is 144 cm². Find the length of its diagonal.
C: 12√2 cm
The side length is √144 = 12 cm. A diagonal is the hypotenuse of a right triangle with both shorter sides equal to 12 cm. By Pythagoras, d² = 12² + 12² = 288, so d = 12√2 cm.
AI-reviewed · calculation checkedOne angle of a rhombus is 60°. The shorter diagonal is 8 cm. Find the length of the longer diagonal.
A: 8√3 cm
The diagonals of a rhombus bisect its angles and meet at right angles. In the resulting 30°–60°–90° triangle, half of the shorter diagonal is 4 cm. The longer half-diagonal is 4√3 cm, so the full longer diagonal is 8√3 cm.
AI-reviewed · calculation checkedFrom a point P outside a circle, PT is a tangent. Two secants pass through the circle in the orders P–M–N and P–Q–R. If PM = 5 cm, PN = 12 cm and PQ = 4.8 cm, find PR and PT respectively, to the nearest 0.1 cm.
C: 12.5 cm and 7.7 cm
For secants from the same external point, PM × PN = PQ × PR = PT². The common product is 5 × 12 = 60 cm². Hence PR = 60/4.8 = 12.5 cm and PT = √60 ≈ 7.7 cm.
AI-reviewed · calculation checkedSimplify cos²x(sec²x + sec²x tan²x).
C: sec²x
Factor out sec²x: cos²x sec²x(1 + tan²x). Since cos²x sec²x = 1 and 1 + tan²x = sec²x, the expression equals sec²x wherever the original expression is defined.
AI-reviewed · calculation checkedPQRS is a square of side 8 cm, with its vertices named in order. U lies on PQ, V on QR and W on RS. Given UQ = 4 cm, QV = 6 cm and RW = 2 cm, find the area of triangle UVW.
D: 10 cm²
PU = 4 cm, SW = 6 cm and RV = 2 cm. The trapezium PUWS has area ½ × (4 + 6) × 8 = 40 cm². The corner triangles UQV and VRW have areas ½ × 4 × 6 = 12 cm² and ½ × 2 × 2 = 2 cm². Hence area UVW = 64 − 40 − 12 − 2 = 10 cm².
AI-reviewed · calculation checkedFind the total surface area of a solid cylinder whose base radius is 4 cm and height is 5 cm.
B: 72π cm²
The two circular ends have total area 2πr² = 32π cm². The curved surface has area 2πrh = 40π cm². Adding them gives 72π cm².
AI-reviewed · calculation checkedA family spends 3% of its income on electricity, 59% on food, 20% on transport, 11% on education and 7% on extended family. What angles represent education and food, respectively, in a pie chart?
D: 39.6° and 212.4°
A full circle is 360°. Education takes 11/100 × 360° = 39.6°, and food takes 59/100 × 360° = 212.4°.
AI-reviewed · calculation checkedFifty boxes, each containing 50 balls, were inspected. The numbers of defective balls per box were 4, 5, 6, 7, 8 and 9, with corresponding numbers of boxes 2, 7, 17, 10, 8 and 6. Find the mean (to one decimal place) and median number of defective balls per box, respectively.
A: 6.7 and 6
The frequencies total 50. The weighted total is 4×2 + 5×7 + 6×17 + 7×10 + 8×8 + 9×6 = 333, so the mean is 333/50 = 6.66, or 6.7 to one decimal place. There are 9 values below 6 and 26 values at or below 6. Both the 25th and 26th values are 6, so the median is 6.
AI-reviewed · calculation checkedSimplify (4¾ − 6¼) ÷ (4⅕ × 1¼).
B: −2/7
The numerator is19/4−25/4=−3/2. The denominator is(21/5)(5/4)=21/4. Therefore(−3/2)÷(21/4)=−2/7.
AI-reviewed · calculation checkedFind the highest common factor of a²bx + abx² and a²b − b³.
A: b
Factor both expressions: a²bx + abx² = abx(a + x), and a²b − b³ = b(a − b)(a + b). The only polynomial factor common to both expressions is b. The other factors involve independent variables and are not shared by both expressions.
AI-reviewed · calculation checkedEvaluate 241.34 × (3 × 10⁻³)² and give the answer to four significant figures.
D: 0.002172
First square the entire bracket: (3 × 10⁻³)² = 9 × 10⁻⁶. Multiplying by 241.34 gives 0.00217206. The first four significant digits are 2, 1, 7 and 2; the next digit is 0, so the answer is 0.002172. Leading zeros do not count as significant digits.
AI-reviewed · calculation checkedAt what annual simple-interest rate would ₦100 deposited for five years earn ₦7.50 interest?
A: 1½%
For simple interest, I = Prt/100 when r is the annual percentage rate. Hence 7.50 = 100 × r × 5/100 = 5r, so r = 1.5% per year. This is option A, written as 1½%.
AI-reviewed · calculation checkedThree children share a basket of mangoes. The first takes ¼ of the mangoes and the second takes ¾ of the remainder. What fraction of the original basket does the third child receive?
A: 3/16
After the first child takes ¼, the fraction left is ¾. The second child takes ¾ of that remainder, leaving ¼ of ¾ for the third child: ¼ × ¾ = 3/16. As a check, ¼ + 9/16 + 3/16 = 1.
AI-reviewed · calculation checkedThree brothers share a profit. The first receives ⅓ of the total and the second receives ⅔ of the remainder. The third receives the remaining ₦12,000. What was the total profit?
B: ₦54,000
After the first brother receives ⅓, the remainder is ⅔. The second takes ⅔ of that remainder, leaving ⅓ × ⅔ = 2/9 of the original profit. Therefore 2/9 of the total is ₦12,000, so the total is ₦12,000 × 9/2 = ₦54,000.
AI-reviewed · calculation checkedIf a² + b² = 16 and 2ab = 7, find all possible values of a − b.
A: 3 and −3
Use (a − b)² = a² + b² − 2ab = 16 − 7 = 9. Taking both square roots gives a − b = 3 or −3. Both signs are possible because exchanging a and b preserves the given equations and reverses their difference.
AI-reviewed · calculation checkedDivide x³ − 2x² − 5x + 6 by x − 1.
A: x² − x − 6
Polynomial division gives the quotient x² − x − 6 with zero remainder. Check by multiplying: (x − 1)(x² − x − 6) = x³ − x² − 6x − x² + x + 6 = x³ − 2x² − 5x + 6.
AI-reviewed · calculation checkedSolve x − 8√x + 15 = 0 over the real numbers.
C: 9 and 25
Because √x is real, x ≥ 0. Substitute u = √x, so u ≥ 0 and x = u². Then u² − 8u + 15 = (u − 3)(u − 5) = 0, giving u = 3 or 5. Squaring gives x = 9 or 25. Both satisfy the original equation: 9 − 24 + 15 = 0 and 25 − 40 + 15 = 0.
AI-reviewed · calculation checkedA rectangular lawn has perimeter 24 m and area 35 m². Find its width, taking the width as the shorter side.
A: 5 m
If the sides are l and w, then 2(l + w) = 24, so l + w = 12, and lw = 35. The positive numbers 5 and 7 have sum 12 and product 35. Thus the shorter side, or width, is 5 m. The perimeter check is 2(5 + 7) = 24 m.
AI-reviewed · calculation checkedSimplify x/(x + y) + y/(x − y) − x²/(x² − y²), where x ≠ y and x ≠ −y.
B: y²/(x² − y²)
Use the common denominator (x + y)(x − y) = x² − y². The numerator becomes x(x − y) + y(x + y) − x² = x² − xy + xy + y² − x² = y². Therefore the result is y²/(x² − y²). The restrictions keep every original denominator nonzero.
AI-reviewed · calculation checkedFind the sum of the first twenty terms of the arithmetic progression log a, log(a²), log(a³), …, where a > 0 and the logarithms have the same valid base.
D: log(a²¹⁰)
The terms are log a, 2 log a, 3 log a, and so on. Their sum is (1 + 2 + … + 20) log a = [20 × 21/2] log a = 210 log a = log(a²¹⁰).
AI-reviewed · calculation checkedFind the sum of the first 18 terms of the progression 3, 6, 12, ….
D: 3(2¹⁸ − 1)
This is a geometric progression with first term 3 and common ratio 2. The sum is S₁₈ = 3(2¹⁸ − 1)/(2 − 1) = 3(2¹⁸ − 1). Multiplying the sum by 2 and subtracting the original sum also leaves 3 × 2¹⁸ − 3.
AI-reviewed · calculation checkedIn trapezium PQRS shown, QR is parallel to PS. Its area is 73.5 cm² and its perpendicular height is 10.5 cm. Find PS if QR is one-third of PS.
A sector of a circle has radius 10.5 cm and central angle 48°. Calculate its perimeter to one decimal place.
