GCE Mathematics 1990 past questions and answers

Simplify: 125^−1/3 × 49^−1/2 × 10⁰

1. A.350
2. B.35
3. C.1/35
4. D.1/350
5. E.0

If 3^2x = 27, what is x?

1. A.1
2. B.1.5
3. C.4.5
4. D.18
5. E.40.5

Express 0.00562 in standard form

1. A.5.62 × 10⁻³
2. B.5.62 × 10⁻²
3. C.0.562 × 10⁻²
4. D.5.62 × 10²
5. E.5.62 × 10³

Given that 1/3 log₁₀P = 1, find the value of P.

1. A.1/10
2. B.3
3. C.10
4. D.100
5. E.1000

Simplify: (log√8)/(log 8)

1. A.1/3
2. B.1/2
3. C.1/3 log√2
4. D.1/3 log√8
5. E.1/2 log√2

Evaluate, using logarithm table, log(0.65)²

1. A.1.6258
2. B.0.6272
3. C.0.6258
4. D.1̄.6258
5. E.1.6272

If log x = 2̄.3675 and log y = 0.9750, what is the value of x + y? Correct to three significant figures.

1. A.1.18
2. B.1.31
3. C.9.03
4. D.9.44
5. E.9.46

While doing his Physics practicals, Idowu recorded a reading as 1.12cm instead of 1.21cm. Calculate his percentage error.

1. A.1.17%
2. B.6.38%
3. C.7.44%
4. D.8.05%
5. E.9.00%

Find the 4th term of an A.P. whose first term is 2 and the common difference is 0.5.

1. A.0.5
2. B.2.5
3. C.3.5
4. D.4
5. E.4.5

From the graph, determine the roots of the equation y = 2x² + x − 6.

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

From the graph of y = 2x² + x − 6, find the values of x for which y = 5

1. A.−2.5, 2
2. B.−2.6, 2.1
3. C.−2.0, 1.5
4. D.−2.0, 2.0
5. E.0

An arc of length 22cm subtends an angle of θ at the centre of the circle. What is the value of θ if the radius of the circle is 15cm? (Take π = 22/7)

1. A.70°
2. B.84°
3. C.96°
4. D.156°
5. E.168°

Find the sum of the first five terms of the G.P. 2, 6, 18, ...

1. A.484
2. B.243
3. C.242
4. D.130
5. E.1

Let J be the set of positive integers. If H = {x : x ∈ J, x² < 3 and x ≠ 0}, then

1. A.H = {1}
2. B.H is an infinite set
3. C.H = {0, 1, 2}
4. D.H = { }
5. E.J ⊂ H

In a class of 80 students, every student had to study Economics or Geography, or both Economics and Geography. If 65 students studied Economics and 50 studied Geography, how many studied both subjects?

1. A.15
2. B.30
3. C.35
4. D.45
5. E.50

The area shaded with horizontal lines is the solution set of the inequalities:

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

The area shaded with vertical lines is the solution set of the inequalities:

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

In the diagram below, O is the centre of the circle with radius 10cm, and ∠ABC = 30°. Calculate, correct to 1 decimal place, the length of arc AC. (Take π = 3.14)

1. A.5.2cm
2. B.10.5cm
3. C.13.2cm
4. D.20.6cm
5. E.31.4cm

Factorize x² + 4x − 192

1. A.(x − 4)(x + 48)
2. B.(x − 48)(x + 4)
3. C.(x − 12)(x + 16)
4. D.(x − 12)(x − 16)
5. E.(x + 12)(x + 16)

Factorize 2e² − 3e + 1

1. A.(2e − 1)(e − 1)
2. B.(e + 1)(2e + 1)
3. C.(2e + 3)(e − 2)
4. D.(2e − 3)(e − 1)
5. E.(e² − 3)(2e − 1)

Solve the equation 7y² = 3y.

1. A.y = 3 or 7
2. B.y = 0 or 7
3. C.y = 0 or 3/7
4. D.y = 0 or 9
5. E.y = 0 or 10

Use the graph to answer this question. Locate the points where y = x − 3 intersects y = x² − 2x − 3

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

Find the value of m which makes x² + 8x + m a perfect square.

1. A.2
2. B.4
3. C.8
4. D.12
5. E.16

Solve the equation 2a² − 3a − 27 = 0.

1. A.3/2, 9
2. B.−2/3, 9
3. C.3, 9/2
4. D.−3, −9/2
5. E.−3, 9/2

A sector of a circle of radius 7cm has an area of 44cm². Calculate the angle of the sector, correct to the nearest degree. (Take π = 22/7)

1. A.6°
2. B.26°
3. C.52°
4. D.103°
5. E.206°

Calculate the surface area of a sphere of radius 7cm. (Take π = 22/7)

1. A.86cm²
2. B.154cm²
3. C.616cm²
4. D.1434cm²
5. E.4312cm²

If the radius of the parallel of latitude 30°N is equal to the radius of the parallel of latitude θ°S, what is the value of θ?

