49 questions. Try each one, then open the answer to check yourself and read the explanation.
Rationalize the expression 1/(√2 + √5)
Answer: A (⅓(√5 − √2))
Multiply top and bottom by √2 − √5: the bottom becomes 2 − 5 = −3, so the result is (√2 − √5)/(−3) = ⅓(√5 − √2).
Simplify 3 − 2 ÷ 4/5 + 1/2
Answer: D (1)
Divide first: 2 ÷ 4/5 = 5/2. Then 3 − 5/2 + 1/2 = 1.
If ₦560.70 is shared in the ratio 7:2:1, what is the smallest share?
Answer: E (₦56.07.)
There are 7 + 2 + 1 = 10 parts, and the smallest share is 1 part: 560.70 ÷ 10 = ₦56.07.
In the above diagram, angle QPR = 90°, angle PSR = 90° and PR = 5 units. Find the length of QS.
Answer: B (5 cot 25° sin 65°)
In triangle PQR, <R = 65°, so PS = PR sin 65° = 5 sin 65°. In triangle PQS, tan 25° = PS/QS, so QS = 5 sin 65°/tan 25° = 5 cot 25° sin 65°.
Seven years ago, the age of a father was three times that of his son, but in six years time the age of the son will be half that of the father. Representing the present ages of the father and son by x and y respectively, the two equations relating x and y are
Answer: B (3y − x = 14; x − 2y = 6)
x − 7 = 3(y − 7) gives 3y − x = 14. In six years, 2(y + 6) = x + 6 gives x − 2y = 6.
The factors of 6x − 5 − x2 are
Answer: D ((x − 5)(1 − x))
6x − 5 − x² = −(x² − 6x + 5) = −(x − 5)(x − 1) = (x − 5)(1 − x).
The solution of the quadratic equation bx2 + cx + a = 0, is given by
Answer: A (x = (−c ± √(c2 − 4ab))/2b)
In the usual formula the coefficient of x² is now b, of x is c, and the constant is a. So x = (−c ± √(c² − 4ab))/2b.
The graphical method of solving the equation x3 + 3x2 + 4x − 28 = 0 is by drawing the graphs of the curves
Answer: D (y = x2 + 3x + 4 and y = 28/x)
Rearranging gives x³ + 3x² + 4x = 28; dividing by x gives x² + 3x + 4 = 28/x. The solutions are where these two graphs cross.
Write the equation 2log2 x − xlog2(1 + y) = 3 in a form not involving logarithms.
Answer: C (x2 = 8(1 + y)x)
2log₂x = log₂x² and xlog₂(1 + y) = log₂(1 + y)ˣ. So log₂[x²/(1 + y)ˣ] = 3, giving x² = 2³(1 + y)ˣ = 8(1 + y)ˣ.
Find α and β such that x3/8 × y−6/7 × (y9/7/x−45/8)1/9 = xα/yβ
Answer: A (α = 1, β = 5/7)
The bracket to the power 1/9 gives y^(1/7) × x^(5/8). Then x^(3/8 + 5/8) = x¹ and y^(−6/7 + 1/7) = y^(−5/7), so the result is x/y^(5/7): α = 1, β = 5/7.
Which of the following lines is not parallel to the line 3y + 2x + 7 = 0?
Answer: D (3y − 2x + 7 = 0)
Parallel lines have the same ratio of x to y coefficients (2 : 3). All options keep 2 : 3 except 3y − 2x + 7 = 0, whose gradient is +2/3 instead of −2/3.
The diagram below is the distance-time graph of a vehicle. Find its average speed in kilometres per hour during the journey.
Answer: E (84km/hr.)
The distance goes from 50km to 155km, i.e. 105km, in 75 minutes = 1.25 hours. Average speed = 105 ÷ 1.25 = 84km/hr.
What is the least possible value of 9/(1 + 2x2) if 0 ≤ x ≤ 2?
Answer: C (1)
The fraction is smallest when the denominator is largest, at x = 2: 9/(1 + 8) = 1.
A man bought a wristwatch of ₦150 but was only able to sell it for ₦120. Find the loss per cent on the transaction.
Answer: C (20%)
The loss is ₦30 on a cost of ₦150: 30/150 × 100 = 20%.
Find the median of the set of numbers 110, 116, 113, 119, 118, 127, 118, 117, 113.
Answer: C (117)
In order: 110, 113, 113, 116, 117, 118, 118, 119, 127. The middle (5th) number is 117.
Simplify (log108 − log104)/(log104 − log102)
Answer: E (1.)
The top is log(8/4) = log 2 and the bottom is log(4/2) = log 2, so the value is 1.
If f(x − 2) = 3x2 + 4x + 1. Find f(1)
Answer: B (40)
f(1) needs x − 2 = 1, so x = 3: 3(9) + 12 + 1 = 40.
