44 questions. Try each one, then open the answer to check yourself and read the explanation.
If 5, 8, 6 and 2 occur with frequencies 3, 2, 4 and 1 respectively, find the product of the modal and the median number.
Answer: A (36)
The mode is 6 (frequency 4). In order the 10 numbers are 2, 5, 5, 5, 6, 6, 6, 6, 8, 8, so the median is 6. Product = 36.
The histogram above represents the weights of students who travelled out of their school for an examination. How many people made the trip?
Answer: C (29)
Add the bar heights: 2 + 3 + 5 + 6 + 8 + 4 + 1 = 29.
A senatorial candidate had planned to visit seven cities prior to a primary election. However, he could only visit four of the cities. How many different itineraries could be considered?
Answer: B (840)
Order matters for an itinerary, so it is 7P4 = 7 × 6 × 5 × 4 = 840.
The pie chart above illustrates the amount of private time a student spends in a week studying various subjects. Find the value of k.
Answer: C (30°)
105° + 75° + 2k + k + 3k = 360°, so 6k = 180° and k = 30°.
What is the mean deviation of 3, 5, 8, 11, 12 and 21?
Answer: A (4.7)
The mean is 60/6 = 10. The absolute deviations are 7, 5, 2, 1, 2, 11, which total 28, so the mean deviation = 28/6 ≈ 4.7.
Marks: 3, 4, 5, 6, 7, 8
Frequency: 5, y − 1, y, 9, 4, 1
The table above gives the frequency distribution of marks obtained by a group of students in a test. If the total mark scored is 200, calculate the value of y.
Answer: C (11)
15 + 4(y − 1) + 5y + 54 + 28 + 8 = 200 gives 9y + 101 = 200, so y = 11.
In how many ways can 6 subjects be selected from 10 subjects for an examination?
Answer: D (210)
10C6 = 10C4 = (10 × 9 × 8 × 7)/(4 × 3 × 2 × 1) = 210.
The cumulative frequency for 5 ≤ x ≤ 12 from the distribution above is
Answer: B (72)
Read f at x = 5, 6, ..., 12 from the graph: 10 + 12 + 10 + 8 + 6 + 8 + 10 + 8 = 72.
Integrate (x2 − √x)/x with respect to x
Answer: A (x2/2 − 2√x + k)
Divide first: x − x−½. Integrating gives x2/2 − 2x½ + k.
Find the value of x for which the function f(x) = 2x3 − x2 − 4x + 4 has a maximum value.
Answer: C (−2/3)
f'(x) = 6x2 − 2x − 4 = 2(3x + 2)(x − 1) = 0 at x = −2/3 or 1. f''(x) = 12x − 2 is negative at x = −2/3, so that is the maximum.
Determine the value of ∫0π/2 (−2cos x)dx.
Answer: A (−2)
∫(−2 cos x)dx = −2 sin x. From 0 to π/2: −2(1 − 0) = −2.
A binary operation ⊕ on real numbers is defined by x⊕y = xy + x + y for any two real numbers x and y. The value of (−3/4)⊕6 is
Answer: A (3/4)
(−3/4)(6) + (−3/4) + 6 = −18/4 − 3/4 + 24/4 = 3/4.
The graph above is represented by
Answer: B (y = x3 + 2x2 − x − 2)
The curve cuts the x-axis at −2, −1 and 1, so y = (x + 2)(x + 1)(x − 1) = x3 + 2x2 − x − 2, which also gives y = −2 at x = 0 as on the graph.
Make L the subject of the formula if d = √(42W/5L)
Answer: B (42W/5d2)
Square both sides: d2 = 42W/(5L), so 5Ld2 = 42W and L = 42W/(5d2).
The solution of the quadratic inequality (x2 + x − 12) ≥ 0 is
Answer: D (x ≥ 3 or x ≤ −4)
x2 + x − 12 = (x + 4)(x − 3), which is zero at −4 and 3 and positive outside them: x ≥ 3 or x ≤ −4.
Factorize 2t2 + t − 15
Answer: B ((t + 3)(2t − 5))
Find two numbers with product 2 × (−15) = −30 and sum 1: 6 and −5. 2t2 + 6t − 5t − 15 = 2t(t + 3) − 5(t + 3) = (t + 3)(2t − 5).
W ∝ L2 and W = 6 when L = 4. If L = √17, find W.
Answer: C (6 3/8)
W = kL2: 6 = 16k, so k = 3/8. Then W = 3/8 × 17 = 51/8 = 6 3/8.
A binary operation Δ is defined by aΔb = a + b + 1 for any real numbers a and b. Find the inverse of the real number 7 under the operation Δ, if the identity element is −1.
Answer: B (−9)
7 Δ a−1 must equal the identity: 7 + a−1 + 1 = −1, so a−1 = −9.
The nth term of the sequence ___, 3, 7, 16, 35, 74, ... is
Answer: A (5.2n−2 − n)
Check 5 × 2n−2 − n: n = 2 gives 3, n = 3 gives 7, n = 4 gives 16, n = 5 gives 35 and n = 6 gives 74.
If f(x) = 3x − 2, P = (2 1; −1 0) and I is 2 × 2 identity matrix, evaluate f(P). (Matrices written row by row.)
Answer: D ((4 3; −3 −2))
f(P) = 3P − 2I = (6 3; −3 0) − (2 0; 0 2) = (4 3; −3 −2).
Find the sum to infinity of the series 2 + 3/2 + 9/8 + 27/32 + ...
Answer: C (8)
a = 2 and r = 3/4, so S = 2/(1 − 3/4) = 8.
