40 questions. Try each one, then open the answer to check yourself and read the explanation.
Subtract 164189 from 186309.
Answer: D (22119)
In base ten, 186309 = 12906 and 164189 = 11276, so the difference is 1630. 1630 = 2 × 729 + 2 × 81 + 1 × 9 + 1, which is 22119.
If 55x + 52x = 7710, find x.
Answer: C (7)
(5x + 5) + (5x + 2) = 77 gives 10x = 70, so x = 7.
Simplify 7 1/12 − 4 3/4 + 2 1/2.
Answer: C (4 5/6)
In twelfths: 85/12 − 57/12 + 30/12 = 58/12 = 4 5/6.
Evaluate (81.81 + 99.44)/(20.09 + 36.16) correct to 3 significant figures.
Answer: B (3.22)
The top is 181.25 and the bottom is 56.25, and 181.25 ÷ 56.25 = 3.222..., which is 3.22 to 3 s.f.
A man bought a second-hand photocopying machine for N34 000. He serviced it at a cost of N2, 000 and then sold it at a profit of 15%. What was the selling price?
Answer: C (N41, 400)
His total cost was N36,000. A 15% profit adds N5,400, so the selling price is N41,400.
A student spent 1/5 of his allowances on books, 1/3 of the remainder on food and kept the rest for contingencies. What fraction was kept?
Answer: B (8/15)
After books, 4/5 is left. Food takes 1/3 × 4/5 = 4/15. He keeps 1 − 1/5 − 4/15 = 8/15.
Solve 52(x−1) × 5x+1 = 0.04.
Answer: D (−1/3)
0.04 = 1/25 = 5−2. Adding the powers: 2x − 2 + x + 1 = −2, so 3x = −1 and x = −1/3.
If log102 = 0.3010 and log107 = 0.8451, evaluate log10280.
Answer: B (2.4471)
280 = 22 × 7 × 10, so log 280 = 2(0.3010) + 0.8451 + 1 = 2.4471.
Simplify (5 + √7)/(3 + √7)
Answer: B (4 − √7)
Multiply top and bottom by 3 − √7: (15 − 5√7 + 3√7 − 7)/(9 − 7) = (8 − 2√7)/2 = 4 − √7.
If x = {n2 + 1 : n is a positive integer and 1 ≤ n ≤ 5}, y = {5n : n is a positive integer and 1 ≤ n ≤ 5}, find x ∩ y.
Answer: A ({5, 10})
x = {2, 5, 10, 17, 26} and y = {5, 10, 15, 20, 25}. The common elements are 5 and 10.
I. S ∩ T ∩ W = S
II. S ∪ T ∪ W = W
III. T ∩ W = S
If S ⊂ T ⊂ W, which of the above statements are true?
Answer: A (I and II)
Since S is inside T and T is inside W, the intersection of all three is S (I true) and the union is W (II true). But T ∩ W = T, not S, so III is false.
If P = √(rs3/t), express r in terms of p, s and t.
Answer: A (p2t/s3)
Square both sides: P2 = rs3/t, so r = P2t/s3.
A polynomial in x whose roots are 4/3 and −3/5 is
Answer: A (15x2 − 11x − 12)
The factors are (3x − 4) and (5x + 3). Multiplying: 15x2 + 9x − 20x − 12 = 15x2 − 11x − 12.
W is directly proportional to U. If W = 5 when U = 3, find U when W = 2/7.
Answer: A (6/35)
W = kU gives k = 5/3. Then U = W/k = 2/7 × 3/5 = 6/35.
Determine the value of x for which (x2 − 1) > 0.
Answer: A (x < −1 or x > 1)
x2 − 1 = (x − 1)(x + 1) is positive when both factors have the same sign, i.e. x < −1 or x > 1.
Find the range of values of x for which 3x − 7 ≤ 0 and x + 5 > 0
Answer: C (−5 < x ≤ 7/3)
3x − 7 ≤ 0 gives x ≤ 7/3, and x + 5 > 0 gives x > −5. Together −5 < x ≤ 7/3.
The sum of the first n terms of the arithmetic progression 5, 11, 17, 23, 29, 35, ... is
Answer: B (n(3n + 2))
a = 5 and d = 6, so Sn = n/2 [10 + 6(n − 1)] = n/2 (6n + 4) = n(3n + 2).
If m * n = n − (m + 2) for any real numbers m and n, find the value of 3 * (−5).
Answer: C (−10)
3 * (−5) = −5 − (3 + 2) = −10.
A binary operation ⊗ defined on the set of integers is such that m ⊗ n = m + n + mn for all integers m and n. Find the inverse of −5 under this operation, if the identity element is 0.
Answer: A (−5/4)
−5 ⊗ a = 0 gives −5 + a − 5a = 0, so −4a = 5 and a = −5/4.
If Q = (9 −2; −7 4), then |Q| is (matrix written row by row)
Answer: C (22)
|Q| = 9 × 4 − (−2)(−7) = 36 − 14 = 22.
If P = (x + 3 x + 2; x + 1 x − 1), evaluate x if |P| = −10. (Matrix written row by row.)
