Save as PDF or print — the answers and worked solutions live online, where your work gets marked free.
Examinations Council of Zambia — past paper
ECZ 2026 GCE Paper 2 — Mathematics
11 questions · downloaded from g12titan.com, where this paper is marked free online
1.
(a)
Express 7x - 3 - 81 - 2x as a single fraction in its simplest form. [3]
(b)
A box contains 2 green balls and 3 blue balls. A ball is selected at random from the box and not replaced. A second ball is then selected.
(b(i))
Draw a tree diagram to illustrate the outcomes. [2]
(b(ii))
What is the probability of selecting two balls of different colours? [3]
2.
(a)
In a group of 40 girls, 25 like mangoes, 20 like bananas, 15 like guavas, 9 like mangoes and bananas, 10 like mangoes and guavas, 7 like bananas and guavas and 4 like all the three types of fruits.
(a(i))
Illustrate this information in a Venn diagram. [2]
(a(ii))
How many girls
(a(ii)(a))
like guavas only, [1]
(a(ii)(b))
like two types of fruits, [1]
(a(ii)(c))
do not like bananas? [1]
(b)
Given that the determinant of matrix M = x - 3-1x + 23 is 5, find the
(b(i))
value of x, [2]
(b(ii))
inverse of matrix M. [2]
3.
(a)
The second and fourth terms of a geometric progression are 6 and 1.5 respectively. Find the
(a(i))
common ratio and the first term, [3]
(a(ii))
8th term, [2]
(a(iii))
sum to infinity of the progression. [2]
(b)
Evaluate ∫1-1 (3 - 4x + 9x2 - 4x3) dx. [3]
4.
(a(i))
Construct a triangle PQR in which PQ = 10cm, QR = 7cm and angle PQR = 50°. [1]
(a(ii))
Measure and write the length PR. [1]
(b)
Within triangle PQR, construct the locus of points which are
(b(i))
6cm from Q, [1]
(b(ii))
2cm from QR, [1]
(b(iii))
equidistant from PR and PQ. [2]
(c)
A point T, within the triangle PQR is such that it is less than or equal to 6cm from Q, more than 2cm from QR and nearer to PR than PQ. Indicate clearly by shading, the region in which T must lie. [2]
5.
(a)
In the diagram, OA = 2a, AB = 2b, M is the midpoint of OA and AN = 2NB.
Express in terms of a and/or b
(a(i))
OB, [1]
(a(ii))
MB, [1]
(a(iii))
ON, [1]
(a(iv))
MN. [2]
(b)
Solve the equation x2 + 6x + 1 = 0, giving your answers correct to 2 decimal places. [5]
6.
(a)
Simplify 2x2 - 50x + 5. [2]
(b)
Study the following pseudocode.
Draw a flowchart corresponding to the pseudocode above. [5]
7.
(a)
The variables x and y are connected by the equation y = x3 - 5x2 - x + 5 as shown in the table.
x-2-1012345.2y-21050-9-16q5.2
(a(i))
Calculate the value of q. [1]
(a(ii))
Using a scale of 2cm to represent 1 unit on the x-axis for -2 ≤ x ≤ 6 and 2cm to represent 5 units on the y-axis for -25 ≤ y ≤ 10, draw the graph of y = x3 - 5x2 - x + 5. [3]
(a(iii))
Use your graph to solve the equations
(a(iii)(a))
x3 - 5x2 - x + 5 = 0, [2]
(a(iii)(b))
x3 - 5x2 - x + 5 = 2x - 15. [3]
(b)
Find the equation of a tangent to the curve y = x3 - x2 - 15x + 3 at the point (1, -12). [3]
8.
(a)
In the triangle, AB = 125cm, BC = 57cm and angle ABC = 91.2°.
Calculate
(a(i))
AC, [5]
(a(ii))
the area of triangle ABC, [2]
(a(iii))
the shortest distance from B to AC. [2]
(b)
Solve the equation tan θ = 0.7 for 180° ≤ θ ≤ 360°. [1]
(c)
Simplify 52x2 y25a2 b2 ÷ 13x2 y310a3 b2. [2]
9.
The frequency table shows the distribution by age of 70 people in a town.
Age years0 < x ≤ 1010 < x ≤ 2020 < x ≤ 3030 < x ≤ 4040 < x ≤ 5050 < x ≤ 60Frequency4161820102
(a)
Calculate the standard deviation. [6]
(b(i))
Using the information in the frequency table, complete the following relative cumulative frequency table.
Age years≤ 0≤ 10≤ 20≤ 30≤ 40≤ 50≤ 60Cumulative frequency042038586870Relative cumulative frequency0.000.060.290.54 [1]
(b(ii))
Using a scale of 2cm to represent 10 units on the x-axis for 0 ≤ x ≤ 60 and 2cm to represent 0.1 unit on the y-axis for 0 ≤ y ≤ 1, draw a smooth relative cumulative frequency curve. [3]
(b(iii))
Showing your method clearly, use your graph to estimate the interquartile range. [2]
10.
A man intends to make two types of bricks, type A and type B for sale. He intends to make at least 100 type A bricks and at least 50 type B bricks. He wants to make at most 600 bricks altogether. The number of type B bricks must not be more than two times the number of type A bricks.
(a)
Let x be the number of type A bricks and y the number of type B bricks. Write four inequalities which satisfy the above conditions. [4]
(b)
Using a scale of 2cm to represent 100 bricks on each axis, draw the x-axis for 0 ≤ x ≤ 700 and y-axis for 0 ≤ y ≤ 700 respectively and shade the unwanted region to show clearly the region in which the solution of the inequalities lie. [4]
(c)
Given that the profit on the sale of a type A brick is K10.00 and on a type B brick is K15.00, how many bricks of each type should he make to maximise profit? [2]
(d)
Calculate the maximum profit. [2]
11.
Study the diagram below and answer the questions that follow.
(a)
Triangle ABC is mapped onto triangle A1B1C1 by a single transformation. Describe fully this transformation. [3]
(b)
An enlargement maps triangle ABC onto triangle A2B2C2. Find the centre of the enlargement and scale factor. [3]
(c)
Triangle ABC is mapped onto triangle A3B3C3 by a shear. Find the
(c(i))
matrix of this transformation, [2]
(c(ii))
shear factor. [1]
(d)
A transformation with the matrix 3001 maps triangle ABC onto triangle A4B4C4, not on the diagram. Find the coordinates of A4, B4, and C4. [3]