22 reviewed questions with answers and explanations.
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Three 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 checked