GCE Mathematics 1989 past questions and answers

If 3 log a + 5 log a − 6 log a = log 64, what is a?

1. A.4
2. B.6
3. C.8
4. D.16
5. E.32.

If the second and fourth terms of a G.P. are 8 and 32 respectively, what is the sum of the first four terms?

1. A.28
2. B.40
3. C.48
4. D.60
5. E.68.

In the diagram, TPS is a straight line, PQRS is a parallelogram with base QR and height 8cm. |QR| = 6cm and the area of triangle QST is 52cm². Find the area of △QPT.

1. A.39cm²
2. B.28cm²
3. C.24cm²
4. D.21cm²
5. E.19cm²

In the diagram, O is the centre of the circle, if ∠QRS = 62°, find the value of ∠SQR.

1. A.14°
2. B.28°
3. C.31°
4. D.45°
5. E.90°

Factorize 35 − 2b − b²

1. A.(35 − 2b)(b − 1)
2. B.(7 + b)(5 − b)
3. C.(3 + 7b)(5 − b)
4. D.(35 − b)(3b + 7)
5. E.(7 + b)(5 + b)

The graph below is that of a quadratic function y = 2x² − x − 6. Use the information provided by the graph. The roots of the equation are

1. A.1.5, 2
2. B.−1.5, 2
3. C.1, −6
4. D.1, 6
5. E.−1, 6

The graph below is that of a quadratic function y = 2x² − x − 6. Use the information provided by the graph. Find the minimum value of y.

1. A.y = 2
2. B.y = 0
3. C.y = 1.5
4. D.y = −6
5. E.y = −8

P and Q are two places on the same circle of latitude 79°S. P is on longitude 68°E while Q is on longitude 22°W. The angular distance between P and Q is

1. A.12°
2. B.45°
3. C.48°
4. D.90°
5. E.92°

Evaluate 0.009 ÷ 0.012, leaving your answer in standard form.

1. A.7.5 × 10²
2. B.7.5 × 10¹
3. C.7.5 × 10⁻¹
4. D.7.5 × 10⁻²
5. E.7.5 × 10⁻³

Factorize the following expression; 2x² + x − 15

1. A.(2x + 5)(x − 3)
2. B.(2x − 5)(x + 3)
3. C.(2x − 5)(x − 3)
4. D.(2x − 3)(x + 5)
5. E.(2x + 5)(x + 3)

The angle of elevation of X from Y is 30°. If |XY| = 40m, how high is X above the level of Y?

1. A.10m
2. B.20m
3. C.20√3m
4. D.40m
5. E.50m

Instead of recording the number 1.23cm of the radius of a tube, a student recorded 1.32cm. Find the percentage error, correct to one decimal place.

1. A.6.8%
2. B.7.3%
3. C.9.6%
4. D.14.4%
5. E.15.4%

In the diagram below, ∠SPR = p and ∠SQR = 2x. Find x in terms of p.

1. A.x = p + 2
2. B.x = p − 2
3. C.x = p²
4. D.x = p/2
5. E.x = 2p

In the diagram above ABDE and FCDE are parallelograms. If |FC| = 12.2cm and the height |PC| = 4.0cm, calculate the area of parallelogram ABDE.

1. A.3.05cm²
2. B.8.20cm²
3. C.16.20cm²
4. D.24.40cm²
5. E.48.80cm²

Using the graph above, if 0° < x < 60°, what is the value of x for which sin x = cos x?

1. A.0°
2. B.20°
3. C.30°
4. D.45°
5. E.50°

The Venn diagram below shows the number of students who studied Physics, Chemistry, and Mathematics in a certian school. How many students took at least two of the three subjects?

1. A.165
2. B.160
3. C.155
4. D.135
5. E.85.

An arc of a circle radius 7cm is 14cm long. What angle does the arc subtend at the centre of circle? (Take π = 22/7)

1. A.25.7°
2. B.44°
3. C.51.43°
4. D.98°
5. E.114.55°

If 5 times a certain integer is subtracted from twice the square of the integer, the result is 63. Find the integer.