Answer: D (5)
|P| = (x + 3)(x − 1) − (x + 2)(x + 1) = (x2 + 2x − 3) − (x2 + 3x + 2) = −x − 5. So −x − 5 = −10 and x = 5.
Find the acute angle between the straight lines y = x and y = √3x
Answer: A (15°)
y = x makes 45° with the x-axis (tan 45° = 1) and y = √3x makes 60° (tan 60° = √3). The angle between them is 60° − 45° = 15°.
In the figure above, TS // XY and XY = TY, ∠STZ = 34°, ∠TXY = 47°, find the angle marked n.
Answer: B (52°)
XY = TY, so ∠XTY = 47° and the exterior angle ∠TYZ = 94°. ∠TZY = ∠STZ = 34° (alternate angles). In triangle TYZ, n = 180° − 94° − 34° = 52°.
If the hypotenuse of a right-angled isosceles triangle is 2 cm, what is the area of the triangle?
Answer: B (1 cm2)
Each equal side x satisfies 2x2 = 4, so x = √2. Area = ½ × √2 × √2 = 1 cm2.
A chord is drawn 5 cm away from the centre of a circle of radius 13 cm. Calculate the length of a chord.
Answer: D (24cm)
Half the chord = √(132 − 52) = √144 = 12cm, so the chord is 24cm.
Find the locus of a particle which moves in the first quadrant so that it is equidistant from the lines x = 0 and y = 0 (where k is a constant)
Answer: B (x − y = 0)
Being equidistant from both axes means x = y, i.e. the line x − y = 0 (in the first quadrant).
What is the value of p if the gradient of the line joining (−1, p) and (p, 4) is 2/3?
Answer: D (2)
(4 − p)/(p + 1) = 2/3 gives 12 − 3p = 2p + 2, so 5p = 10 and p = 2.
What is the value of r if the distance between the points (4, 2) and (1, r) is 3 units?
Answer: B (2)
(1 − 4)2 + (r − 2)2 = 9 gives (r − 2)2 = 0, so r = 2.
Find the value of sin 45° − cos 30°.
Answer: D ((√2 − √3)/2)
sin 45° = √2/2 and cos 30° = √3/2, so the difference is (√2 − √3)/2.
A cliff on the bank of a river is 300 metres high. If the angle of depression of a point on the opposite side of the river is 60°, find the width of the river.
Answer: C (100√3 m)
The angle of elevation from the point is 60°, so width = 300/tan 60° = 300/√3 = 100√3 m.
If y = 3 cos 4x, dy/dx equals
Answer: D (−12 sin 4x)
dy/dx = 3 × (−sin 4x) × 4 = −12 sin 4x.
If s = (2 + 3t)(5t − 4), find ds/dt when t = 4/5 secs.
Answer: C (22 units per sec.)
s = 15t2 − 2t − 8, so ds/dt = 30t − 2 = 24 − 2 = 22 at t = 4/5.
What value of x will make the function x(4 − x) a maximum?
Answer: C (2)
y = 4x − x2 gives dy/dx = 4 − 2x = 0 at x = 2, and the second derivative −2 is negative, so this is a maximum.
The distance travelled by a particle from a fixed point is given as s = (t3 − t2 − t + 5) cm. Find the minimum distance that the particle can cover from the fixed point.
Answer: B (4.0 cm)
ds/dt = 3t2 − 2t − 1 = (3t + 1)(t − 1) = 0 at t = 1 (for t > 0), and the second derivative is positive there. s = 1 − 1 − 1 + 5 = 4.0 cm.
Evaluate ∫sec2θ dθ
Answer: B (tan θ + k)
The derivative of tan θ is sec2θ, so the integral is tan θ + k.
The pie chart above represents 400 fruits on display in a grocery store. How many apples are in the store?
Answer: B (50)
The apple angle is x = 360° − (100° + 100° + 115°) = 45°. 45/360 × 400 = 50 apples.
5, 8, 6 and k occur with frequencies 3, 2, 4 and 1 respectively and have a mean of 5.7. Find the value of k.
Answer: C (2)
(15 + 16 + 24 + k)/10 = 5.7 gives 55 + k = 57, so k = 2.
What is the mean deviation of x, 2x, x + 1 and 3x, if their mean is 2?
Answer: A (0.5)
(7x + 1)/4 = 2 gives x = 1, so the numbers are 1, 2, 2, 3. The absolute deviations from 2 are 1, 0, 0, 1, so the mean deviation = 2/4 = 0.5.
In how many ways can 9 people be seated if 3 chairs are available?
Answer: B (504)
Order matters on the chairs: 9P3 = 9 × 8 × 7 = 504.
Marks: 2, 3, 4, 5, 6, 7, 8, 9
No. of students: 3, 4, 1, 0, 4, 5, 2, 1
The table above shows the distribution of marks of students in a test. Find the probability of passing the test if the pass mark is 5.
Answer: A (3/5)
There are 20 students and 0 + 4 + 5 + 2 + 1 = 12 scored 5 or more, so the probability is 12/20 = 3/5.