D: 29.8 cm
The arc length is (48/360) × 2π × 10.5 = 2.8π cm, approximately 8.8 cm. The perimeter includes both radii, so it is 10.5 + 10.5 + 2.8π ≈ 29.8 cm.
AI-reviewed · calculation checkedIn the figure, P lies on XZ and Q lies on XY. The marked angles XQP and XZY are equal. Given XP = 4 cm, XQ = 5 cm and PQ = QY = 3 cm, find ZY.
Find the length of a side of a rhombus whose diagonals are 6 cm and 8 cm.
B: 5 cm
The diagonals of a rhombus bisect each other at right angles. Half-diagonals of 3 cm and 4 cm form the legs of a right triangle whose hypotenuse is a side of the rhombus. The side length is √(3² + 4²) = √25 = 5 cm.
AI-reviewed · calculation checkedIf cos θ = 12/13, find 1 + cot² θ.
A: 169/25
Use sin² θ = 1 − cos² θ = 1 − 144/169 = 25/169. Therefore cot² θ = cos² θ/sin² θ = 144/25, and 1 + cot² θ = 169/25. Equivalently, 1 + cot² θ = 1/sin² θ. Squared values make the result independent of the sign of sin θ.
AI-reviewed · calculation checkedIn triangle XYZ shown, angle YXZ = 30°, angle XYZ = 105° and XY = 8 cm. Calculate YZ.
OPRS is a rectangle with OP = 11 cm. A semicircle is attached outside it along diameter PR. Inside the rectangle, triangle OTS is right-angled at T, with OT = 6 cm and ST = 8 cm. Find the area of the rectangle and semicircle together, excluding triangle OTS. Use π = 22/7.
A: 125 2/7 cm²
OS=√(6²+8²)=10 cm, so the rectangle area is 10×11=110 cm² and the semicircle radius is 5 cm. Its area is(1/2)(22/7)(25)=275/7 cm². Subtract triangle area (1/2)(6)(8)=24 cm² to obtain 110+275/7−24=125 2/7 cm².
AI-reviewed · calculation checkedTwo distinct numbers are drawn at random without replacement from 1, 2, 3 and 4. What is the probability that their sum is even?
C: 1/3
The six equally likely unordered pairs are {1,2}, {1,3}, {1,4}, {2,3}, {2,4} and {3,4}. Only {1,3} and {2,4} have even sums, so the probability is 2/6 = 1/3. An even sum requires two odd numbers or two even numbers.
AI-reviewed · calculation checkedAn integer is selected uniformly at random from 41 to 56 inclusive. What is the probability that it is a multiple of 9?
A: 1/8
There are 56 − 41 + 1 = 16 integers in the inclusive range. The multiples of 9 are 45 and 54, so there are two favourable outcomes. The probability is 2/16 = 1/8.
AI-reviewed · calculation checkedEvaluate (2,700,000 × 0.03) / 18,000.
A: 4.5 × 10⁰
2,700,000 × 0.03 = 81,000. Dividing by 18,000 gives 4.5. Since 10⁰ = 1, this is 4.5 × 10⁰, option A. An exponent of 1 would instead give 45.
AI-reviewed · calculation checkedWhich list contains all the distinct prime factors of 2,520?
C: 2, 3, 5, 7
Repeated division gives 2,520 = 2³ × 3² × 5 × 7. The distinct prime factors are therefore 2, 3, 5 and 7. The number 9 is not prime because 9 = 3 × 3.
AI-reviewed · calculation checkedFind the absolute difference between 0.007685 rounded to three significant figures and the same number rounded to four decimal places.
A: 10⁻⁵
To three significant figures, 0.007685 rounds to 0.00769. To four decimal places, it rounds to 0.0077. Their absolute difference is 0.007700 − 0.007690 = 0.000010 = 10⁻⁵. The leading zeros do not count as significant figures.
AI-reviewed · calculation checkedFind p in terms of q if log₃(p) + 3log₃(q) = 3, with p > 0 and q > 0.
A: (3/q)³
Use 3log₃(q) = log₃(q³), then combine the logarithms to obtain log₃(pq³) = 3. Therefore pq³ = 3³ = 27, giving p = 27/q³ = (3/q)³. Positivity is required for the original real logarithms.
AI-reviewed · calculation checkedFind all real values of y satisfying 9ʸ − 4(3ʸ) + 3 = 0.
D: 0 and 1
Since 9ʸ = (3ʸ)², substitute u = 3ʸ. The equation becomes u² − 4u + 3 = (u − 1)(u − 3) = 0. Thus 3ʸ = 1 or 3, giving y = 0 or 1. The function 3ʸ is one-to-one, so there are no other real solutions.
AI-reviewed · calculation checkedMake R the subject of S = √[(2R + T)/(2RT)], using the principal square root and values for which the original expression is defined.
B: R = T/[2(TS² − 1)]
Square both sides: S² = (2R + T)/(2RT). Multiply by 2RT to obtain 2RTS² = 2R + T. Collect the terms containing R: 2R(TS² − 1) = T. Therefore R = T/[2(TS² − 1)]. The original formula requires R and T to be nonzero and S ≥ 0; the rearranged denominator requires TS² ≠ 1.
AI-reviewed · calculation checkedThe cost of dinner for a group of students is partly constant and partly varies directly with the number of students. It costs ₦74.00 for 20 students and ₦96.00 for 30 students. Find the cost for 15 students.
B: ₦63.00
The extra 10 students cost ₦96 − ₦74 = ₦22, so the variable cost is ₦2.20 per student. The fixed cost is ₦74 − 20 × ₦2.20 = ₦30. For 15 students, the total is ₦30 + 15 × ₦2.20 = ₦63.00.
AI-reviewed · calculation checkedIf f(x) = 2x² − 5x + 3, find f(x + 1).
A: 2x² − x
Replace every x in the definition with x + 1: f(x + 1) = 2(x + 1)² − 5(x + 1) + 3. Expand to get 2x² + 4x + 2 − 5x − 5 + 3 = 2x² − x.
AI-reviewed · calculation checkedFactorize 4a² + 12ab − c² + 9b².
B: (2a + 3b − c)(2a + 3b + c)
Group the first, second and fourth terms: 4a² + 12ab + 9b² = (2a + 3b)². The expression is therefore (2a + 3b)² − c². Applying the difference-of-squares identity gives (2a + 3b − c)(2a + 3b + c).
AI-reviewed · calculation checkedFind K and L, respectively, if ½(3y − 4x)² = 8x² + Kxy + Ly².
A: (−12, 9/2)
Expand the square before multiplying by one half: ½(9y² − 24xy + 16x²) = 8x² − 12xy + (9/2)y². Comparing coefficients gives K = −12 and L = 9/2.
AI-reviewed · calculation checkedSolve 2x⁻¹ − 3y⁻¹ = 4 and 4x⁻¹ + y⁻¹ = 1 for x and y, respectively, where x and y are nonzero.
D: (2, −1)
Substitute u = 1/x and v = 1/y. The equations become 2u − 3v = 4 and 4u + v = 1. Add the first equation to three times the second to obtain 14u = 7, so u = 1/2. Then v = 1 − 4(1/2) = −1. Thus x = 2 and y = −1. Substitution gives 1 + 3 = 4 and 2 − 1 = 1, checking both equations.
AI-reviewed · calculation checkedWhat value of Q makes 4x² + 5x + Q a perfect square?
A: 25/16
Write the perfect square as (2x + c)² = 4x² + 4cx + c². Matching the coefficient of x gives 4c = 5, so c = 5/4 and Q = c² = 25/16. Hence the expression is (2x + 5/4)².
AI-reviewed · calculation checkedFor positive a, b and c, find the range of r satisfying r/a + r/b + r/c > 1.
A: r > abc/(bc + ac + ab)
Factor out r: r(1/a + 1/b + 1/c) > 1. Since a, b and c are positive, their reciprocal sum is positive, so division keeps the inequality direction unchanged. Therefore r > 1/(1/a + 1/b + 1/c) = abc/(bc + ac + ab).
AI-reviewed · calculation checkedOn the curve shown, at which labelled points is the gradient zero?
If −8, m, n and 19 are consecutive terms of an arithmetic progression, find (m, n).
A: (1, 10)
There are three equal steps from −8 to 19. The common difference is [19 − (−8)]/3 = 27/3 = 9. The terms are −8, 1, 10, 19, so m = 1 and n = 10.
AI-reviewed · calculation checkedIn triangle PQR, H lies on PQ and K lies on PR, with HK parallel to QR. PH = 4 cm and HQ = 3 cm. Find KR : PR.
B: 3 : 7
PQ = 4 + 3 = 7 cm. Similar triangles PHK and PQR give PK/PR = PH/PQ = 4/7. The remaining part is KR/PR = 1 − 4/7 = 3/7, so KR : PR = 3 : 7.