Evaluate (3 −2; −7 5) + 2(−2 4; 3 −1) (matrices written row by row)
Answer: B ((−1 6; −1 3))
2(−2 4; 3 −1) = (−4 8; 6 −2). Adding: (3 − 4, −2 + 8; −7 + 6, 5 − 2) = (−1 6; −1 3).
Find y, if √12 − √147 + y√3 = 0.
Answer: A (5)
√12 = 2√3 and √147 = 7√3, so 2√3 − 7√3 + y√3 = 0 and y = 5.
If x10 = 12145 find x.
Answer: C (184)
12145 = 1 × 125 + 2 × 25 + 1 × 5 + 4 = 184.
Evaluate ((0.5625)2 − (0.4375)2)/0.04 correct to 3 significant figures.
Answer: B (3.13)
Difference of two squares: (0.5625 + 0.4375)(0.5625 − 0.4375) = 1 × 0.125 = 0.125. 0.125/0.04 = 3.125 ≈ 3.13.
Find the value of x for which 2(32x−1) = 162.
Answer: A (5/2)
32x−1 = 81 = 34, so 2x − 1 = 4 and x = 5/2.
If log102 = x, express log1012.5 in terms of x.
Answer: D (2 − 3x)
12.5 = 100/8, so log 12.5 = log 100 − 3 log 2 = 2 − 3x.
A man made a profit of 5% when he sold an article for N60 000.00. How much would he have to sell the article to make a profit of 26%?
Answer: B (N72 000)
Cost price = 60,000 ÷ 1.05. To make 26% profit: 60,000 × 1.26/1.05 = 60,000 × 1.2 = N72,000.
Given: P = {1, 3, 5, 7, 9, 11} and Q = {2, 4, 6, 8, 10, 12}. Determine the relationship between P and Q.
Answer: A (P∩Q = ∅)
P holds only odd numbers and Q only even numbers, so they have no common element.
Evaluate 101122 − 10122.
Answer: C (11000002)
10112 = 11 and 1012 = 5. 121 − 25 = 96 = 64 + 32 = 11000002.
The sum of the ages of Musa and Lawal is 28 years. After sharing a certain sum of money in the ratio of their ages, Musa gets N600 and Lawal N800. How old is Lawal?
Answer: B (16 years)
Their ages are in the ratio 600 : 800 = 3 : 4, so Lawal is 4/7 × 28 = 16 years.
If X = {all perfect squares less than 40} and Y = {all odd numbers from 1 to 15}, find X∩Y.
Answer: D ({1, 9})
X = {1, 4, 9, 16, 25, 36} and Y = {1, 3, 5, ..., 15}. The common elements are 1 and 9.
Calculate the length of an arc of a circle of diameter 14 cm, which subtends an angle of 90° at the centre of the circle.
Answer: A (7π/2 cm)
r = 7, so the arc = 90/360 × 2π × 7 = 7π/2 cm.
If the lines 3y = 4x − 1 and qy = x + 3 are parallel to each other, the value of q is
Answer: D (3/4)
The gradients are 4/3 and 1/q. Parallel lines have equal gradients, so 1/q = 4/3 and q = 3/4.
In the parallelogram PQRS above, find angle SQR.
Answer: A (100°)
∠QSR = 50° (alternate to the 50° at Q). ∠PSR = 180° − 30° = 150°, so ∠SRQ = 30° (co-interior angles). Then ∠SQR = 180° − 50° − 30° = 100°.
The volume of a hemispherical bowl is 718⅔ cm3. Find its radius. [π = 22/7]
Answer: C (7.0 cm)
(2/3) × 22/7 × r3 = 2156/3, so r3 = 2156/3 × 3/2 × 7/22 = 343 and r = 7cm.
A particle P moves between points S and T such that angle SPT is always constant. Find the locus of P.
Answer: A (It is a semi-circle with ST as diameter.)
Points that see ST at a fixed angle lie on a circular arc through S and T; of the choices, the semicircle on ST as diameter is the arc (the case where the angle is 90°).
If the lines 2y − kx + 2 = 0 and y + x − k/2 = 0 intersect at (1, −2), find the value of k.
Answer: C (−2)
Put (1, −2) into the first line: −4 − k + 2 = 0, so k = −2. It also fits the second line: −2 + 1 + 1 = 0.
A man 40 m from the foot of a tower observes the angle of elevation of the tower to be 30°. Determine the height of the tower.
Answer: A (40√3/3 m)
height = 40 tan 30° = 40/√3 = 40√3/3 m.
Find the locus of points equidistant from two straight lines y − 5 = 0 and y − 3 = 0.
Answer: B (y − 4 = 0)
The lines y = 5 and y = 3 are parallel; points halfway between them lie on y = 4, i.e. y − 4 = 0.
What is the value of k if the mid-point of the line joining (1 − k, −4) and (2, k + 1) is (−k, k)?
Answer: A (−3)
x: (3 − k)/2 = −k gives k = −3. y: (k − 3)/2 = k also gives k = −3.
Find the size of each exterior angle of a regular octagon.
Answer: B (45°)
Exterior angles add to 360°, so each is 360° ÷ 8 = 45°.
Find the value of (tan 60° − tan 30°)/(tan 60° + tan 30°)
Answer: D (1/2)
tan 60° = √3 and tan 30° = 1/√3: (√3 − 1/√3)/(√3 + 1/√3) = (2/√3)/(4/√3) = 1/2.
The area of a square is 144 sq cm. Find the length of the diagonal.
Answer: B (12√2 cm)
The side is √144 = 12cm, and the diagonal of a square is side × √2 = 12√2 cm.