A sector of a circle is bounded by two radii 7cm long and an arc of length 6cm. Find the area of the sector.
Answer: C (21cm2)
Area of a sector = ½ × radius × arc length = ½ × 7 × 6 = 21cm².
Given that p:q = 1/3 : 1/2 and q:r = 2/5 : 4/7, find p:r.
Answer: B (7:15)
p : q = 2 : 3 and q : r = 14 : 20 = 7 : 10. Make q the same: p : q = 14 : 21 and q : r = 21 : 30, so p : r = 14 : 30 = 7 : 15.
If sin θ = (m − n)/(m + n) find the value of 1 + tan2θ
Answer: D ((m2 + n2 + 2mn)/4mn)
1 + tan²θ = sec²θ = 1/cos²θ. cos²θ = 1 − (m − n)²/(m + n)² = 4mn/(m + n)², so 1 + tan²θ = (m + n)²/4mn = (m² + n² + 2mn)/4mn.
The currency used in a country is called 'Matimalik' (M) and is of base seven. A lady in that country bought 4 bags of rice at M 56 per bag and 3 tins of milk at M 4 per tin. What is the total cost of the items she bought?
Answer: D (M341(7))
56₇ = 41 in base ten, so the rice costs 4 × 41 = 164 and the milk 3 × 4 = 12, total 176. In base seven 176 = 3 × 49 + 4 × 7 + 1 = 341₇.
Solve for x, if (2/x)/(1/p2 + 1/q2) = m
Answer: C (2p2q2/m(p2 + q2))
1/p² + 1/q² = (p² + q²)/p²q², so the left side is 2p²q²/(x(p² + q²)) = m. Hence x = 2p²q²/m(p² + q²).
Express 37.05 × 0.0042 in standard form
Answer: E (1.5561 × 10-1)
37.05 × 0.0042 = 0.15561 = 1.5561 × 10⁻¹.
Find the values of x for which the expression (x − 3)(x − 2)/(x2 + x − 2) is undefined
Answer: A (1, −2)
A fraction is undefined when its denominator is zero: x² + x − 2 = (x + 2)(x − 1) = 0, so x = 1 or −2.
A stone Q is tied to a point P vertically above Q by an inelastic string of length 2 metres. How high does the stone rise when the string is inclined at an angle 60° to the vertical?
Answer: D (1 metre)
The stone is now 2 cos 60° = 1m below P instead of 2m, so it has risen 2 − 1 = 1 metre.
Express 130 kilometres per second in metres per hour.
Answer: B (468 × 106)
130km/s = 130 000m/s, and an hour has 3600 seconds: 130 000 × 3600 = 468 000 000 = 468 × 10⁶ m per hour.
Determine the real numbers x satisfying x(3/x + x) > 2
Answer: C (All real numbers x)
Expanding gives 3 + x² > 2, i.e. x² > −1, which is true for every real x (x ≠ 0 so that 3/x is defined).
In the above figure where PQRTU is a circle, |ST| = |RS| and angle TSR = 52°. Find the angle marked m.
Answer: E (116°)
Triangle STR is isosceles, so <STR = (180° − 52°)/2 = 64° and <PTR = 180° − 64° = 116°. The angle m at U stands on the same chord PR in the same segment, so m = 116°.
The quantity y is partly constant and partly varies inversely as the square of x. With P and Q as constants, a possible relationship between x and y is.
Answer: E (y = Q + P/x2)
A constant part Q plus a part that varies inversely as x² (P/x²) gives y = Q + P/x².
A world congress of mathematicians was held in Nice in 1970 with 800 people participating. There were 300 from Europe, 200 from America, 150 from Asia, 45 from Africa and 105 from Australia. Representing the above on a Pie Chart, the angle of the sector representing the participants from Asia is
Answer: B (67½°)
Asia's angle = 150/800 × 360° = 67½°.
If K is a constant, which of the following equations best describes the parabola shown above?
Answer: B (x = y2 − K)
The parabola opens to the right and is symmetric about the x-axis, so x is a function of y². Its vertex is on the negative x-axis, which fits x = y² − K with K > 0.
The diameter of a metal rod is measured as 23.40cm to four significant figures. What is the maximum error in the measurement?
Answer: D (0.005cm)
The last figure is in the hundredths place, so the reading can be off by at most half of 0.01cm, i.e. 0.005cm.
In a regular polygon of n sides, each interior angle is 144°. Find n
Answer: C (10)
Each exterior angle is 180° − 144° = 36°, and the exterior angles add to 360°, so n = 360 ÷ 36 = 10.