AI-reviewed · calculation checkedIn triangle XYZ, angles Y and Z are each 30° and XZ = 3 cm. Find YZ.
C: 3√3 cm
Angle X is 180° − 30° − 30° = 120°. By the sine rule, YZ/sin 120° = XZ/sin 30°. Thus YZ = 3(√3/2)/(1/2) = 3√3 cm. The side opposite 120° must be the longest, consistent with this result.
AI-reviewed · calculation checkedTriangle RST is right-angled at S. Its perpendicular sides have lengths x and y, and its hypotenuse has length z. If x : y = 5 : 12 and z = 52 cm, find its perimeter.
D: 120 cm
Let the perpendicular sides be5k and12k. By Pythagoras, the hypotenuse is13k. Since13k=52, k=4. The sides are20,48 and52 cm, giving perimeter120 cm.
AI-reviewed · calculation checkedAn aeroplane flies horizontally at a height of 10 km toward a landmark on level ground. At two positions before reaching it, the angles of depression are 35° and 55°, respectively. Find the distance between the observation positions.
D: 10(cot 35° − cot 55°) km
The angle of depression equals the angle of elevation from the landmark. At angle θ, the horizontal distance to the landmark is 10/tan θ = 10 cot θ. As the aircraft approaches, this distance decreases. The distance flown is therefore 10(cot 35° − cot 55°) km, approximately 7.28 km. This uses the constant-height, level-ground geometry described in the question.
AI-reviewed · calculation checkedThe average age of a family of 21 people is 14 years. Excluding the grandfather, the average age of the remaining 20 people is 12 years. How old is the grandfather?
D: 54 years
The total age of all 21 people is 21 × 14 = 294 years. The total for the other 20 people is 20 × 12 = 240 years. The grandfather is therefore 294 − 240 = 54 years old.
AI-reviewed · calculation checkedTwo numbers are chosen at random from the three numbers1,3 and6. Find the probability that their sum is not odd.
C: 1/3
The three possible pairs are(1,3), (1,6) and(3,6). Their sums are4,7 and9. Only one sum is even, so the probability is1/3.
AI-reviewed · calculation checkedEvaluate 0.24633 / 0.0306, correct to one decimal place.
D: 8.1
The exact quotient is 8.05: 0.0306 × 8.05 = 0.24633. To round to one decimal place, inspect the hundredths digit, which is 5. Round the tenths digit up, giving 8.1.
AI-reviewed · calculation checkedTaiwo and Kehinde own a store in the ratio 11:9. Kehinde sells two-thirds of her share to Taiwo for ₦720. What is the value of the whole store?
B: ₦2,400
Kehinde owns 9/(11 + 9) = 9/20 of the store. She sells (2/3) × (9/20) = 3/10 of the whole store for ₦720. Therefore the whole store is worth ₦720 ÷ (3/10) = ₦2,400.
AI-reviewed · calculation checkedGiven log₁₀(2) = 0.3010 and log₁₀(3) = 0.4771, evaluate log₁₀(4.5) using these values, without logarithm tables.
C: 0.6532
Write 4.5 = 9/2 = 3²/2. Then log₁₀(4.5) = 2log₁₀(3) − log₁₀(2) = 2(0.4771) − 0.3010 = 0.9542 − 0.3010 = 0.6532. This uses the supplied rounded logarithms.
AI-reviewed · calculation checkedAn 800-page book is 18 mm thick. Calculate the thickness of one leaf in metres, expressed in standard form. Treat the thickness as that of the paper leaves alone.
D: 4.50 × 10⁻⁵ m
Each leaf has two pages, so 800 pages correspond to 400 leaves. The thickness of one leaf is 18/400 = 0.045 mm. Dividing by 1,000 converts millimetres to metres: 0.045/1,000 = 0.000045 m = 4.50 × 10⁻⁵ m.
AI-reviewed · calculation checkedIf 1/p = (a² + 2ab + b²)/(a − b) and 1/q = (a + b)/(a² − 2ab + b²), find p/q. Assume a ≠ b and a ≠ −b.
B: 1/(a² − b²)
Factor the expressions: 1/p = (a + b)²/(a − b) and 1/q = (a + b)/(a − b)². Therefore p/q = (1/q)/(1/p) = [(a + b)/(a − b)²] × [(a − b)/(a + b)²] = 1/[(a − b)(a + b)] = 1/(a² − b²). The stated exclusions keep the original expressions and their reciprocals defined.
AI-reviewed · calculation checkedx varies inversely as the cube root of y. Given x = 1 when y = 8, find y when x = 3.
C: 8/27
Inverse variation gives x = k/∛y. When x = 1 and y = 8, k = 1 × ∛8 = 2. For x = 3, ∛y = 2/3. Cubing both sides gives y = (2/3)³ = 8/27.
AI-reviewed · calculation checkedIf g(y) = (y − 3)/11 + 11/(y² − 9), find g(y + 3), where y ≠ 0 and y ≠ −6.
A: y/11 + 11/[y(y + 6)]
Replace every occurrence of y in the definition by y + 3. The first term becomes [(y + 3) − 3]/11 = y/11. The second denominator becomes (y + 3)² − 9 = y² + 6y = y(y + 6). Hence g(y + 3) = y/11 + 11/[y(y + 6)]. This is defined when y is neither 0 nor −6.
AI-reviewed · calculation checkedSolve 3x − 5y − 3 = 0 and 2y − 6x + 5 = 0. Give the ordered pair (x, y).
D: (19/24, −1/8)
Double the first equation to obtain 6x − 10y = 6. The second equation is −6x + 2y = −5. Adding gives −8y = 1, so y = −1/8. Substituting in 3x − 5y = 3 gives 3x = 19/8 and x = 19/24. The ordered pair is (19/24, −1/8).
AI-reviewed · calculation checkedSimplify 1/(x² + 5x + 6) + 1/(x² + 3x + 2), where x ≠ −1, −2, −3.
C: 2/[(x + 1)(x + 3)]
Factor the denominators as (x + 2)(x + 3) and (x + 1)(x + 2). Over their common denominator, the numerator is (x + 1) + (x + 3) = 2(x + 2). Cancelling x + 2 gives 2/[(x + 1)(x + 3)]. The original restriction x ≠ −2 still applies, along with x ≠ −1 and x ≠ −3.
AI-reviewed · calculation checkedSimplify (4a² − 49b²)/(2a² + 5ab − 7b²), where the denominator is nonzero.
D: (2a − 7b)/(a − b)
Factor the numerator as (2a − 7b)(2a + 7b), a difference of two squares. Factor the denominator as (2a + 7b)(a − b). Cancel the common nonzero factor 2a + 7b to obtain (2a − 7b)/(a − b), option D. The original restrictions remain: a ≠ b and 2a + 7b ≠ 0.
AI-reviewed · calculation checkedSolve x² − x − 1 = 0, giving the roots to one decimal place.
A: x = 1.6 and x = −0.6
The quadratic formula gives x = (1 ± √5)/2. The roots are approximately 1.618 and −0.618, which round to 1.6 and −0.6. Their exact sum is 1 and their exact product is −1.
AI-reviewed · calculation checkedThe solutions of x² − 2x − 1 = 0 are the x-coordinates where two graphs intersect. If one graph is y = 2 + x − x², which is the second graph?
A: y = 1 − x
At a solution of x² − 2x − 1 = 0, we have x² = 2x + 1. Substituting into y = 2 + x − x² gives y = 2 + x − (2x + 1) = 1 − x. Conversely, equating 2 + x − x² and 1 − x gives the required quadratic.
AI-reviewed · calculation checkedWhich of these can be the exterior angle of a regular polygon? (i) 35°; (ii) 18°; (iii) 115°.
B: ii only
For a regular polygon with n sides, each exterior angle is 360°/n, where n is an integer of at least 3. We get 360/35 = 72/7 and 360/115 = 72/23, neither an integer. But 360/18 = 20, so only (ii) is possible.
AI-reviewed · calculation checkedA man is 14√3 metres horizontally from a tree on level ground. From a point 2 metres above the ground, he measures an angle of elevation of 30° to the top of the tree. How tall is the tree?
D: 16 m
The height above the observation point is 14√3 × tan 30° = 14√3 × (1/√3) = 14 m. Add the observation height of 2 m: the tree is 14 + 2 = 16 m tall.
AI-reviewed · calculation checkedA metal pipe is 10 cm long, has an external diameter of 12 cm and a wall thickness of 1 cm. Find the volume of metal in the pipe.
B: 110π cm³
The outer radius is 12/2 = 6 cm, and the inner radius is 6 − 1 = 5 cm. Subtract the inner cylinder from the outer cylinder: volume = π(6² − 5²) × 10 = π(36 − 25) × 10 = 110π cm³.
