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 2023 O-Level Paper 2 — Mathematics
12 questions · downloaded from g12titan.com, where this paper is marked free online
1.
(a)
Express 23m - 2 - 12m - 3 as a single fraction in its lowest terms. [3]
(b)
Solve the equation 3x2 + 13x - 7 = 0, giving your answers correct to 2 decimal places. [5]
2.
(a)
Given that matrix K = 89x1x, find the
(a(i))
value of x for which the determinant of K is 2, [2]
(a(ii))
inverse of K. [2]
(b)
A survey carried out in a certain district showed that out of 215 teachers, 170 taught Mathematics, 100 taught Additional Mathematics, 41 taught Computer Studies, 65 taught Mathematics and Additional Mathematics, 30 taught Additional Mathematics and Computer Studies, 26 taught Computer Studies and Mathematics and 21 taught all the three subjects.
(b(i))
Illustrate this information in a Venn diagram. [2]
(b(ii))
How many teachers taught
(b(ii)(a))
none of the subjects carried out in the survey, [1]
(b(ii)(b))
Additional Mathematics or Computer Studies but not Mathematics, [1]
(b(ii)(c))
two subjects only? [1]
3.
(a)
Find the equation of the curve for which dydx = 12x3 - 4x + 5 at the point (0, -2). [3]
(b)
The nth term of a geometric progression is given by 18012n. Find the
(b(i))
first 3 terms, [2]
(b(ii))
common ratio, [2]
(b(iii))
sum to infinity of the progression. [3]
4.
(a)
Construct a trapezium ABCD in which AB = 10 cm, BC = 8 cm, AD = 7 cm, angle ABC = 60° and angle BAD is 90°. [1]
(b)
Measure and write the length of CD. [1]
(c)
Within the trapezium ABCD, draw the locus of points which are
(c(i))
2 cm from CD, [1]
(c(ii))
9 cm from B, [1]
(c(iii))
equidistant from AB and AD. [2]
(d)
A point E, within the trapezium ABCD, is such that it is greater than or equal to 2 cm from CD, less than or equal to 9 cm from B and nearer to AD than AB. Indicate, by shading, the region in which E must lie. [2]
5.
(a)
Simplify 10p3 q48m4 n2 ÷ 25p4 q34m4 n2. [2]
(b)
Study the following pseudocode.
Start
Enter E, S
IF S < E THEN
Display 'error message'
ELSE probability = ES
END IF
Display probability
Stop
Draw a corresponding flowchart for the pseudocode above. [5]
6.
(a)
A box contains 4 black and 5 blue identical pens. Two pens are picked from the box at random one after the other without replacement. Find the probability that the pens picked are
(a(i))
both blue, [2]
(a(ii))
of the same colour. [3]
(b)
The diagram shows a trapezium OABC in which OA = 2a, OC = 3c, AB = 3OC and D is the midpoint of AC.
Express in terms of a and/or c
(b(i))
AC, [1]
(b(ii))
OD, [1]
(b(iii))
OB, [1]
(b(iv))
BD. [2]
7.
(a)
The points A, B, C and D are on the surface of the earth as shown in the diagram.
(Take π = 3.142 and R = 6 370 km)
(a(i))
Find the difference in latitude between points A and D. [2]
(a(ii))
Calculate in kilometres, the distance between
(a(ii)(a))
A and B along latitude 50° N, [2]
(a(ii)(b))
B and C along longitude 40° E. [2]
(b)
The following diagram is a frustum of a rectangular pyramid with a base 15 cm long and 6 cm wide. The top of the frustum is 10 cm long and 4 cm wide. Its perpendicular height is 8 cm.
(b(i))
Find the height of the pyramid. [2]
(b(ii))
Calculate the volume of the frustum. [4]
8.
Study the diagram below and answer the questions that follow.
(a)
Describe fully a single transformation which maps triangle ABC onto triangle A1B1C1. [3]
(b)
Triangle ABC is mapped onto triangle A2B2C2 (not on the diagram) by an enlargement with centre (1, 0) and scale factor 2. Find the coordinates of A2, B2 and C2. [3]
(c)
Given that triangle A1B1C1 is mapped onto triangle A3B3C3 by a stretch, find the
(c(i))
invariant line, [1]
(c(ii))
scale factor. [2]
(d)
Triangle ABC is mapped onto triangle A4B4C4 by a shear. Find the matrix of this transformation. [3]
9.
A man intends to buy goats and cows for sale. He decides to buy not more than 50 goats and not more than 40 cows. He plans to buy at least 60 animals altogether.
(a)
Given that x represents the number of goats and y the number of cows, write three inequalities which represent these conditions. [4]
(b)
Using a scale of 2 cm to represent 10 animals on each axis, draw the x and y axes for 0 ≤ x ≤ 80 and 0 ≤ y ≤ 80 respectively and shade the unwanted region to show clearly the region where the solution of the inequalities lie. [4]
(c)
The profit on the sale of a goat is K100.00 and on a cow profit is K800.00. How many animals of each type should he buy to make maximum profit? [2]
(d)
What is the maximum profit? [2]
10.
(a)
The values of x and y are connected by the equation y = x3 - 9x + 5. Some of the corresponding values of x and y are given in the table below. x-3.5-3-2-10122.533.5y-6.45q135-3-5-1.9516.4
(a(i))
Calculate the value of q. [1]
(a(ii))
Using a scale of 2 cm to represent 1 unit on the x-axis and 2 cm to represent 5 units on the y-axis for -4 ≤ x ≤ 4 and -10 ≤ y ≤ 20, draw the graph of y = x3 - 9x + 5. [3]
(a(iii))
Use your graph to find the solutions of the equations
(a(iii)(a))
x3 - 9x + 5 = 0, [2]
(a(iii)(b))
x3 - 9x + 5 = -3x + 6. [3]
(b)
Evaluate ∫2-1 (3x2 - 2x + 1) dx. [3]
11.
(a)
In the following triangle, PQ = 41.6 m, PR = 70 m and angle PQR = 105°.
Calculate
(a(i))
angle QPR, [5]
(a(ii))
the area of triangle PQR, [2]
(a(iii))
the shortest distance from Q to PR. [2]
(b)
Solve the equation tan θ = 2.75 for 180° ≤ θ ≤ 270°. [1]
(c)
Simplify 3d2 - 27d + 3. [2]
12.
The following table shows the age distribution of 30 people in a village. Age in years0<x≤1010<x≤2020<x≤3030<x≤4040<x≤5050<x≤60Frequency269742
(a)
Calculate the standard deviation. [6]
(b(i))
Using the table above, copy and complete the relative cumulative frequency table below. Age in years≤0≤10≤20≤30≤40≤50≤60Cumulative frequency02817242830Relative cumulative frequency0.000.070.271.00 [1]
(b(ii))
Using a scale of 2 cm to represent 10 units on the horizontal axis and 2 cm to represent 0.1 units on the vertical axis for 0 ≤ x ≤ 60 and 0.0 ≤ y ≤ 1.0, draw a smooth relative cumulative frequency curve. [3]
(b(iii))
Showing your method clearly, use your graph to estimate the 70th percentile. [2]