49 questions. Try each one, then open the answer to check yourself and read the explanation.
Evaluate (212)3 − (121)3 + (222)3
Answer: C ((1020)3)
In base ten: 212₃ = 23, 121₃ = 16 and 222₃ = 26, so the value is 23 − 16 + 26 = 33. 33 = 27 + 6 = 1020₃.
If Musa scored 75 in Biology instead of 57, his average mark in four subjects would have been 60. What was his total mark?
Answer: C (222)
With 75 his total would be 4 × 60 = 240. He actually scored 18 less in Biology, so his total is 240 − 18 = 222.
Divide the L.C.M. of 48, 64 and 80 by their H.C.F.
Answer: D (60.)
48 = 2⁴ × 3, 64 = 2⁶, 80 = 2⁴ × 5. L.C.M. = 2⁶ × 3 × 5 = 960 and H.C.F. = 2⁴ = 16. 960 ÷ 16 = 60.
Find the smallest number by which 252 can be multiplied to obtain a perfect square.
Answer: D (7.)
252 = 2² × 3² × 7. Only the 7 is unpaired, so multiplying by 7 gives a perfect square (1764 = 42²).
Find the reciprocal of (2/3)/(1/2 + 1/3)
Answer: B (5/4)
½ + ⅓ = 5/6, so the fraction is 2/3 ÷ 5/6 = 4/5. Its reciprocal is 5/4.
Three boys shared some oranges. The first received 1/3 of the oranges, the second received 2/3 of the remainder. If the third boy received the remaining 12 oranges, how many oranges did they share?
Answer: B (54)
After the first boy, ⅔ of the oranges remain. The second takes ⅔ of that, leaving ⅓ × ⅔ = 2/9 for the third boy. 2/9 of the total = 12, so the total is 54.
If P = 18, Q = 21, R = −6 and S = −4, calculate ((P − Q)3 + S2)/R3
Answer: B (11/216)
(18 − 21)³ = −27 and (−4)² = 16, so the top is −11. R³ = (−6)³ = −216, and −11/−216 = 11/216.
Simplify (0.0324 × 0.00064)/(0.48 × 0.012)
Answer: C (3.6 × 10-3)
The top is 0.000020736 and the bottom 0.00576. Dividing gives 0.0036 = 3.6 × 10⁻³.
Udoh deposited ₦150.00 in the bank. At the end of 5 years the simple interest on the principal was ₦55.00. At what rate per annum was the interest paid?
Answer: B (7⅓%)
R = 100 × I/(P × T) = 100 × 55/(150 × 5) = 22/3 = 7⅓%.
A number of pencils were shared out among Bisi, Sola and Tunde in the ratio of 2:3:5 respectively. If Bisi got 5, how many were shared out?
Answer: B (25)
Bisi's 2 parts out of 10 are 5 pencils, so 1 part is 2.5 and the total is 25.
The ages of Tosan and Isa differ by 6 and the product of their ages is 187. Write their ages in the form (x,y), where x > y.
Answer: C ((17, 11))
x = y + 6 and y(y + 6) = 187, so y² + 6y − 187 = 0 = (y + 17)(y − 11). Ages are positive, so y = 11 and x = 17.
In 1984, Ike was 24 years old and his father was 45 years old. In what year was Ike exactly half his father's age?
Answer: B (1981)
The age gap is always 21 years. Ike is half his father's age when Ike is 21 and his father 42, which was 3 years before 1984: 1981.
Simplify (1/(√5 + √3) − 1/(√5 − √3)) × 1/√3
Answer: D (−1)
Over the common denominator 5 − 3 = 2, the bracket is ((√5 − √3) − (√5 + √3))/2 = −√3. Multiplying by 1/√3 gives −1.
Find n if log24 + log27 − log2 n = 1
Answer: B (14)
log₂(28/n) = 1, so 28/n = 2 and n = 14.
Simplify (9⅓ × 27−½)/(3−⅙ × 3−⅔)
Answer: B (1)
The top is 3^(2/3) × 3^(−3/2) = 3^(−5/6) and the bottom is 3^(−1/6 − 2/3) = 3^(−5/6). They are equal, so the value is 1.
If x varies directly as y3 and x = 2 when y = 1, find x when y = 5.
Answer: D (250.)
x = ky³, and x = 2 at y = 1 gives k = 2. At y = 5, x = 2 × 125 = 250.