1. A.21
2. B.9
3. C.7
4. D.4
5. E.3

If 3^y = 243, find the value of y.

1. A.2
2. B.3
3. C.4
4. D.5
5. E.6.

Points X and Y are respectively 20km North and 9km East of a point O. What is the bearing of Y from X? Correct to the nearest degree.

1. A.024°
2. B.114°
3. C.156°
4. D.204°
5. E.336°

In the diagram below, O is the centre of the circle. Calculate the length of the chord AB if |OA| = 5cm, |OD| = 3cm and ∠AOD = ∠BOD.

1. A.3cm
2. B.4cm
3. C.5cm
4. D.8cm
5. E.15cm

Calculate the surface area of a hollow cylinder which is closed at one end, if the base radius is 3.5cm and the height 8cm. (Take π = 22/7)

1. A.126.5cm²
2. B.165cm²
3. C.176cm²
4. D.214.5cm²
5. E.253.5cm²

Solve the following equation: 6x² − 7x − 5 = 0

1. A.x = ½ or x = −2½
2. B.x = 1/3 or x = 2½
3. C.x = 5/3 or x = −½
4. D.x = −12/3 or x = ½
5. E.x = 5/6 or x = −1

Use the pie chart, which shows the distribution of the population of a village. What angle represents grown up girls? Correct to 1 decimal place.

1. A.37.5°
2. B.26.7°
3. C.25.7°
4. D.20°
5. E.3.75°

Use the pie chart, which shows the distribution of the population of a village. What percentage of the population are married? Correct to 1 decimal place.

1. A.14.8%
2. B.20.4%
3. C.30.5%
4. D.35.2%
5. E.36.5%

If A = (a,b,c), B = (a,b,c,d,e) and C = (a,b,c,d,e,f). Find (A∪B) ∩ (A∪C).

1. A.(a,b,c,d)
2. B.(a,b,c,d,e)
3. C.(a,b,c,d,e,f)
4. D.(a,b,c)
5. E.ϕ

For a class of 30 students, the scores in a Mathematics test out of 10 marks were as follows:
4, 5, 7, 2, 3, 6, 5, 5, 8, 9,
5, 4, 2, 3, 7, 9, 8, 7, 7, 7,
3, 4, 5, 5, 2, 3, 6, 7, 7, 2.
What is the mode of the scores?

1. A.3
2. B.4
3. C.5
4. D.6
5. E.7

For a class of 30 students, the scores in a Mathematics test out of 10 marks were as follows:
4, 5, 7, 2, 3, 6, 5, 5, 8, 9,
5, 4, 2, 3, 7, 9, 8, 7, 7, 7,
3, 4, 5, 5, 2, 3, 6, 7, 7, 2.
What is the median score?

1. A.3
2. B.4
3. C.5
4. D.6
5. E.7

For a class of 30 students, the scores in a Mathematics test out of 10 marks were as follows:
4, 5, 7, 2, 3, 6, 5, 5, 8, 9,
5, 4, 2, 3, 7, 9, 8, 7, 7, 7,
3, 4, 5, 5, 2, 3, 6, 7, 7, 2.
What is the range of the distribution?

1. A.2
2. B.7
3. C.8
4. D.9
5. E.10

In the diagram below, O is the centre of the circle. If ∠PAQ = 75°, what is the value of ∠PBQ?

1. A.15°
2. B.75°
3. C.105°
4. D.150°
5. E.210°

Simplify: 9^−1/2/27^2/3

1. A.1
2. B.½
3. C.1/9
4. D.1/18
5. E.1/27

In the diagram below, what is the size of angle ACB?

1. A.40°
2. B.50°
3. C.60°
4. D.80°
5. E.100°

With reference to the graph above, which of the following solution sets is represented by △OPQ?