AI-reviewed · calculation checkedLet x be the mean and y the median of 11, 12, 13, 14, 15, 16, 17, 18, 19, 21. Find x/y, correct to one decimal place.
D: 1.0
The ten values sum to 156, so the mean is x = 156/10 = 15.6. The middle two values are 15 and 16, so the median is y = 15.5. Thus x/y = 15.6/15.5 = 156/155 ≈ 1.00645, which rounds to 1.0 to one decimal place.
AI-reviewed · calculation checkedA score distribution is given by these (score, frequency) pairs: (0, 7), (1, 11), (2, 6), (3, 7), (4, 7), (5, 5), (6, 3). Find the mode and the median, respectively.
B: (1, 2)
Score 1 has the largest frequency, 11, so the mode is 1. There are 46 observations; the median is the average of the 23rd and 24th values. The cumulative frequencies through scores 0, 1 and 2 are 7, 18 and 24. Both middle values are therefore 2, so the median is 2. The answer is (1, 2).
AI-reviewed · calculation checkedTwo fair six-sided dice are thrown independently. What is the probability that their total score is at least 10?
A: 1/6
There are 6 × 6 = 36 equally likely ordered outcomes. A total of 10 occurs in three ways: (4,6), (5,5), (6,4). A total of 11 occurs in two ways, and 12 in one way. Thus the probability is (3 + 2 + 1)/36 = 6/36 = 1/6.
AI-reviewed · calculation checkedFind the least length of a rod which can be cut into exactly equal strips, each of either 40cm or 48cm in length.
B: 240 cm
The length must be a multiple of both 40 and 48. Since 40 = 2³ × 5 and 48 = 2⁴ × 3, their least common multiple is 2⁴ × 3 × 5 = 240 cm. This gives six 40 cm strips or five 48 cm strips. A length of 480 cm also works, but is not the least.
AI-reviewed · calculation checkedRound each number to two significant figures, then evaluate (0.02174 × 1.2047) / 0.023789.
C: 1.1
First round each input: 0.02174 becomes 0.022, 1.2047 becomes 1.2, and 0.023789 becomes 0.024. Now calculate (0.022 × 1.2) / 0.024 = 0.0264 / 0.024 = 1.1. The instruction requires rounding the inputs before doing the multiplication and division.
AI-reviewed · calculation checkedIf the length of a square is increased by 20% while its width is decreased by 20% to form a rectangle, what is the ratio of the area of the rectangle to the area of the square?
D: 24:25
Let the original side length be s. The rectangle has length 1.2s and width 0.8s, so its area is 0.96s². Compared with the original area s², the ratio is 0.96:1 = 96:100 = 24:25. The area decreases by 4%.
AI-reviewed · calculation checkedTwo brothers invested a total of ₦5,000.00 on a farm project. The farm yield was sold for ₦ 15, 000.00 at the end of the season. If the profit was shared in the ratio 2:3, what is the difference in the amount of profit received by the brothers?
A: ₦2,000.00
The profit is ₦15,000 − ₦5,000 = ₦10,000. There are 2 + 3 = 5 equal shares, so each share is ₦2,000. The brothers receive ₦4,000 and ₦6,000, a difference of ₦2,000.
AI-reviewed · calculation checkedPeter earns ₦20 per week for the first 20 weeks and ₦36 per week for the next 24 weeks. His average weekly wage for all 52 weeks is ₦30. Find his average weekly wage for the remaining 8 weeks.
A: ₦37.00
Total annual wages are 52 × ₦30 = ₦1,560. The first 44 weeks account for 20 × ₦20 + 24 × ₦36 = ₦400 + ₦864 = ₦1,264. The remaining eight weeks therefore account for ₦296, giving an average of ₦296 ÷ 8 = ₦37 per week.
AI-reviewed · calculation checkedBy selling 20 oranges for ₦1.35, a trader makes a profit of 8%. What is the percentage gain or loss if he sells the same 20 oranges for ₦1.10?
C: 12%
An 8% profit means ₦1.35 is 108% of the cost. The cost is ₦1.35 ÷ 1.08 = ₦1.25. Selling for ₦1.10 gives a loss of ₦0.15. The percentage loss is (₦0.15 ÷ ₦1.25) × 100 = 12%.
AI-reviewed · calculation checkedInstead of writing 35/6 correct to 3 significant figures, a student writes it correct to 3 decimal places. Express the difference between these two rounded values in standard form.
B: 3.0 × 10⁻³
35/6 = 5.8333… rounds to 5.83 at three significant figures and 5.833 at three decimal places. Their difference is 5.833 − 5.830 = 0.003 = 3.0 × 10⁻³. Standard form requires a coefficient from 1 up to, but not including, 10. Option A has the same numerical value but is not written in the requested standard form.
AI-reviewed · calculation checkedSimplify [log₂(6) − log₂(3)] / [log₂(8) − 2log₂(1/2)] without using tables.
A: 1/5
The numerator is log₂(6/3) = log₂(2) = 1. The denominator is log₂(8) − 2log₂(1/2) = 3 − 2(−1) = 5. Therefore the expression is 1/5, option A.
AI-reviewed · calculation checkedSimplify [(2√14)(3√21)] / [(7√24)(2√98)] without using tables.
D: 3√2/28
Combine the coefficients and square roots: (6/14)√[(14 × 21)/(24 × 98)] = (3/7)√(1/8). Since √(1/8) = √2/4, the result is 3√2/28. The numbers before the radical signs are multiplying coefficients; all the radicals here are square roots.
AI-reviewed · calculation checkedThe formula Q = 15 + 0.5n gives the cost Q, in naira, of feeding n people for a week. Find, in kobo, the extra cost of feeding one additional person.
D: 50 kobo
For one extra person, the increase is Q(n + 1) − Q(n) = [15 + 0.5(n + 1)] − [15 + 0.5n] = 0.5 naira. Since one naira equals 100 kobo, the extra cost is 50 kobo.
AI-reviewed · calculation checkedIf P varies inversely as V and V varies directly as R², find the relationship between P and R, given that R = 7 when P = 2.
B: PR² = 98
Combining the variations gives P proportional to 1/R², so PR² is constant. Substituting P = 2 and R = 7 gives PR² = 2 × 49 = 98.
AI-reviewed · calculation checkedMake y the subject of Z = x² + 1/y³, for real values with Z ≠ x².
C: y = 1/∛(Z − x²)
Subtract x² to obtain Z − x² = 1/y³. Taking reciprocals gives y³ = 1/(Z − x²), so y = 1/∛(Z − x²). This requires Z ≠ x². A cube root does not distribute over subtraction: ∛(Z − x²) is not generally ∛Z − ∛(x²).
AI-reviewed · calculation checkedFind the values of m for which x² + 2(m + 1)x + m + 3 is a perfect square.
C: 1, −2
A monic perfect square with middle term 2(m + 1)x is [x + (m + 1)]². Its constant term must satisfy m + 3 = (m + 1)². This gives m² + m − 2 = (m + 2)(m − 1) = 0, so m = −2 or 1. These give (x − 1)² and (x + 2)² respectively.
AI-reviewed · calculation checkedFind the two values of y that satisfy the simultaneous equations x + y = 5 and x² − 2y² = 1.
C: −12, 2
From x + y = 5, x = 5 − y. Substitution gives (5 − y)² − 2y² = 1, so 25 − 10y − y² = 1. Rearranging gives y² + 10y − 24 = (y + 12)(y − 2) = 0. Thus y = −12 or 2; the matching x values are 17 and 3.
AI-reviewed · calculation checkedSolve the inequality x − 1 > 4(x + 2).
B: x < −3
Expand the bracket: x − 1 > 4x + 8. Rearranging gives −3x > 9. Dividing by −3 reverses the inequality, so x < −3. The boundary x = −3 is excluded because the original inequality is strict.
AI-reviewed · calculation checkedSimplify (x² − y²)/(2x² + xy − y²), where the original denominator is nonzero.
C: (x − y)/(2x − y)
Factor the numerator as (x − y)(x + y). The denominator is (2x − y)(x + y), since expanding gives 2x² + xy − y². Cancel x + y to obtain (x − y)/(2x − y). The original restrictions remain: x + y ≠ 0 and 2x − y ≠ 0.
AI-reviewed · calculation checkedIn triangle PTS, Q lies on PT and R lies on PS. QR is parallel to TS, and QR:TS = 2:3. Find the ratio of the area of triangle PQR to the area of trapezium QRST.
Triangle XYZ is right-angled at Y. T lies on XZ and YT is perpendicular to XZ. Given XT = 9 cm and TZ = 16 cm, find YZ.
Rectangle PQRS has width 2h cm and height 3h cm. Its shaded top strip and two side strips are each 2 cm thick. The unshaded rectangular opening reaches the bottom edge, as shown. If the shaded area is 72 cm², find h in centimetres.