Factorize completely 8a + 125ax3
Answer: B (a(2 + 5x)(4 − 10x + 25x2))
Take out a: a(8 + 125x³) = a(2³ + (5x)³). A sum of cubes factorises as (2 + 5x)(4 − 10x + 25x²).
If y = x/(x − 3) + x/(x + 4) find y when x = −2
Answer: A (−3/5)
y = −2/(−5) + (−2)/2 = 2/5 − 1 = −3/5.
Find all real numbers x which satisfy the inequality 1/3(x + 1) − 1 > 1/5(x + 4).
Answer: D (x > 11.)
Multiply by 15: 5(x + 1) − 15 > 3(x + 4), so 5x − 10 > 3x + 12, 2x > 22 and x > 11.
Factorize x2 + 2a + ax + 2x
Answer: D ((x + 2)(x + a).)
Group the terms: x(x + 2) + a(x + 2) = (x + 2)(x + a).
Solve the equation 3x2 + 6x − 2 = 0
Answer: B (x = −1 ± √15/3)
x = (−6 ± √(36 + 24))/6 = (−6 ± 2√15)/6 = −1 ± √15/3.
Simplify 1/(5x + 5) + 1/(7x + 7)
Answer: D (12/(35x + 35))
1/(5(x + 1)) + 1/(7(x + 1)) = (7 + 5)/(35(x + 1)) = 12/(35x + 35).
The curve y = −x2 + 3x + 4 intersects the co-ordinate axes at
Answer: D ((0, 4), (4, 0), (−1, 0).)
At x = 0, y = 4. For y = 0, x² − 3x − 4 = (x − 4)(x + 1) = 0, so x = 4 or −1. The points are (0, 4), (4, 0) and (−1, 0).
Factorize (4a + 3)2 − (3a − 2)2
Answer: C ((a + 5)(7a + 1))
Difference of two squares: (4a + 3 + 3a − 2)(4a + 3 − 3a + 2) = (7a + 1)(a + 5).
If 5(x + 2y) = 5 and 4(x + 3y) = 16, find 3(x + y)
Answer: B (1)
x + 2y = 1 and x + 3y = 2, so y = 1 and x = −1. Then 3^(x + y) = 3⁰ = 1.
Make v the subject of the formula s = √(6/v − w/2)
Answer: D (v = 12/(2s2 + w))
Square: s² = 6/v − w/2, so 6/v = (2s² + w)/2 and v = 12/(2s² + w).
Find the values of x which satisfy the equation 16x − 5 × 4x + 4 = 0
Answer: C (0 and 1)
Put y = 4ˣ: y² − 5y + 4 = 0, so y = 1 or 4. Then 4ˣ = 1 gives x = 0 and 4ˣ = 4 gives x = 1.
If a/b = c/d = k, find the value of (3a2 − ac + c2)/(3b2 − bd + d2) in terms of k.
Answer: D (k2)
Put a = kb and c = kd. The top becomes k²(3b² − bd + d²), so the fraction is k².
At what points does the straight line y = 2x + 1 intersect the curve y = 2x2 + 5x − 1?
Answer: A ((−2, −3) and (½, 2))
2x² + 5x − 1 = 2x + 1 gives 2x² + 3x − 2 = 0 = (2x − 1)(x + 2), so x = ½ or −2. Then y = 2 or −3.
A regular polygon of n sides has 160° as the size of each interior angle. Find n.
Answer: A (18)
Each exterior angle is 180° − 160° = 20°, and they add to 360°, so n = 360 ÷ 20 = 18.
If cos θ = a/b, find 1 + tan2θ
Answer: A (b2/a2)
1 + tan²θ = sec²θ = 1/cos²θ = b²/a².
In the diagram below, PQ and RS are chords of a circle centre O which meet at T outside the circle. If TP = 24cm, TQ = 8cm and TS = 12cm, find TR.
Answer: A (16cm)
For two secants from an outside point, TP × TQ = TS × TR. So 24 × 8 = 12 × TR and TR = 16cm.
The angle of elevation of the top of a vertical tower 50 metres high from a point X on the ground is 30°. From a point Y on the opposite side of the tower, the angle of elevation of the top of the tower is 60°. Find the distance between the points X and Y.
Answer: D (115.47m.)
X is 50/tan 30° = 86.60m from the foot and Y is 50/tan 60° = 28.87m from it, on opposite sides. XY = 86.60 + 28.87 = 115.47m.
A girl walks 45 metres in the direction 050° from a point Q to a point X. She then walks 24 metres in the direction 140° from X to a point Y. How far is she then from Q?
