50 questions. Try each one, then open the answer to check yourself and read the explanation.
Which of the following is in descending order?
Answer: C (9/10, 17/20, 4/5, 3/4)
As decimals, 9/10 = 0.9, 17/20 = 0.85, 4/5 = 0.8 and 3/4 = 0.75. Only option C goes from largest to smallest.
Evaluate 2700 000 × 0.03 ÷ 18 000
Answer: A (4.5 × 100)
2,700,000 × 0.03 = 81,000, and 81,000 ÷ 18,000 = 4.5 = 4.5 × 100.
The prime factors of 2520 are
Answer: C (2, 3, 5, 7)
2520 = 2³ × 3² × 5 × 7, so its prime factors are 2, 3, 5 and 7 (9 is not prime).
If 12e = X7, find X where e = 12
Answer: A (20)
12 in base 12 is 1 × 12 + 2 = 14 in base 10. In base 7, 14 = 2 × 7 + 0, which is 207.
Simplify ∛((64r−6)½)
Answer: D (2/r)
The square root then cube root is the 1/6 power: (64r−6)1/6 = 2r−1 = 2/r.
What is the difference between 0.007685 correct to three significant figures and 0.007685 correct to four places of decimal?
Answer: A (10−5)
To 3 s.f. it is 0.00769; to 4 d.p. it is 0.0077. The difference is 0.0077 − 0.00769 = 0.00001 = 10−5.
If a : b = 5 : 8, x : y = 25 : 16, evaluate a/x : b/y
Answer: D (2 : 5)
(a/x) ÷ (b/y) = (a/b) × (y/x) = 5/8 × 16/25 = 2/5. So the ratio is 2 : 5.
Oke deposited N800.00 in the bank at the rate of 12½% simple interest. After some time the total amount was one and half times the principal. For how many years was the money left in the bank?
Answer: B (4)
The amount is 1.5 × 800 = N1,200, so the interest is N400. Each year earns 12½% of 800 = N100, so it took 4 years.
If the surface area of a sphere is increased by 44%, find the percentage increase in its diameter.
Answer: D (20)
Surface area is proportional to the diameter squared. The area is multiplied by 1.44, so the diameter is multiplied by √1.44 = 1.2, a 20% increase.
Simplify 4 − 1/(2 − √3)
Answer: D (2 − √3)
Rationalise: 1/(2 − √3) = (2 + √3)/(4 − 3) = 2 + √3. Then 4 − (2 + √3) = 2 − √3.
Find p in terms of q if log3p + 3log3q = 3.
Answer: A ((3/q)3)
log3(pq³) = 3, so pq³ = 27. Then p = 27/q³ = (3/q)³.
What are the values of y which satisfy the equation 9y − 4 × 3y + 3 = 0?
Answer: D (0 and 1.)
Let x = 3y: x² − 4x + 3 = 0, so x = 1 or 3. Then 3y = 1 gives y = 0 and 3y = 3 gives y = 1.
Make R the subject of the formula S = √((2R + T)/2RT)
Answer: B (R = T/(2(TS2 − 1)))
Square both sides: 2RTS² = 2R + T. Collect R: R(2S²T − 2) = T, so R = T/(2(TS² − 1)).
Find the value of the expression (32/81)x3 − (64/27)x2 − (81/16)x−3 when x = −3/4
Answer: A (10½)
x³ = −27/64, x² = 9/16 and x−3 = −64/27. The terms are −1/6, −4/3 and +12, which add up to 10½.
The cost of a dinner for a group of students is partly constant and partly varies directly as the number of students. If the cost is N74.00 when the number of students is 20, and N96.00 when the number is 30, find the cost when they are 15 students.
Answer: B (N63.00)
C = a + ks. From 74 = a + 20k and 96 = a + 30k, k = 2.2 and a = 30. For 15 students C = 30 + 2.2 × 15 = N63.
If f(x) = 2x2 − 5x + 3, find f(x + 1)
Answer: A (2x2 − x)
f(x + 1) = 2(x + 1)² − 5(x + 1) + 3 = 2x² + 4x + 2 − 5x − 5 + 3 = 2x² − x.
Solve for a positive number x such that 2(x³ − x² − 2x) = 1
Answer: C (2)
2 to a power equals 1 only when the power is 0: x³ − x² − 2x = x(x − 2)(x + 1) = 0. The positive solution is x = 2.
Simplify (324 − 4x2)/(2x + 18)
Answer: D (−2(x − 9))
324 − 4x² = (18 − 2x)(18 + 2x). Cancelling 2x + 18 leaves 18 − 2x = −2(x − 9).
Factorize completely y3 − 4xy + xy3 − 4y
Answer: C (y(1 + x)(y + 2)(y − 2))
Group: y³(1 + x) − 4y(1 + x) = (1 + x)(y³ − 4y) = y(1 + x)(y + 2)(y − 2).