1. A.75°
2. B.60°
3. C.45°
4. D.30°
5. E.0°

A cylindrical container closed at both ends, has a radius of 7cm and height 5cm. (Take π = 22/7)
Find the total surface area of the container.

1. A.35cm³
2. B.154cm³
3. C.220cm³
4. D.528cm³
5. E.770cm³

A cylindrical container closed at both ends, has a radius of 7cm and height 5cm. (Take π = 22/7)
What is the volume of the container?

1. A.35cm³
2. B.154cm³
3. C.220cm³
4. D.528cm³
5. E.770cm³

In the diagram below, |PQ| = |PR| = |RS| and ∠RPS = 32°. Find the value of ∠QPR.

1. A.72°
2. B.64°
3. C.52°
4. D.32°
5. E.26°

In the diagram below, WXYZ is a rhombus and ∠WYX = 20°. What is the value of ∠XZY?

1. A.20°
2. B.30°
3. C.45°
4. D.60°
5. E.70°

In the diagram below, PS||RQ, |RQ| = 6.4cm and perpendicular PH = 3.2cm. Find the area of SQR.

1. A.5.12cm²
2. B.9.60cm²
3. C.10.24cm²
4. D.20.48cm²
5. E.40.96cm²

Use the diagram below. Which angle is equal to ∠VWZ?

1. A.∠VXZ
2. B.∠VYZ
3. C.∠XZW
4. D.∠YXZ
5. E.∠VYW

Use the diagram below. Which triangle is equal in area to ΔVWZ?

1. A.ΔVXZ
2. B.ΔVYZ
3. C.ΔXYV
4. D.ΔWYV
5. E.ΔWYZ

In the diagram below, PQRS is a cyclic quadrilateral, ∠PSR = 86° and ∠QPR = 38°. Calculate ∠PRQ.

1. A.58°
2. B.53°
3. C.48°
4. D.43°
5. E.38°

In the diagram below, O is the centre of the circle. If ∠QOR = 290°, find the size of ∠QPR.

1. A.110°
2. B.70°
3. C.55°
4. D.35°
5. E.20°

In the diagram, PQ is a tangent at T to the circle ABT. ABC is a straight line and TC bisects ∠BTQ. Find x.

1. A.20°
2. B.30°
3. C.35°
4. D.40°
5. E.50°

Use Mathematical tables to evaluate (cos 40° − sin 30°).

1. A.−0.2660
2. B.−0.0266
3. C.0.0266
4. D.0.2660
5. E.1.5320

If the shadow of a pole 7m high is ½ its length, what is the angle of elevation of the sun, correct to the nearest degree?

1. A.90°
2. B.63°
3. C.60°
4. D.26°
5. E.0°

From the top of a building 10m high, the angle of depression of a stone lying on the horizontal ground is 69°. Calculate, correct to 1 decimal place, the distance of the stone from the foot of the building.

1. A.3.6m
2. B.3.8m
3. C.6.0m
4. D.9.3m
5. E.26.1m

The bearing of a point X from a point Y is 074°. What is the bearing of Y from X?

1. A.106°
2. B.148°
3. C.164°
4. D.254°
5. E.286°

Using the graph above, find the value of sin 270° + 2 cos 270°.

1. A.−2
2. B.−1
3. C.0
4. D.1
5. E.2

The annual salary of Mr. Johnson Mohammed for 1989 was N12,000.00. He spent this on agricultural projects, education of his children, food items, savings, maintenance and miscellaneous items as shown in the pie chart.
How much did he spend on food items?

1. A.N9,700.00
2. B.N6,700.00
3. C.N4,000.00
4. D.3,700.00
5. E.2,300.00

The annual salary of Mr. Johnson Mohammed for 1989 was N12,000.00. He spent this on agricultural projects, education of his children, food items, savings, maintenance and miscellaneous items as shown in the pie chart.
How much money did he invest in agriculture.

1. A.N1,200.00
2. B.N1,440.00
3. C.N2,000.00
4. D.N2,910.00
5. E.N4,000.00

The table shows the amount of money (in naira) collected through voluntary donations in a secondary school.
Amount (in naira): 3, 6, 9, 12, 15, 18
No. of Students: 3, 9, 6, 15, 3, 12
What is the mode?

1. A.N3
2. B.N6
3. C.N9
4. D.N12
5. E.15

The table shows the amount of money (in naira) collected through voluntary donations in a secondary school.
Amount (in naira): 3, 6, 9, 12, 15, 18
No. of Students: 3, 9, 6, 15, 3, 12
Find the median of distribution

1. A.N3.00
2. B.N9.00
3. C.N12.00
4. D.N15.00
5. E.N18.00

Two groups of male students cast their vote on a particular proposal. The results are as follows:
Group A: In favour 128, Against 32
Group B: In favour 96, Against 48
If a student in favour of the proposal is selected for a post, what is the probability that he is from group A?