Answer: C (51m)
The two directions differ by 140° − 50° = 90°, so the angle at X is a right angle. QY = √(45² + 24²) = √2601 = 51m.
The figure is a solid with the trapezium PQRS as its uniform cross-section. Find its volume
Answer: C (816m3)
The trapezium has parallel sides PQ = 6m and RS = 11m, 12m apart, so its area is ½(6 + 11) × 12 = 102m². The solid is 8m long, giving 102 × 8 = 816m³.
PQ and PR are tangents from P to a circle centre O as shown in the figure above. If QRP = 34°, find the angle marked x.
Answer: C (68°)
By the alternate segment theorem the angle QRP between the tangent and the chord equals the angle at the circumference on QR, 34°. The angle at the centre x is twice that: 68°.
An arc of a circle of radius 6cm is 8cm long. Find the area of the sector.
Answer: B (24cm2)
Area of a sector = ½ × radius × arc length = ½ × 6 × 8 = 24cm².
In ΔXYZ above, determine the cosine of angle Z
Answer: B (29/36)
The sides next to Z are XZ = 3 and YZ = 6, and the side opposite is XY = 4. cos Z = (3² + 6² − 4²)/(2 × 3 × 6) = 29/36.
In the figure above ΔPQT is isosceles. PQ = QT. SRQ = 35°, TPQ = 20° and PQR is a straight line. Calculate TSR.
Answer: C (75°)
<QTP = 20° too, so <PQT = 140° and <TQR = 40°. In triangle SQR, <QSR = 180° − 40° − 35° = 105°, so <TSR = 180° − 105° = 75°.
Find the total surface area of a solid cone of radius 2√3cm and slanting side 4√3cm.
Answer: D (36πcm2)
Total surface area = πr² + πrl = π(12) + π(2√3)(4√3) = 12π + 24π = 36πcm².
If U and V are two distinct fixed points and W is a variable point such that UWV is a right angle, what is the locus of W?
Answer: D (A circle with the line UV as the diameter.)
The angle in a semicircle is a right angle, so every W with angle UWV = 90° lies on the circle with UV as diameter.
In the figure above, PQ∥ST, RS∥UV. If PQR = 35° and QRS = 65°, find STV.
Answer: A (30°)
Draw a line through R parallel to PQ. It splits the 65° angle at R into 35° (alternate to angle PQR) and 30°. The parallel lines ST and UV carry that 30° across to angle STV.
An open rectangular box externally measures 4m × 3m × 4m. Find the total cost of painting the box externally if it costs ₦2.00 to paint one square metre.
Answer: C (₦136.00)
Painted area = base 4 × 3 = 12m² plus sides 2(4 × 4) + 2(3 × 4) = 56m², total 68m². At ₦2.00 per m² the cost is ₦136.00.
Of the nine hundred students admitted in a university in 1979, the following was the distribution by state:
Anambra 185, Imo 135, Kaduna 90, Kwara 110, Ondo 155, Oyo 225
In a pie chart drawn to represent this distribution, the angle subtended at the centre by Anambra is
Answer: C (74°)
Anambra's angle = 185/900 × 360° = 74°.
Find the median of the numbers 89, 141, 130, 161, 120, 131, 131, 100, 108 and 119.
Answer: B (125)
In order: 89, 100, 108, 119, 120, 130, 131, 131, 141, 161. With ten numbers the median is (120 + 130)/2 = 125.
Find the probability that a number selected at random from 40 to 50 is a prime.
Answer: A (3/11)
There are 11 numbers from 40 to 50, and 41, 43 and 47 are prime. The probability is 3/11.
The people in a city with a population of 0.9 million were grouped according to their ages. Use the diagram below to determine the number of people in the 15 − 29 years group.
Answer: B (26 × 104)
The 15–29 sector is 104°, so the number is 104/360 × 900 000 = 260 000 = 26 × 10⁴.
A man kept 6 black, 5 brown and 7 purple shirts in a drawer. What is the probability of his picking a purple shirt with his eyes closed?
Answer: C (7/18)
There are 18 shirts and 7 are purple, so the probability is 7/18.
The table below gives the scores of a group of students in a Mathematics test.
Score: 1, 2, 3, 4, 5, 6, 7, 8
Frequency: 2, 4, 7, 14, 12, 6, 4, 1
If the mode is m and the number of students who scored 4 or less is s, what is (s, m)?
Answer: A ((27, 4))
The score 4 has the highest frequency (14), so m = 4. Students scoring 4 or less: 2 + 4 + 7 + 14 = 27, so s = 27.