If one factor of x3 − 8−1 is x − 2−1 the other factor is
Answer: D (x2 + 2−1x + 4−1)
This is a difference of cubes: x³ − (½)³ = (x − ½)(x² + ½x + ¼).
Factorize 4a2 − 12ab − c2 + 9b2
Answer: C ((2a − 3b − c)(2a − 3b + c))
4a² − 12ab + 9b² = (2a − 3b)², so the expression is (2a − 3b)² − c² = (2a − 3b − c)(2a − 3b + c).
What are K and L respectively if ½(3y − 4x)2 = (8x2 + Kxy + Ly2)?
Answer: A (−12, 9/2)
(3y − 4x)² = 9y² − 24xy + 16x². Half of it is 8x² − 12xy + (9/2)y², so K = −12 and L = 9/2.
Solve the pair of equation for x and y respectively
2x−1 − 3y−1 = 4
4x−1 + y−1 = 1
Answer: D (2, −1)
Let u = 1/x, v = 1/y: 2u − 3v = 4 and 4u + v = 1. Then v = 1 − 4u, so 14u − 3 = 4 and u = ½, v = −1. So x = 2 and y = −1.
What value of Q will make the expression 4x2 + 5x + Q a complete square?
Answer: A (25/16)
(2x + a)² = 4x² + 4ax + a², so 4a = 5 and a = 5/4. Then Q = a² = 25/16.
Find the range of values of r which satisfies the following inequality, where a, b and c are positive.
r/a + r/b + r/c > 1
Answer: A (r > abc/(bc + ac + ab))
r(bc + ac + ab)/abc > 1. Since a, b, c are positive we can multiply out without changing the sign: r > abc/(bc + ac + ab).
Express
1/(x + 1) − 1/(x − 2)
as a single algebraic fraction
Answer: B (3/((x + 1)(2 − x)))
The result is ((x − 2) − (x + 1))/((x + 1)(x − 2)) = −3/((x + 1)(x − 2)). Changing the sign of the factor (x − 2) gives 3/((x + 1)(2 − x)).
Simplify
(x(x + 1)−½ − (x + 1)½)/(x + 1)½
Answer: B (−1/(x + 1))
Divide each term by (x + 1)½: x/(x + 1) − 1 = (x − x − 1)/(x + 1) = −1/(x + 1).
On the curve above, the points at which the gradient of the curve is equal to zero are
Answer: B (b, e, g, j, m)
The gradient is zero at the turning points, the tops and bottoms of the curve where the tangent is flat. These are b, e, g, j and m.
The sum of the first two terms of a geometric progression is x and the sum of the last two terms is y. If there are n terms in all, then the common ratio is
Answer: D ((y/x)1/(n−2))
x = a + ar and y = arn−2 + arn−1 = rn−2(a + ar). So y/x = rn−2 and r = (y/x)1/(n−2).
If −8, m, n, 19 are in arithmetic progression, find (m, n)
Answer: A (1, 10)
From −8 to 19 is three equal steps: 3d = 27, so d = 9. Then m = 1 and n = 10.
MN is a tangent to the given circle at M, MR and MQ are two chords. If QM̂N is 60° and MN̂Q is 40°, find RM̂Q
Answer: D (20°.)
By the alternate segment theorem angle MRQ = angle QMN = 60°. In triangle MQN, angle MQN = 180° − 60° − 40° = 80°, so angle MQR = 100°. In triangle MRQ, angle RMQ = 180° − 60° − 100° = 20°.
In the diagram above, HK is parallel to QR, PH = 4cm and HQ = 3cm. What is the ratio of KR:PR?
Answer: B (3 : 7)
A line parallel to one side divides the other two sides in the same ratio. So KR : PR = HQ : PQ = 3 : 7.
A regular polygon of (2k + 1) sides has 140° as the size of each interior angle. Find k.
Answer: A (4)
Each exterior angle is 180° − 140° = 40°, so there are 360 ÷ 40 = 9 sides. Then 2k + 1 = 9 and k = 4.
If PST is a straight line and PQ = QS = SR in the above diagram, find y.
Answer: C (72°)
PQ = QS gives angle QSP = 24°, and the exterior angle SQR = 24° + 24° = 48°. SQ = SR gives angle SRQ = 48°, so angle QSR = 84°. On the straight line at S, y = 180° − 24° − 84° = 72°.
In the above diagram PQ is parallel to RS and QS bisects PQ̂R. If PR̂Q is 60°, find x
Answer: B (40°)
Angle PRQ = 60° means arc PQ = 120°. QS bisects angle PQR, so arcs PS and SR are equal, and the parallel chords make arc QR equal to arc PS. These three equal arcs share 240°, so each is 80°. x stands on arc QR, so x = 40°.
PQRS is a rhombus. If PR2 + QS2 = kPQ2, determine k.