1. A.y ≥ x, y ≤ 0, y + x ≥ 7
2. B.y ≤ x, y ≥ 0, y + x ≤ 7
3. C.y ≥ x, y ≥ 0, y + x ≤ 7
4. D.y ≤ x, y ≤ 0, y + x ≤ 7
5. E.y ≤ x, y ≥ 0, y + x ≥ 7

Find the volume of a cone of radius 3.5cm and vertical height 12cm. (Take π = 22/7)

1. A.3.4cm³
2. B.15.5cm³
3. C.21.0cm³
4. D.42.0cm³
5. E.154.0cm³

The table below gives the scores of a group of students in an English Language test
Scores: 2, 3, 4, 5, 6, 7
Number of Students: 2, 4, 7, 2, 3, 2
If a student is chosen at random from the group, what is the probability that he scored at least 6 marks?

1. A.3/20
2. B.1/5
3. C.1/4
4. D.3/10
5. E.¾.

In the diagram above, O is the centre of the circle QRT and PT is a tangent to the circle at T. Calculate the angle x.

1. A.55°
2. B.40°
3. C.35°
4. D.25°
5. E.20°

Which of the following gives the point of intersection of the graphs y = x² and y = x + 6 shown above?

1. A.(0,6), (3,9)
2. B.(−3,0)(2,4)
3. C.(−2,4)(3,9)
4. D.(−2,3)(−3,2)
5. E.(−2,2)(3,6)

A sector is cut off from a circle radius 8.2cm to form a cone, if the radius of the resulting cone is 3.5cm, calculate the curved surface area of the cone. (Take π = 22/7)

1. A.11.25cm²
2. B.12.83cm²
3. C.22.0cm²
4. D.67.2cm²
5. E.90.2cm²

If log_ax = p, express x in terms of a and p.

1. A.x = a + p
2. B.x = a/p
3. C.x = p
4. D.x = ap
5. E.x = a^p

Calculate, correct to three significant figures, the length of an arc subtends an angle of 70° at the centre of the circle radius 4cm. (Take π = 22/7)

1. A.2.44cm
2. B.4.89cm
3. C.9.78cm
4. D.25.1cm
5. E.50.3cm

In the diagram below, O is the centre of the circle. If ∠POR = 114°, calculate ∠PQR.

1. A.123°
2. B.118.5°
3. C.117°
4. D.114°
5. E.54°

The angle of depression of a point on the ground from the top of a building is 20.3°. If the distance from the point to the foot of the building is 40m, calculate the height of the building, correct to one decimal place.

1. A.37.5m
2. B.28.1m
3. C.27.8m
4. D.14.8m
5. E.13.9m

Solve for x: (x² + 2x + 1) = 25

1. A.−6, −4
2. B.6, −4
3. C.6, 4
4. D.−6, 4
5. E.5, 5

Given that log p = 2 log x + 3 log q, which of the following expresses p in terms of x and q?

1. A.p = 5xq
2. B.p = 6xq
3. C.p = x² + q³
4. D.p = 2x + 3q
5. E.p = x²q³

If sin θ = ½ and cos θ = −√3/2, what is the value of θ?

1. A.30°
2. B.60°
3. C.90°
4. D.120°
5. E.150°

What is the probability that 3 customers waiting in a bank will be served in the sequence of their arrival at the bank.

1. A.1/6
2. B.1/3
3. C.½
4. D.2/3
5. E.5/6

In the diagram, O is the centre of the circle. If ∠POQ = 80° and ∠PRQ = 5x, find the value of x.

1. A.4°
2. B.8°
3. C.16°
4. D.20°
5. E.32°

In an A.P, the first term is 2, and the sum of the 1st and 6th terms is 16½. What is the 4th term?

1. A.12
2. B.9½
3. C.8
4. D.7
5. E.5½

(a) Without using Mathematics Tables, find x, given that 6log (x + 4) = log 64 (b) if U = (1,2,3,4,5,6,7,8,9,10) X = (1,2,4,5,7,8,9) Y = (1,2,3,4,7,9) and Z = (2,3,4,7,9), what is X∩Y∩Z?