Simplify 3/(2x − 1) + (2 − x)/(x − 2)
Answer: C ((4 − 2x)/(2x − 1))
(2 − x)/(x − 2) = −1, so the sum is 3/(2x − 1) − 1 = (3 − 2x + 1)/(2x − 1) = (4 − 2x)/(2x − 1).
The solution to the quadratic equation 5 + 3x − 2x2 = 0 is
Answer: E ((5/2, −1).)
5 + 3x − 2x² = (5 − 2x)(1 + x) = 0, so x = 5/2 or −1.
A housewife bought 3 kilograms of garri at ₦13.00 per kg. She deposited ₦160.00 for half a cow and bought 24 oranges at 10k each. She came back home with ₦20.60. She therefore left home with
Answer: B (₦222.00)
She spent ₦39.00 + ₦160.00 + ₦2.40 = ₦201.40 and had ₦20.60 left, so she started with ₦222.00.
The solution to the simultaneous equations
3x + 5y = 4
4x + 3y = 5 is
Answer: B ((13/11, 1/11))
Multiply the first by 4 and the second by 3 and subtract: 11y = 1, so y = 1/11. Then 3x = 4 − 5/11 = 39/11 and x = 13/11.
The area of the curved surface of the cone generated by the sector of a circle radius 6cm and arc length 22cm (π = 22/7) is
Answer: E (66sq.cm.)
The curved surface of the cone is the sector itself: area = ½ × radius × arc = ½ × 6 × 22 = 66sq.cm.
The square base of a pyramid of side 3cm has height 8cm. If the pyramid is cut into two parts by a plane parallel to the base midway between the base and the vertex, the volumes of the two sections are
Answer: A ((21cm3, 3cm3))
Whole volume = ⅓ × 3² × 8 = 24cm³. The top part is a similar pyramid with half the height, so its volume is 24 × (½)³ = 3cm³, leaving 21cm³ for the bottom part.
Simplify f½ g2h⅓ ÷ f5/2g0 h7/3
Answer: A ((g/fh)2)
Subtract the powers: f^(½ − 5/2) = f⁻², g^(2 − 0) = g², h^(⅓ − 7/3) = h⁻². So the result is g²/(f²h²) = (g/fh)².
Determine the mean monthly salary of 50 employees of a company from the following frequency distribution:
Monthly salary: ₦200.00, ₦325.00, ₦100.00, ₦120.00, ₦60.00, ₦80.00
Frequency: 10, 5, 20, 2, 10, 3
Answer: B (₦134.10)
Σfx = 2000 + 1625 + 2000 + 240 + 600 + 240 = 6705. Mean = 6705 ÷ 50 = ₦134.10.
If the function y = 5x is graphed, what would be its intercept on the y-axis?
Answer: E (0)
The y-intercept is the value of y when x = 0: y = 5 × 0 = 0. The line passes through the origin.
PQRS is a parallelogram with area 50 square cm and the side PQ is 10cm long. T is a point on RS and TF is the altitude of the triangle TPQ. Find TF.
Answer: E (5cm.)
TF is also the height of the parallelogram on base PQ, so 10 × TF = 50 and TF = 5cm.
A side of a rhombus is 2cm in length. An angle of the rhombus is 60°. What is the length of the diagonal facing this angle?
Answer: A (2cm)
The two 2cm sides with a 60° angle between them and the diagonal form an equilateral triangle, so the diagonal is 2cm.
A variable y is inversely proportional to x2, when y = 10, x = 2. What is y when x = 10?
Answer: D (0.4)
y = k/x². 10 = k/4 gives k = 40, so at x = 10, y = 40/100 = 0.4.
Two distinct sectors in the same circle subtend 100° and 30° respectively at the centre of the circle. Their corresponding arcs are in the ratio
Answer: D (10:3)
In the same circle, arc length is proportional to the angle at the centre: 100 : 30 = 10 : 3.
The arithmetic mean of the ages of 30 pupils in a class is 15.3 years. One boy leaves the class and one girl is enrolled, and the new average age of 30 pupils in the class becomes 15.2 years. How much older is the boy than the girl?
Answer: D (3 years)
The total age falls from 30 × 15.3 = 459 to 30 × 15.2 = 456 years, so the boy is 3 years older than the girl.
In the above figure, QRST is a rectangle; PT∥QM, angle P = 60°. Find angle MUR.
Answer: A (150°)
PT ∥ QM, so <MQR = <P = 60° (corresponding angles). In right triangle QUR, <QUR = 30°, so <MUR = 180° − 30° = 150°.
TP and TQ are tangents to a circle O and R is a point on the circumference of the circle as shown, in the figure. If angle PTQ = 45°, what is the magnitude of the angle PRQ?
Answer: D (67½°)
Tangents from T are equal, so <TPQ = <TQP = (180° − 45°)/2 = 67½°. By the alternate segment theorem <PRQ = <TQP = 67½°.