1. A.8/19
2. B.16/35
3. C.10/19
4. D.4/7
5. E.4/5

Two groups of male students cast their vote on a particular proposal. The results are as follows:
Group A: In favour 128, Against 32
Group B: In favour 96, Against 48
If a student is chosen at random, what is probability that he is against the proposal?

1. A.3/19
2. B.4/19
3. C.5/19
4. D.9/19
5. E.10/19

(a) Simplify (0.016 × 0.084)/0.48 leaving your answer in standard form.
(b) Eight wooden poles are to be used for pillars and the lengths of the poles form an Arithmetic Progression (A.P). If the second pole is 2m and the sixth is 5m, give the lengths of the poles, in order.

Show on a graph, the area which gives the solution set of the inequalities:
y − 2x ≤ 4, 3y + x ≥ 6, y ≥ 7x − 9.

(a) Prove that the sum of the angles of a triangle is two right angles
(b) The side AB of a triangle ABC is produced to a point D. The bisector of ACB cuts AB at E. Prove that ∠CAE + ∠CBD = 2∠CEB.

A carpenter was told to make a rectangular desk with top, of dimension 50cm by 40cm. The carpenter actually made the desk 60cm by 35cm.
(a) Calculate the percentage error in the: (i) length and the breadth; (ii) area of the table top.
(b) Find the product of the two errors in (a)(i).

The feet of two vertical poles of height 3m and 7m are in line with a point P on the ground, the smaller pole being between the taller pole and P and at a distance of 20m from P. The angle of elevation of the top (T) of the taller pole from the top (R) of the smaller pole is 30°. Calculate the: (i) distance RT; (ii) distance of the foot of the taller pole from P, correct to three significant figures; (iii) angle of elevation of T from P, correct to one decimal place.

(a) Find the volume of a right solid cone of base radius 4cm and perpendicular height 6cm. (Take π = 3.142)
(b) A hemispherical tank of diameter 10m is filled by water issuing from a pipe of radius 20cm at 2m per second. Calculate, correct to three significant figures, the time, in minutes, it takes to fill the tank.

The weights to the nearest kilogram of a group of 50 students in a College of Technology are given below:
65, 70, 60, 46, 51, 55, 59, 63, 68, 53
47, 53, 72, 53, 67, 62, 64, 70, 57, 56
73, 56, 48, 51, 58, 63, 65, 62, 49, 64
53, 59, 63, 50, 48, 72, 67, 56, 61, 64
66, 52, 49, 62, 71, 58, 53, 69, 63, 59
(a) Prepare a grouped frequency table with class intervals 45 − 49, 50 − 54, 55 − 59, etc.
(b) Using an assumed mean of 62 or otherwise, calculate the mean and the standard deviation of the grouped data, correct to one decimal place.

ABCD is a trapezium in which AB||DC, |AB| = 8cm, ∠ABC = 60°, |BC| = 5.5cm and |BD| = 8.3cm.
(a) Using a ruler and a pair of compasses only, construct (i) the trapezium ABCD; (ii) a rectangle PQCD, where P, Q are two points on AB.
(b) Measure |AC| and |QB|

(a) If a number is chosen at random from the integers 5 to 25 inclusive, find the probability that the number is a multiple of 5 or 3.
(b) A bag contains 10 balls that differ only in colour; 4 are blue and the others are red. Two balls are picked, one after the other with replacement. What is the probability that (i) both are red? (ii) both are the same colour?

The following is an incomplete table for the relation y = 2x² − 5x + 1
x: −3, −2, −1, 0, 1, 2, 3, 4, 5
y: _, _, 8, 1, −2, _, _, _, 26
(a) Copy and complete the table.
(b) Using a scale of 2cm to 1 unit on the x axis and 2cm to 10 units on the y axis, draw a graph of the relation y = 2x² − 5x + 1 for −3 ≤ x ≤ 5.
(c) Using the same scale and axes, draw the graph of y = x + 6.
(d) Estimate from your graphs, correct to one decimal place: (i) the least value of y and the value of x for which it occurs; (ii) the solution of the equation 2x² − 5x + 1 = x + 6.

In a class of 40 students, 25 speak Hausa, 16 speak Igbo, 21 speak Yoruba and each of the students speaks at least one of these three languages. If 8 speak Hausa and Igbo, 11 speak Hausa and Yoruba and 6 speak Igbo and Yoruba
(a) Draw a Venn diagram to illustrate this information, using x to represent the number of students that speak all three;
(b) calculate the value of x.

An aeroplane flies from a town P (lat. 40°N, 38°E) to another town Q (lat. 40°N, 22°W). It later flies to a third town T (28°N, 22°W). Calculate the: (a) distance between P and Q along their parallel of latitude; (b) distance between Q and T along the line of longitudes. (c) average speed at which the aeroplane will fly from P to T via Q, if the journey takes 12 hours, correct to three significant figures. (Take the radius of the earth to be 6400km and π = 3.142)