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Examinations Council of Zambia — past paper
ECZ 2025 GCE Paper 2 — Mathematics
12 questions · downloaded from g12titan.com, where this paper is marked free online
1.
(a)
Simplify a2 b32c × ac22b ÷ b2 c2a. [2]
(b)
For the geometric progression 60, 30, 15, ..., find the
(b(i))
6th term, [2]
(b(ii))
geometric mean of 15 and 3.75, [2]
(b(iii))
sum to infinity. [3]
2.
(a)
Express 5x - 1 - 2x - 2 as a single fraction in its simplest form. [3]
(b)
Find the equation of a tangent to the curve y = x3 - 5x2 + 8x - 4 at x = 1. [3]
3.
(a)
Given that matrix P = 4-211, find the
(a(i))
determinant of matrix P, [2]
(a(ii))
inverse of matrix P. [2]
(b)
A box contains 6 white circuit breakers and 4 black circuit breakers, all identical in appearance. A circuit breaker is selected at random from the box and not replaced. A second circuit breaker is then selected. Find the probability that both circuit breakers selected are
(b(i))
white, [2]
(b(ii))
of different colours. [3]
4.
(a)
Solve the equation 2x2 = 7 - 10x, giving your answers correct to 2 decimal places. [5]
(b)
At Kamulima Farming Community, 33 farmers planted sorghum (S), 25 planted groundnuts (G), 38 planted maize (M), 15 planted both sorghum and groundnuts, 14 planted both sorghum and maize, 13 planted both groundnuts and maize and 6 farmers planted all the three types of crops.
(b(i))
Illustrate this information on a Venn diagram. [2]
(b(ii))
How many farmers
(b(ii)(a))
were in this community altogether, [1]
(b(ii)(b))
planted one type of crop only, [1]
(b(ii)(c))
planted two different types of crops only? [1]
5.
(a(i))
Construct triangle ABC in which AB = 10 cm, AC = 8 cm and angle BAC = 100°. [1]
(a(ii))
Measure and write the length of BC. [1]
(b)
On your diagram, draw the locus of points within triangle ABC which are
(b(i))
7 cm from C, [1]
(b(ii))
4 cm from AB, [1]
(b(iii))
equidistant from AC and BC. [2]
(c)
A point P, within triangle ABC, is such that it is less than or equal to 7 cm from C, less than or equal to 4 cm from AB and nearer to AC than BC. Indicate, by shading, the region in which P must lie. [2]
6.
(a)
In the diagram, AD = 3a, AB = 2b, AB : BC = 1 : 2, BX : BD = 1 : 4 and E is the midpoint of CD.
Express in terms of a and/or b
(a(i))
BD, [1]
(a(ii))
DX, [1]
(a(iii))
CD, [1]
(a(iv))
AE. [2]
(b)
Study the following flowchart.
Write a pseudocode corresponding to the flowchart program above. [5]
7.
The frequency distribution table shows the amount of money spent by customers at a particular shop.
Amount K0 < x ≤ 2020 < x ≤ 4040 < x ≤ 6060 < x ≤ 8080 < x ≤ 100Frequency52055182
(a)
Calculate the standard deviation. [6]
(b(i))
Using the information in the frequency distribution table, complete the cumulative frequency table below.
Amount K≤ 0≤ 20≤ 40≤ 60≤ 80≤ 100Cumulative frequency05100 [1]
(b(ii))
Using a horizontal scale of 2 cm to represent 20 units on the x-axis for 0 ≤ x ≤ 100 and a vertical scale of 2 cm to represent 10 units on the y-axis for 0 ≤ y ≤ 100, draw a smooth cumulative frequency curve. [3]
(b(iii))
Showing your method clearly, use your graph to estimate the semi-interquartile range. [2]
8.
(a)
A garden is in the form of a quadrilateral PQRS with PQ = 5 m, QR = 8 m, RS = 7 m and PS = 6 m.
Given that angle PQR = 140° and angle PRS = 18°, calculate
(a(i))
PR, [5]
(a(ii))
the area of triangle PRS, [2]
(a(iii))
the shortest distance from S to PR. [2]
(b)
Solve the equation 10 tan θ = 50 for 0° ≤ θ ≤ 90°. [1]
(c)
Simplify 9a - c81a2 - c2. [2]
9.
Study the following diagram and answer the questions that follow.
(a)
Describe fully a single transformation that maps triangle PQR onto triangle P1Q1R1. [3]
(b)
An enlargement maps triangle PQR onto triangle P2Q2R2. Find the
(b(i))
centre of enlargement, [2]
(b(ii))
scale factor. [1]
(c)
Triangle PQR is mapped onto triangle P3Q3R3 by a single transformation. Find the matrix of this transformation. [3]
(d)
The transformation with 1-201 maps triangle PQR onto triangle P4Q4R4, not shown on the diagram. Find the coordinates of triangle P4Q4R4. [3]
10.
(a)
The values of x and y are connected by the equation y = x3 - 6x2 + 9x. Some corresponding values of x and y correct to 1 decimal place where appropriate are given in the table below.
x-1.5-10123455.5y-30.4p042042034.4
(a(i))
Calculate the value of p. [1]
(a(ii))
Using a scale of 2 cm to represent 1 unit on the x-axis for -2 ≤ x ≤ 6 and 2 cm to represent 10 units on the y-axis for -40 ≤ y ≤ 40, draw the graph of y = x3 - 6x2 + 9x. [3]
(a(iii))
Use your graph to solve the equations
(a(iii)(a))
x3 - 6x2 + 9x = 0, [2]
(a(iii)(b))
x3 - 6x2 + 9x = 5x - 10. [3]
(b)
Evaluate ∫41 (5 + 4x - x2) dx. [3]
11.
(a)
In the diagram, the points T and R lie on the equator while P and Q are on latitude 40° N. [π = 3.142 and R = 6 370 km]
(a(i))
Find the difference in longitude between points T and R. [1]
(a(ii))
Find the distance PQ along latitude 40° N in kilometres. [3]
(a(iii))
Calculate the circumference of the circle of latitude 40° N in kilometres. [2]
(b)
In the following diagram, the dimensions of the top and bottom rectangles of the frustum of a right pyramid are 2 cm by 6 cm and 5 cm by 15 cm respectively.
Given that the height of the frustum is 10 cm, calculate its volume. [6]
12.
A woman intends to order apples and oranges for sale.
(a)
Let x represent the number of apples and y the number of oranges. Write the inequalities which represent each of the conditions described below.
(a(i))
The total number of apples and oranges should not exceed 70. [1]
(a(ii))
The number of oranges should not be more than twice that of apples. [1]
(a(iii))
She has to order not more than 40 apples and at least 10 oranges. [2]
(b)
Using a scale of 2 cm to represent 10 units on both axes, draw x and y axes from 0 to 80 and shade the unwanted region to indicate clearly the region where the solution of the inequalities lie. [4]
(c(i))
The profit on an apple is K2.50 and on an orange it is K1.50. Find the number of apples and oranges she should order for her to make maximum profit. [2]
(c(ii))
Calculate this maximum profit. [2]