Daylight lasts from 5:30 a.m. to 7:00 p.m. Express the daylight and darkness durations as angles of Earth’s rotation, in that order, using 360° for 24 hours.
C: 202°30′, 157°30′
Daylight lasts 13 hours 30 minutes, or 13.5 hours. Darkness lasts 24 − 13.5 = 10.5 hours. Earth rotates through 360°/24 = 15° per hour. The daylight angle is 13.5 × 15 = 202.5° = 202°30′, and the darkness angle is 10.5 × 15 = 157.5° = 157°30′. The two angles add to 360°.
AI-reviewed · calculation checkedA figure is selected at random from a square, a rectangle, a rhombus, a kite and a trapezium. Find the probability that it is a parallelogram.
A: 3/5
A square, a rectangle and a rhombus each have two pairs of parallel opposite sides. The ordinary kite and trapezium do not. There are three qualifying figures among five equally likely choices, so the probability is 3/5.
AI-reviewed · calculation checkedEvaluate (212)₃ − (121)₃ + (222)₃. All subscripts indicate base three.
C: (1020)₃
Convert to base ten: (212)₃ = 2×9 + 1×3 + 2 = 23; (121)₃ = 1×9 + 2×3 + 1 = 16; (222)₃ = 2×9 + 2×3 + 2 = 26. Then 23 − 16 + 26 = 33. Since 33 = 1×27 + 0×9 + 2×3 + 0, the result is (1020)₃, option C. Base three uses only the digits 0, 1 and 2.
AI-reviewed · calculation checkedFind the reciprocal of (2/3) ÷ (1/2 + 1/3).
B: 5/4
First add the denominator fractions: 1/2 + 1/3 = 5/6. The given expression is (2/3) ÷ (5/6) = (2/3) × (6/5) = 4/5. Its reciprocal is 5/4, option B. The question asks for the reciprocal, not just the value of the original expression.
AI-reviewed · calculation checkedIf P = 18, Q = 21, R = −6 and S = −4, calculate [(P − Q)³ + S²]/R³.
B: 11/216
Substitute carefully: P − Q = −3, so (P − Q)³ = −27, while S² = (−4)² = 16. The numerator is −27 + 16 = −11. The denominator is R³ = (−6)³ = −216. Dividing two negative numbers gives 11/216, option B.
AI-reviewed · calculation checkedUdoh deposited ₦150.00 in a bank. After 5 years, the simple interest earned was ₦55.00. What was the annual interest rate?
B: 7⅓%
For simple interest, I = Prt/100, where r is the annual percentage rate. Therefore r = 100I/(Pt) = (100 × 55)/(150 × 5) = 22/3 = 7⅓%. The correct answer is B. Use the interest earned, ₦55, rather than the final balance.
AI-reviewed · calculation checkedThe ages of Tosan and Isa differ by 6 years, and their product is 187. Write the ages as (x, y), where x > y.
C: (17, 11)
Let the younger age be y. Then the older age is y + 6, so y(y + 6) = 187. Rearranging gives y² + 6y − 187 = (y − 11)(y + 17) = 0. An age cannot be negative, so y = 11 and x = 17. Check: 17 − 11 = 6 and 17 × 11 = 187. The ages are (17, 11), option C.
AI-reviewed · calculation checkedFactorize completely: 8a + 125ax³.
B: a(2 + 5x)(4 − 10x + 25x²)
First factor out a: a(8 + 125x³). Since 8 = 2³ and 125x³ = (5x)³, use u³ + v³ = (u + v)(u² − uv + v²). This gives a(2 + 5x)(4 − 10x + 25x²), option B. Expanding the two factors inside a gives 8 + 125x³.
AI-reviewed · calculation checkedFind all real numbers x satisfying (x + 1)/3 − 1 > (x + 4)/5.
D: x > 11
Multiply both sides by 15, a positive number, so the inequality direction stays the same: 5(x + 1) − 15 > 3(x + 4). Thus 5x − 10 > 3x + 12, giving 2x > 22 and x > 11. The boundary x = 11 is excluded because the inequality is strict. Answer D.
AI-reviewed · calculation checkedFactorize x² + 2a + ax + 2x.
D: (x + 2)(x + a)
Reorder the terms: x² + ax + 2x + 2a. Group them as x(x + a) + 2(x + a), then take out the common factor x + a. This gives (x + 2)(x + a), option D. Expanding this product returns x² + ax + 2x + 2a.
AI-reviewed · calculation checkedSolve 3x² + 6x − 2 = 0.
B: x = −1 ± √15/3
Divide by 3 and complete the square: x² + 2x = 2/3, so (x + 1)² = 5/3. Taking both square roots gives x = −1 ± √(5/3) = −1 ± √15/3, option B. Both values satisfy the original equation.
AI-reviewed · calculation checkedFor x ≠ −1, simplify 1/(5x + 5) + 1/(7x + 7).
D: 12/(35x + 35)
Factor each denominator: 5x + 5 = 5(x + 1) and 7x + 7 = 7(x + 1). The common denominator is 35(x + 1), giving (7 + 5)/[35(x + 1)] = 12/(35x + 35), option D. The restriction x ≠ −1 comes from the original denominators.
AI-reviewed · calculation checkedAt which points does the curve y = −x² + 3x + 4 intersect the coordinate axes?
D: (0, 4), (4, 0), (−1, 0)
On the y-axis, x = 0, so y = 4 and the intercept is (0, 4). On the x-axis, y = 0, giving x² − 3x − 4 = 0. Factorizing gives (x − 4)(x + 1) = 0, so x = 4 or −1. The three intercepts are (0, 4), (4, 0) and (−1, 0), option D.
AI-reviewed · calculation checkedFactorize (4a + 3)² − (3a − 2)².
C: (a + 5)(7a + 1)
Use U² − V² = (U − V)(U + V), with U = 4a + 3 and V = 3a − 2. Then U − V = a + 5 and U + V = 7a + 1. Therefore the factorization is (a + 5)(7a + 1), option C. Be careful: subtracting 3a − 2 adds 2 to the constant term.
AI-reviewed · calculation checkedFor x ≠ 2 and x ≠ −2, simplify 1/(x − 2) + 1/(x + 2) + 2x/(x² − 4).
D: 4x/(x² − 4)
Because x² − 4 = (x − 2)(x + 2), use x² − 4 as the common denominator. The numerator becomes (x + 2) + (x − 2) + 2x = 4x. Thus the expression is 4x/(x² − 4), option D. Keep x ≠ ±2 because the original expression is undefined there.
AI-reviewed · calculation checkedMake v the subject of S = √(6/v − w/2), using the principal square root. Assume S ≥ 0 and 2S² + w ≠ 0.
C: v = 12/(2S² + w)
Square both sides to obtain S² = 6/v − w/2. Multiply by 2: 2S² = 12/v − w. Hence 2S² + w = 12/v and v = 12/(2S² + w), option C. The denominator must be nonzero. Substituting back gives √(S²) = S because S ≥ 0.
AI-reviewed · calculation checkedFind all real x satisfying 16ˣ − 5 × 4ˣ + 4 = 0.
C: 0 and 1
Write 16ˣ = (4ˣ)² and substitute u = 4ˣ. Then u² − 5u + 4 = (u − 1)(u − 4) = 0. Therefore 4ˣ = 1 or 4, giving x = 0 or 1, option C. The exponential function 4ˣ is one-to-one, so these are all the real solutions.
AI-reviewed · calculation checkedFor real numbers with b ≠ 0 and d ≠ 0, suppose a/b = c/d = k. Express (3a² − ac + c²)/(3b² − bd + d²) in terms of k.
D: k²
The equal ratios imply a = kb and c = kd. Substitute into the numerator: 3(kb)² − (kb)(kd) + (kd)² = k²(3b² − bd + d²). Divide by the denominator to get k², option D.
AI-reviewed · calculation checkedAt which points does the line y = 2x + 1 intersect the curve y = 2x² + 5x − 1?
A: (−2, −3) and (1/2, 2)
At an intersection, the two expressions for y are equal: 2x + 1 = 2x² + 5x − 1. Hence 2x² + 3x − 2 = (2x − 1)(x + 2) = 0, so x = 1/2 or −2. Substituting into y = 2x + 1 gives y = 2 or −3 respectively. The points are (−2, −3) and (1/2, 2), option A.
AI-reviewed · calculation checkedIf cos θ = a/b and a and b are nonzero, express 1 + tan² θ in terms of a and b.
A: b²/a²
The identity 1 + tan² θ = sec² θ gives 1 + tan² θ = 1/cos² θ. Substituting cos θ = a/b gives 1/(a²/b²) = b²/a², option A. The nonzero condition ensures that cos θ and these divisions are defined.
