26 reviewed questions with answers and explanations.
Practice selection: full-paper coverage has not been confirmed.
Arrange 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.