The table below shows the weights, to the nearest kilogram, of twelve students in a Further Mathematics class.
Weight in kg: 55, 57, 59, 61, 63
Number of students: 2, 1, 2, 4, 3
(a) Draw a bar chart to illustrate the above information (b) What is (i) the mode (ii) the median of the distribution? (c) Calculate the mean weight, correct to the nearest kilogram.

The diagram shows a wooden structure in the form of a cone, mounted on a hemispherical base. The vertical height of the cone is 24cm and the base radius 7cm. Calculate, correct to three significant figures, the surface area of the structure (Take π = 22/7)

Using a ruler and a pair of compasses only, construct a triangle ABC, given that |AB| = 8.4cm, |BC| = 6.5cm and ∠ABC = 30°. Construct the locus: (a) l₁ of points equidistant from AB and BC, and within the angle ABC (b) l₂ of points equidistant from B and C. Locate the point of intersection P of l₁ and l₂, measure |AP|.

(a) If cos α = 0.6717, use Mathematical Table to find: (i) α (ii) sin α (b) The angle of depression of a point P on the ground, from the top T of a building is 23.6°. If the distance from P to the foot of the building is 50 cm, calculate the height of the building, correct to the nearest metre.

(a) In the diagram below, O is the centre of the circle radius 3.2cm. If ∠PRQ = 42°, calculate, correct to two decimal places, the area of the: (i) minor sector POQ; (ii) shaded part (b) if the sector POQ in (a) is used to form the curved surface of a cone with vertex O, calculate the base radius of the cone, correct to one decimal place.

When a stone is thrown vertically upwards, its distance d metres after t seconds is given by the formula d = 60t − 10t². Draw the graph of d = 60t − 10t² for values of t from 1 to 5 seconds using 2cm to 1 unit on the t axis and 2cm to 20 units on the d axis. (a) Using your graph (i) how long does it take to reach a height of 70 metres? (ii) determine the height of the stone after 5 seconds (iii) after how many seconds does it reach its maximum height? (b) Determine the slope of the graph when t = 4 seconds.

(a)(i) Prove that the angle which an arc of a circle subtends at the centre is twice that which it subtends at any point on the remaining part of the circumference. (ii) In the diagram above, O is the centre of the circle and PT is a diameter. If ∠PTQ = 22° and ∠TOR = 98°, calculate ∠QSR. (b) ABCD is a cyclic quadrilateral and the diagonals AC and BD intersect at H. If ∠DAC = 41° and ∠AHB = 70°, calculate ∠ACB.

(a) In an A.P. the difference between the 8th and the 4th terms is 20 and the 8th term is 1½ times the fourth term. What is the: (i) common difference, (ii) first term of the sequence? (b) The value of a machine depreciates each year by 5% of its value at the beginning of that year. If its value when new on 1st January, 1980 was N10,250.00, what was its value in January, 1989 when it was 9 years old? Give your answer, correct to three significant figures

(a) A pair of fair dice each numbered 1 to 6 is tossed. Find the probability of getting a sum of at least 9, (b) If the probability that a civil servant owns a car is 1/6, find the probability that: (i) two civil servants, A and B, selected at random, each owns a car, (ii) of two civil servants, C and D selected at random, only one owns a car, (iii) of three civil servants, X, Y and Z, selected at random, only one owns a car.

(a) Triangle PQR is right-angled at Q, PQ = 3a cm and QR = 4a cm. Determine PR in terms of a. (b) Ayo travels a distance of 24km from X on a bearing of 60° to Y. He then travels a distance of 18km to a point Z and Z is 30km from X. (i) Draw the diagram to show the positions of X, Y and Z (ii) What is the bearing of Z from Y? (iii) Calculate the bearing of X from Z.

(a) Derive an equation whose coefficients are integers and which has roots of 1/2 and −7. (b) Three years ago a father was four times as old as his daughter is now. The product of their present ages is 430. Calculate the present ages of the daughter and the father.