AI-reviewed · calculation checkedA vertical tower is 50 m high. Points X and Y lie on level ground on opposite sides of its base, with X, the base and Y on one straight line. The angles of elevation of the top from X and Y are 30° and 60° respectively. Find XY to two decimal places.
D: 115.47 m
Let O be the tower's base. From X, tan 30° = 50/XO, so XO = 50√3 ≈ 86.60 m. From Y, tan 60° = 50/YO, so YO = 50/√3 ≈ 28.87 m. The points are on opposite sides, so XY = XO + YO ≈ 115.47 m, option D. The old answer incorrectly added these two distances to obtain 101.03 m.
AI-reviewed · calculation checkedThe solid is a prism with trapezium PQRS as its uniform cross-section. PQ and RS are parallel and measure 6 m and 11 m. QR is perpendicular to them and measures 12 m. The prism extends 8 m perpendicular to this cross-section. Find its volume.
An arc of a circle of radius 6 cm is 8 cm long. Find the area of its sector.
B: 24 cm²
For arc length s and radius r, the sector area is A = rs/2. Here A = (6 × 8)/2 = 24 cm², option B. Equivalently, θ = s/r = 4/3 radians and A = r²θ/2. The 8 cm measurement is the curved arc length, not a chord.
AI-reviewed · calculation checkedIn triangle XYZ shown, XY = 4, XZ = 3 and YZ = 6. Determine cos Z.
Find the total surface area, including the circular base, of a right circular cone with radius 2√3 cm and slant height 4√3 cm.
D: 36π cm²
The total surface area is the curved area plus the base: πrl + πr². With r = 2√3 and l = 4√3, the curved area is π(2√3)(4√3) = 24π cm² and the base area is π(2√3)² = 12π cm². The total is 36π cm², option D. The π factor does not cancel.
AI-reviewed · calculation checkedAn open rectangular box has external length 4 m, width 3 m and height 4 m. Find the total cost of painting its outside, including the base, at ₦2.00 per square metre.
C: ₦136.00
The open box has no top. Its outside consists of a 4 × 3 = 12 m² base, two 4 × 4 = 16 m² sides and two 3 × 4 = 12 m² sides. Total area = 12 + 2(16) + 2(12) = 68 m². At ₦2.00 per m², the cost is 68 × 2 = ₦136.00, option C.
AI-reviewed · calculation checkedAn integer is selected uniformly at random from 40 to 50 inclusive. What is the probability that it is prime?
A: 3/11
There are 50 − 40 + 1 = 11 integers in the inclusive range. The primes are 41, 43 and 47: exactly three numbers. In particular, 49 = 7 × 7 is not prime. Therefore the probability is 3/11, option A.
AI-reviewed · calculation checkedThe people in a city with a population of 0.9 million are grouped by age in the pie chart. The sector angles are: under 15 years, 116°; 15–29 years, 104°; 30–44 years, 64°; 45–59 years, 52°; 60 years and over, 24°. How many people are in the 15–29 years group?
A group of students obtained the following scores in a Mathematics test. Each pair gives (score, frequency): (1, 2), (2, 4), (3, 7), (4, 14), (5, 12), (6, 6), (7, 4), (8, 1). If m is the mode and S is the number of students who scored 4 or less, what is (S, m)?
A: (27, 4)
The mode is the score with the greatest frequency. Score 4 occurs 14 times, more than any other score, so m = 4. The number scoring 4 or less is S = 2 + 4 + 7 + 14 = 27. Therefore (S, m) = (27, 4).
AI-reviewed · calculation checkedArrange 6/7, 13/15 and 0.865 in ascending order.
A: 6/7 < 0.865 < 13/15
6/7 = 0.857142...; 0.865; 13/15 = 0.866666... . Only A is ascending.
AI-reviewed · calculation checkedEvaluate 100 + 1/100 + 3/1000 + 27/10000, correct to two decimal places.
A: 100.02
100 + 0.01 + 0.003 + 0.0027 = 100.0157, which rounds to 100.02.
AI-reviewed · calculation checkedSimplify 1/2 + 1/(2 + 1/(2 − 1/(4 + 1/5))).
C: 169/190
Work from the innermost fraction: 4 + 1/5 = 21/5, so 2 - 1/(21/5) = 2 - 5/21 = 37/21. Next, 2 + 1/(37/21) = 2 + 21/37 = 95/37. The whole expression is therefore 1/2 + 37/95 = 95/190 + 74/190 = 169/190, option C.
AI-reviewed · calculation checkedJohn gives one third of his money to Janet, who initially has ₦105. John then has one quarter of Janet’s new total. How much money did John have originally?
A: ₦45
Let original money be x. Remaining 2x/3 = (105 + x/3)/4. Thus 8x = 315 + x and x = 45. Check: John keeps 30, Janet has 120; 30 is one quarter of 120.
AI-reviewed · calculation checkedFind x if log₉(x) = 1.5.
B: 27.0
log base 9 of x = 1.5 gives x = 9^(3/2) = 27. Only B matches.
AI-reviewed · calculation checked22½% of one Nigerian naira has the same value as 17⅒% of one unit of a foreign currency M. Express the value of one M in naira.
C: 1 M = 1 18/57 naira
22.5% of one naira has the same value as 17.1% of one M. Therefore one M = 22.5/17.1 naira = 75/57 naira = 1 18/57 naira.
AI-reviewed · calculation checkedFind the values of p for which x² − (p − 2)x + 2p + 1 = 0 has equal roots.
A: 0, 12
Equal roots require discriminant zero: (p-2)^2 - 4(2p+1) = p^2 - 12p = p(p-12). Thus p is 0 or 12, A.
AI-reviewed · calculation checkedThe cost per head of a meal is partly constant and partly inversely proportional to the number of people. It is 30 kobo for 100 people and 60 kobo for 40 people. Find the cost per head for 50 people.
D: 50 kobo
Cost per head C = a + b/n. Subtracting the conditions gives 30 = b(1/40 - 1/100), so b = 2000 and a = 10. At 50 people, C = 10 + 2000/50 = 50 kobo.
AI-reviewed · calculation checkedFactorise 9 − (x² − 3x − 1)².
A: −(x − 4)(x + 1)(x − 1)(x − 2)
9 - (x^2 - 3x - 1)^2 = (-x^2 + 3x + 4)(x^2 - 3x + 2) = -(x-4)(x+1)(x-1)(x-2).
AI-reviewed · calculation checkedFactorise abx² + 8y − 4bx − 2axy.
A: (ax − 4)(bx − 2y)
Group as bx(ax-4) - 2y(ax-4), giving (ax-4)(bx-2y). Expansion gives abx^2 - 4bx - 2axy + 8y.
AI-reviewed · calculation checkedIf the quadratic expression 3x² − 7x + R is a perfect square, find R.
D: 49/12
Complete the square: 3x^2 - 7x + R = 3(x - 7/6)^2 when R = 49/12. The discriminant check gives 49 - 12R = 0.
AI-reviewed · calculation checkedSolve the simultaneous equations 2x + y = 4 and x² + xy = −12. Give both ordered pairs (x, y).
A: (6, -8); (-2,8)
From y = 4 - 2x, x^2 + xy = -12 becomes x^2 - 4x - 12 = 0. Hence x = 6 or -2, paired with y = -8 or 8 respectively.
AI-reviewed · calculation checkedSolve the simultaneous equations 2x − 3y + 10 = 5 and 10x − 6y = 5.
A: x = 2 1/2, y = 3 1/3
First rearrange 2x - 3y + 10 = 5 to get 2x - 3y = -5. Double it: 4x - 6y = -10. Subtract this from 10x - 6y = 5 to obtain 6x = 15, so x = 5/2. Substituting gives y = 10/3. Thus x = 2 1/2 and y = 3 1/3, option A.
AI-reviewed · calculation checkedIn triangle XYZ, XY = 13 cm, YZ = 9 cm and XZ = 11 cm. If angle XYZ is θ, find cos θ.
E: 43/78
The angle at Y is opposite XZ = 11. Cosine rule gives (13^2 + 9^2 - 11^2)/(2*13*9) = 129/234 = 43/78.
AI-reviewed · calculation checkedGoals scored — Number of matches 0 — 3 1 — 5 2 — 7 3 — 4 4 — 1 5 — 0
Using the frequency data, find the mean number of goals per match and the mode, respectively.
B: (1.75, 2)
For goals 0,1,2,3,4,5 and frequencies 3,5,7,4,1,0: total frequency 20, weighted sum 35, mean 1.75, mode 2.
AI-reviewed · calculation checkedThe corresponding lengths of two similar rectangular blocks are in the ratio 2:3. If the larger block has volume 351 cm³, find the volume of the smaller block.
E: 104.00 cm³
Similar solids have volume ratio equal to the cube of their length ratio. Smaller volume = 351*(2/3)^3 = 104 cubic centimetres.