Answer: D (4.)
The diagonals of a rhombus bisect each other at right angles, so PQ² = (PR/2)² + (QS/2)². Multiplying by 4, PR² + QS² = 4PQ², so k = 4.
In XYZ, Ŷ = Ẑ = 30° and XZ = 3cm. Find YZ.
Answer: C (3√3cm)
Angle X = 120°. By the sine rule YZ/sin 120° = 3/sin 30°, so YZ = 3 × (√3/2) ÷ ½ = 3√3cm.
In ΔPQR, the bisector of QP̂R meets QR at S. The line PQ is produced to V and the bisector of VQ̂S meets PS produced at T. If QP̂R = 46° and QŜT = 75°, calculate QT̂S
Answer: A (41°)
Angle QPS = 23° and angle QSP = 180° − 75° = 105°, so the exterior angle VQS = 23° + 105° = 128° and angle TQS = 64°. In triangle QST, angle QTS = 180° − 75° − 64° = 41°.
If PQR is a straight line with QS = QR, calculate TP̂Q, if QT||SR and TQ̂S = 3y°
Answer: C (20⅔°)
QS = QR, so angle QRS = (180° − 56°)/2 = 62°. QT is parallel to SR, so angle TQP = 62° (corresponding angles). On the straight line at Q, 62° + 3y + 56° = 180°, so y = 20⅔°, the angle TPQ.
If x : y = 5 : 12 and z = 52cm, find the perimeter of the triangle
Answer: D (120cm)
The triangle is right-angled with legs in ratio 5 : 12, so the sides are 5k, 12k, 13k. 13k = 52 gives k = 4, and the perimeter is 30k = 120cm.
The pilot of an aeroplane, flying 10km above the ground in the direction of a landmark, views the landmark to have angles of depression of 35° and 55°. Find the distance between the two points of observation.
Answer: D (10(cot 35° − cot 55°))
At height 10km, the horizontal distance to the landmark is 10 cot θ for a depression angle θ. The two positions are 10 cot 35° and 10 cot 55° away, so the distance between them is 10(cot 35° − cot 55°).
4 sin2x − 3 = 0, find x if 0 ≤ x ≤ 90°
Answer: C (60°)
sin²x = 3/4, so sin x = √3/2 (positive in this range). Then x = 60°.
A square tile has side 30cm. How many of these tiles will cover a rectangular floor of length 7.2m and width 4.2m?
Answer: A (336)
The floor is 720cm by 420cm. That is 24 tiles along and 14 tiles across, so 24 × 14 = 336 tiles.
A cylindrical metal pipe 1m long has an outer diameter of 7.2cm and an inner diameter of 2.8cm. Find the volume of metal used for the cylinder.
Answer: B (1,100πcm3)
Volume of metal = π/4 × (D² − d²) × length = π/4 × (51.84 − 7.84) × 100 = π/4 × 44 × 100 = 1,100π cm³.
OXYZW is a pyramid with a square base such that OX = OY = OZ = OW = 5cm and XY = XW = YZ = WZ = 6cm. Find the height OT.
Answer: D (√7)
T is the centre of the base, half a diagonal from each corner: half of 6√2 = 3√2. OT² = 5² − (3√2)² = 25 − 18 = 7, so OT = √7cm.
In preparing rice cutlets, a cook used 75g of rice, 40g of margarine, 105g of meat and 20g of bread crumbs. Find the angle of the sector which represents meat in a pie chart.
Answer: D (157.5°)
The total is 240g. Meat's angle = 105/240 × 360° = 157.5°.
In a class of 30 students, the marks scored in an examination are displayed in the following histogram: What percentage of the students scored more than 40%?
Answer: C (46⅔%)
The bars above 40 marks show 6 + 6 + 2 = 14 students. 14/30 × 100 = 46⅔%.
In a family of 21 people, the average age is 14 years. if the age of the grandfather is not counted, the average age drops to 12 years. What is the age of the grandfather?
Answer: D (54 years)
Total age = 21 × 14 = 294. Without him, 20 × 12 = 240. The grandfather is 294 − 240 = 54 years old.
If n is the median and m is the mode of the following set of numbers, 2, 4,2.1, 1.6, 2.6, 2.6, 3.7, 2.1, 2.6, then (n, m) is
Answer: A ((2.6, 2.6))
In order the nine numbers are 1.6, 2, 2.1, 2.1, 2.6, 2.6, 2.6, 3.7, 4. The middle (5th) one is 2.6 and the most common is 2.6, so (n, m) = (2.6, 2.6).
Two numbers are chosen at random from three numbers 1, 3, 6. Find the probability that the sum of the two is not odd.
Answer: C (1/3)
The possible pairs are 1 + 3 = 4, 1 + 6 = 7 and 3 + 6 = 9. Only one of the three sums is even, so the probability is 1/3.