AI-reviewed · calculation checkedThe bearing of a bird from a hunter is N72°E. What is the bearing of the hunter from the bird?
B: S72°W
N72 degrees E is a whole-circle bearing of 72 degrees. The reverse bearing is 252 degrees, expressed as S72 degrees W.
AI-reviewed · calculation checkedIn triangle XYZ, Y, K and Z lie on a straight line, angle XKZ = 90°, XK = 15 cm, XZ = 25 cm and YK = 8 cm. Find the area of triangle XYZ.
An integer is chosen uniformly at random from 1 to 10 inclusive. What is the probability that it is prime or a multiple of 3?
B: 3/5
Primes in 1..10 are 2,3,5,7. Multiples of three are 3,6,9. Their union is 2,3,5,6,7,9: six outcomes out of ten, so 3/5. Count 3 only once; 1 is not prime.
AI-reviewed · calculation checkedFind the area of a regular hexagon inscribed in a circle of radius 8 cm.
B: 96√3 cm²
The six triangles formed by joining the centre to the vertices are equilateral, each with side 8. Each has area sqrt(3)*8^2/4 = 16sqrt(3). Total = 96sqrt(3) square centimetres, B.
AI-reviewed · calculation checkedM, N, Q and P lie in that order on a circle. Lines MN and PQ meet at X, with M–N–X and P–Q–X in that order. Lines NQ and MP meet at Y, with N–Q–Y and M–P–Y in that order. Angle MNQ = 86° and angle NQP = 122°. Find x = angle NXQ and y = angle QYP.
A: (28°, 36°)
In triangle NXQ, angle XNQ = 180° − 86° = 94° and angle NQX = 180° − 122° = 58°. Thus x = 180° − 94° − 58° = 28°. Opposite angles of cyclic quadrilateral MNQP sum to 180°, so angle MPQ = 180° − 86° = 94°. In triangle PQY, angle QPY = 180° − 94° = 86° and angle PQY = 180° − 122° = 58°. Therefore y = 180° − 86° − 58° = 36°.
AI-reviewed · calculation checkedIf cos θ = √3/2 and θ is acute (0° < θ < 90°), calculate cot(90° − θ) / sin² θ.
A: 4√3/3
For an acute angle, cos(theta) = sqrt(3)/2 implies theta = 30 degrees. cot(90-theta)/sin^2(theta) = cot(60)/sin^2(30) = (1/sqrt(3))/(1/4) = 4sqrt(3)/3.
AI-reviewed · calculation checkedGHIJKLMN is a cube of side length a. Find the length of HN shown in the diagram.
PQRS is a trapezium of area 14 cm², with PQ parallel to RS. If PQ = 4 cm and RS = 3 cm, find the area of triangle SQR in cm².
B: 6.0
Trapezium area = (4+3)h/2 = 14, hence height h = 4. Triangle SQR uses base SR = 3 and the same perpendicular height, giving area 3*4/2 = 6 square centimetres, B.
AI-reviewed · calculation checkedTwo points X and Y are both at latitude 60°S, with longitudes 147°E and 153°W respectively. Find, to the nearest kilometre, the shorter distance between them along this parallel of latitude. Take the equatorial circumference 2πR = 4 × 10⁴ km.
E: 3,333 km
The longitude difference measured the long way is 147° + 153° = 300°. The shorter separation across the date line is 360° − 300° = 60°. The circumference of the 60°S parallel is 40000 cos 60° = 20000 km. The distance is (60/360) × 20000 = 3333.333… km, which rounds to 3,333 km.
AI-reviewed · calculation checkedIn the diagram, the circle has radius 3 units. Two radii enclose an angle of 120°. Find the area of the shaded minor segment.
P sold a bicycle to Q at a profit of 10%. Q then sold it to R for ₦209 at a loss of 5%. How much did the bicycle originally cost P?
A: ₦200
Q sold the bicycle for 95% of the amount Q paid. Therefore Q paid ₦209 ÷ 0.95 = ₦220. This was 110% of P’s original cost, so P paid ₦220 ÷ 1.10 = ₦200, option A. Check: ₦200 × 1.10 × 0.95 = ₦209.
AI-reviewed · calculation checkedIf the price of each orange were raised by ½ kobo, a customer spending ₦2.40 would be able to buy 16 fewer oranges. What is the price of an orange before the increase?
A: 2½ kobo
Convert the budget to 240 kobo. Let the original price be p kobo. Then 240/p − 240/(p + 0.5) = 16. Combining the fractions gives p(p + 0.5) = 7.5, or (p − 2.5)(p + 3) = 0. A price must be positive, so p = 2.5 kobo, option A. The customer could buy 96 oranges before the increase and 80 afterwards: 16 fewer.
AI-reviewed · calculation checkedBola chooses an integer at random from 1 through 300 inclusive, with each integer equally likely. What is the probability that it is divisible by 4?
B: 1/4
The multiples of 4 are 4, 8, 12, …, 300. There are 300 ÷ 4 = 75 such integers among 300 equally likely choices. The probability is 75/300 = 1/4, option B.
AI-reviewed · calculation checkedWithout logarithm tables, evaluate (log base 3 of 27 − log base 1/4 of 64) / (log base 3 of 1/81).
C: −3/2
Since 3³ = 27, the first logarithm is 3. Since (1/4)⁻³ = 64, the second is −3. Since 3⁻⁴ = 1/81, the denominator is −4. The fraction is therefore [3 − (−3)]/(−4) = 6/(−4) = −3/2, option C.
AI-reviewed · calculation checkedFactorise 6x² − 14x − 12.
C: 2(x - 3) (3x + 2)
First take out 2: 2(3x² − 7x − 6). Split the middle term as −9x + 2x, then group: 2[3x(x − 3) + 2(x − 3)] = 2(x − 3)(3x + 2), option C. Expanding gives 6x² − 14x − 12.
AI-reviewed · calculation checkedIn a racing competition, Musa covered 5x km in the first hour and (x + 10) km in the next hour. He finished behind Ngozi, who covered 118 km in those two hours. Which inequality describes the possible positive values of x?
E: 0 < x < 18
Musa’s total distance is 5x + (x + 10) = 6x + 10 km. Because it is less than 118 km, 6x + 10 < 118, giving x < 18. His first-hour distance is positive, so x > 0. Together, 0 < x < 18, option E.
AI-reviewed · calculation checkedTunde and Shola can finish a piece of work together in 18 days. Tunde can do it alone in x days, while Shola would take 15 days longer. Which equation is satisfied by x?
C: x² - 21x − 270 = 0
Their daily work rates are 1/x and 1/(x + 15). Together, 1/x + 1/(x + 15) = 1/18. Multiplying by 18x(x + 15) gives 18(2x + 15) = x(x + 15), hence x² − 21x − 270 = 0, option C. Its positive solution is x = 30; working with Shola’s 45-day rate gives 1/30 + 1/45 = 1/18.
AI-reviewed · calculation checkedWhich quadratic equation has roots 1 − √13 and 1 + √13?
E: x² − 2x − 12 = 0
The roots add to 2. Their product is (1 − √13)(1 + √13) = 1 − 13 = −12. A monic quadratic with these roots is x² − (sum)x + product = 0, so x² − 2x − 12 = 0, option E.
AI-reviewed · calculation checkedFind a factor which is common to all three binomial expressions 4a² − 9b², a³ + 27b³ and (4a + 6b)².
E: none
First, 4a² − 9b² = (2a − 3b)(2a + 3b), while (4a + 6b)² = 4(2a + 3b)². Their only possible shared nonconstant linear factor is 2a + 3b. But a³ + 27b³ = (a + 3b)(a² − 3ab + 9b²), which is not divisible by 2a + 3b. Therefore none of the listed factors divides all three expressions: option E.
AI-reviewed · calculation checkedA triangle has sides (x + 4) cm, x cm and (x − 4) cm. If the cosine of its largest angle is 1/5, find x.
A: 24 cm
The largest angle is opposite x + 4. By the cosine rule, 1/5 = [x² + (x − 4)² − (x + 4)²]/[2x(x − 4)] = (x − 16)/[2(x − 4)]. Thus 5x − 80 = 2x − 8, so 3x = 72 and x = 24, option A. The resulting sides 20, 24 and 28 form a valid triangle.
AI-reviewed · calculation checkedFind the integer values of x satisfying −3 < 2 − 5x < 12.
C: −1, 0
Subtract 2 throughout: −5 < −5x < 10. Dividing by −5 reverses both inequalities, giving 1 > x > −2, or −2 < x < 1. The integers strictly between these bounds are −1 and 0, option C. The endpoints −2 and 1 are excluded.
AI-reviewed · calculation checkedTwo independent fair six-sided dice are rolled. What is the probability that both show the same number?
E: 1/6
There are 6 × 6 = 36 equally likely ordered outcomes. Six are doubles: (1,1), (2,2), (3,3), (4,4), (5,5) and (6,6). The probability is 6/36 = 1/6, option E.
AI-reviewed · calculation checkedIn triangle PST, Q lies on PS and R lies on PT. ∠TSP = ∠PRQ, QR = 8 cm, PR = 6 cm and ST = 12 cm. Find SP.
C: 9 cm
Triangles PRQ and PST share the angle at P, and ∠PRQ = ∠PST. They are similar, with PR corresponding to PS and RQ corresponding to ST. Therefore 6/SP = 8/12, so SP = 6 × 12 ÷ 8 = 9 cm.
AI-reviewed · calculation checkedIn triangle XYZ, W lies on YZ and XW is perpendicular to YZ. YZ = 10 cm, XZ = 5 cm and WZ = 4 cm. Calculate XY.
B: 3√5 cm
In right triangle XWZ, XW² = 5² − 4² = 9, so XW = 3 cm. Since YZ = 10 cm and WZ = 4 cm, YW = 6 cm. Thus XY² = 3² + 6² = 45 and XY = 3√5 cm.
AI-reviewed · calculation checkedDiameter range (cm) — Frequency 3.35–3.45 — 3 3.45–3.55 — 6 3.55–3.65 — 7 3.65–3.75 — 4
The grouped distribution describes the diameters of 20 copper spheres. Estimate their mean diameter.
C: 3.56 cm
Use the class midpoints 3.40, 3.50, 3.60 and 3.70 cm. Multiply each by its frequency: 3.40 × 3 = 10.20, 3.50 × 6 = 21.00, 3.60 × 7 = 25.20, and 3.70 × 4 = 14.80. Their sum is 71.20 cm. Divide by the total frequency, 20, to obtain an estimated mean of 3.56 cm, option C. It is an estimate because the individual diameters within each interval are not given.
AI-reviewed · calculation checkedIf M represents the median and D the mode of the measurements 5, 9, 3, 5, 8 then (M,D) is
D: (5,5)
Put the values in ascending order: 3, 5, 5, 8, 9. The median is the middle value, 5. The mode is the most frequent value, also 5. Therefore (M, D) = (5, 5), option D.
AI-reviewed · calculation checkedSolve 4x − 3 = 3x + y = 2y + 5x − 12.
A: x = 5, y = 2
Equating the first two expressions gives y = x − 3. Equating the last two gives 2x + y = 12. Substitute: 2x + x − 3 = 12, so x = 5 and y = 2. Check: all three expressions equal 17.
AI-reviewed · calculation checkedIf x = 1 is a root of x³ − 2x² − 5x + 6 = 0, find the other roots.
C: 3 and –2
Dividing by x − 1 gives x² − x − 6 = (x − 3)(x + 2). Thus the polynomial is (x − 1)(x − 3)(x + 2). Apart from x = 1, the roots are 3 and −2: option C.
AI-reviewed · calculation checkedTriangles PQR and PRS share side PR. In triangle PQR, angle QPR = 90°, angle PQR = 60° and QR = 8 cm. In triangle PRS, angle PRS = 90° and angle RPS = 45°. Find PS.
B: 4√6cm
In right triangle PQR, QR is the hypotenuse. PR = 8 sin 60° = 4√3 cm. In right triangle PRS, PS is the hypotenuse, so cos 45° = PR/PS. Therefore PS = 4√3 ÷ (√2/2) = 4√6 cm.
AI-reviewed · calculation checkedIf (x + 2) and (x − 1) are factors of ℓx³ + 2kx² + 24, find ℓ and k.
A: ℓ = −6, k = −9
A factor x + 2 gives f(−2) = 0, so −8ℓ + 8k + 24 = 0, or ℓ − k = 3. A factor x − 1 gives f(1) = 0, so ℓ + 2k = −24. Substitute ℓ = k + 3 to get 3k = −27. Thus k = −9 and ℓ = −6, option A. Both −2 and 1 make the cubic equal to zero.
AI-reviewed · calculation checkedy varies partly as the square of x and partly as the inverse of the square root of x. Find the expression for y if y = 2 when x = 1 and y = 6 when x = 4.
A: y = 10x²/31 + 52/(31√x)
Write y = ax² + b/√x. The first condition gives a + b = 2. The second gives 16a + b/2 = 6, or 32a + b = 12. Subtracting gives 31a = 10, so a = 10/31 and b = 52/31. Therefore y = 10x²/31 + 52/(31√x), option A. Substitution gives 2 at x = 1 and 6 at x = 4.
AI-reviewed · calculation checkedSimplify [(x − 7)/(x² − 9)] × [(x² − 3x)/(x² − 49)].
D: x/[(x + 3)(x + 7)]
Factor x² − 9 = (x − 3)(x + 3), x² − 3x = x(x − 3), and x² − 49 = (x − 7)(x + 7). Cancel the common factors x − 3 and x − 7 to obtain x/[(x + 3)(x + 7)], option D. The original expression excludes x = −7, −3, 3 and 7; cancellation does not remove those restrictions.
AI-reviewed · calculation checkedIf w varies inversely as V and u varies directly as w³, find the relationship between u and V, given that u = 1 when V = 2.
E: u = 8/V³
Write w = a/V and u = bw³. Substitution gives u = ba³/V³ = k/V³. Since u = 1 when V = 2, k = 1 × 2³ = 8. Therefore u = 8/V³, option E.
AI-reviewed · calculation checkedSolve the simultaneous equations for x x² + y – 8 = 0 y + 5x – 2 = 0
C: 6, -1
From y + 5x − 2 = 0, y = 2 − 5x. Substitute into x² + y − 8 = 0: x² − 5x − 6 = 0. Factor: (x − 6)(x + 1) = 0, so x = 6 or x = −1. These are the two x-values in option C; the corresponding y-values are −28 and 7.
AI-reviewed · calculation checkedFind the sector angles, in order, for a pie chart representing the values 6, 10, 14, 16 and 26.
D: 30°, 50°, 70°, 80°, 130°
The total is 6 + 10 + 14 + 16 + 26 = 72. Each unit represents 360° ÷ 72 = 5°. Multiply each value by 5 to obtain 30°, 50°, 70°, 80° and 130°. They sum to 360°, so option D is correct.
AI-reviewed · calculation checkedThe letters of the word MATRICULATION are cut and put into a box. One of the letter is drawn at random from the box. Find the probability of drawing a vowel.
C: 6/13
MATRICULATION contains 13 letters. Its vowels are A, I, U, A, I and O: six letters. Each letter is equally likely to be drawn, so the probability is 6/13, option C. Repeated vowels each count because each letter is a separate item.
AI-reviewed · calculation checkedCorrect each of the number 59.81789 and 0.0746829 to three significant figures and multiply them, giving your answer to three significant figures.
C: 4.47
First round each number to three significant figures, as requested: 59.81789 becomes 59.8 and 0.0746829 becomes 0.0747. Their product is 4.46706. Rounded to three significant figures, this is 4.47, option C.
AI-reviewed · calculation checkedIf (2/3)^m × (3/4)^n = 256/729, find m and n.
D: m = 4, n = -2
Rewrite the left side as 2^(m − 2n) × 3^(n − m). The right side is 2^8 × 3^(−6). Equate the exponents: m − 2n = 8 and n − m = −6. Solving gives n = −2 and m = 4, option D. Substituting these values gives (16/81) × (16/9) = 256/729.
AI-reviewed · calculation checkedFactorise completely 81a⁴ − 16b⁴.
C: (3a − 2b)(3a + 2b)(9a² + 4b²)
Write the expression as (9a²)² − (4b²)². The difference-of-squares rule gives (9a² − 4b²)(9a² + 4b²). Factor the first bracket again: (3a − 2b)(3a + 2b)(9a² + 4b²), option C. The conjugate factors need opposite signs; repeating the minus factor does not give the original expression.
AI-reviewed · calculation checkedPR is the diameter of a semicircle of radius 3.5 cm. A triangle PRT lies on the opposite side of PR from the semicircle, with angle RPT = angle PRT = 60°. Find the outer perimeter formed by PT, TR and the semicircular arc from R to P. Use π = 22/7.
A: 25 cm
PR = 2 × 3.5 = 7 cm. Both base angles of triangle PRT are 60°, so its third angle is also 60° and PT = TR = 7 cm. The semicircular arc has length πr = (22/7) × 3.5 = 11 cm. The outer perimeter is 7 + 7 + 11 = 25 cm. PR is inside the combined figure and is not part of its outer perimeter.
AI-reviewed · calculation checkedFind the mean of 24.57, 25.63, 24.32, 26.01 and 25.77.
C: 25.26
Add the five values: 24.57 + 25.63 + 24.32 + 26.01 + 25.77 = 126.30. Divide by 5 to obtain 25.26, option C.
AI-reviewed · calculation checked