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Examinations Council of Zambia — past paper
ECZ 2022 O-Level Paper 2 — Mathematics
12 questions · downloaded from g12titan.com, where this paper is marked free online
1.
(a)
Given that the determinant of matrix Q = 2b - 14-3b-5 is 15, find the
(a(i))
value of b, [2]
(a(ii))
inverse of matrix Q. [2]
(b)
The Venn diagram shows the number of learners who took at least one of the three subjects Biology, Chemistry and Physics.
(b(i))
Given that the total number of learners was 170, calculate the value of x. [2]
(b(ii))
How many learners took
(b(ii)(a))
Biology and Physics but not Chemistry, [1]
(b(ii)(b))
two subjects only, [1]
(b(ii)(c))
one subject only? [1]
2.
(a)
Solve the equation 3x2 - 5x - 7 = 0, giving your answers correct to 2 decimal places. [5]
(b)
In the parallelogram OABC, OA = 4a, OC = 6c, AP = (23)AC and M is the midpoint of BC.
(b(i))
Find in terms of a and/or c
(b(i)(a))
AC, [1]
(b(i)(b))
OP, [1]
(b(i)(c))
OM. [1]
(b(ii))
Hence or otherwise, show that the points O, P and M are collinear. [2]
3.
(a)
Simplify 2x + 142x2 - 98. [2]
(b)
Given that the third and sixth terms of a geometric progression are 34 and 332 respectively, find the
(b(i))
first term and the common ratio, [3]
(b(ii))
nth term, [2]
(b(iii))
sum to infinity of the progression. [2]
4.
(a)
Study the flowchart below.
Write a pseudocode corresponding to the flowchart above. [5]
(b)
A box contains identical cards and on each card a letter of the English alphabet is printed. A card is selected at random from the box and not replaced, and a second card is then selected. Find the probability that the two cards selected
(b(i))
have vowels printed, [2]
(b(ii))
one has a vowel and the other a consonant printed. [3]
5.
(a)
Construct a parallelogram ABCD in which AB = 10 cm, AD = BC = 6.5 cm, angle BAD = 110° and angle ABC = 70°. [1]
(b)
Measure and write the length AC. [1]
(c)
Within parallelogram ABCD, construct the locus of points which are
(c(i))
3 cm from AB, [1]
(c(ii))
6 cm from C, [1]
(c(iii))
equidistant from BC and CD. [2]
(d)
A point Z, within parallelogram ABCD, is such that it is less than or equal to 3 cm from AB, less than or equal to 6 cm from C and nearer to CD than BC. Indicate clearly, by shading, the region in which Z must lie. [2]
6.
(a)
Evaluate ∫4-1 (2 + 2x + 6x2) dx. [3]
(b)
Find the equation of the normal to the curve y = x2 - 3x - 4 at the point (2, -6). [3]
7.
A businessman intends to sell two types of markers, permanent and dry erase. He decides that there should be at least 20 permanent markers and at least 10 dry erase markers. In order to make his business stable, he decides to order not more than 70 markers altogether. The order price of a permanent marker is K15.00 and that of a dry erase marker is K20.00. He is prepared to spend not more than K1 200.00 altogether.
(a)
Let x represent the number of permanent markers and y the number of dry erase markers. Write four inequalities which satisfy the above conditions. [5]
(b)
Using a scale of 2 cm to represent 10 markers on each axis, draw 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 permanent marker is K25.00 and on each dry erase marker profit is K30.00.
(c(i))
How many markers of each type should be ordered to make maximum profit? [2]
(c(ii))
Find the maximum profit. [1]
8.
(a)
The variables x and y are connected by the equation y = (x - 3)(x - 5)(x + 2). Some corresponding values of x and y are shown in the table below. x-2-10123456y0243024120-60k
(a(i))
Calculate the value of k. [1]
(a(ii))
Using a scale of 2 cm to represent 1 unit on the x axis for -2 ≤ x ≤ 6 and a scale of 2 cm to represent 10 units on the y axis for -10 ≤ y ≤ 40, draw the graph of y = (x - 3)(x - 5)(x + 2). [3]
(a(iii))
Use your graph to estimate the
(a(iii)(a))
gradient of the curve at the point (-1, 24), [2]
(a(iii)(b))
area bounded by the curve, x = -1, x = 2 and y = 0. [3]
(b)
Express 2x + 5 + 53x - 4 as a single fraction in its simplest form. [3]
9.
(a)
The diagram shows a triangular piece of land ABC.
Given that AB = 1 km, BC = 2.5 km and AC = 2.8 km, calculate the
(a(i))
angle ABC, [5]
(a(ii))
area of triangle ABC, [2]
(a(iii))
shortest distance from B to AC. [2]
(b)
Solve the equation 2 cos θ = -1 for 180° ≤ θ ≤ 360°. [1]
(c)
Simplify 42xyz56 × 32xy40x2 y. [2]
10.
The table shows the heights of 100 plants measured in centimetres on a particular day. Height in cm0<x≤1010<x≤2020<x≤3030<x≤4040<x≤5050<x≤6060<x≤70Frequency38142223228
(a)
Calculate the standard deviation. [6]
(b(i))
Using the table above, copy and complete the cumulative frequency table below. Height in cm≤0≤10≤20≤30≤40≤50≤60≤70Cumulative frequency031125100 [1]
(b(ii))
Using a scale of 2 cm to represent 10 units on both axes for 0 ≤ x ≤ 70 and 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]
11.
(a)
The diagram shows the points P(20° N, 80° E), Q(40° S, 80° E) and R(40° S, 30° E) on the surface of the earth. [Take π as 3.142 and R = 6370 km]
(a(i))
Find the difference in longitudes between points R and Q. [2]
(a(ii))
Calculate the distance PQ in kilometres. [2]
(a(iii))
Find the length of the circle of latitude 40° S in kilometres. [2]
(b)
The diagram shows a bin BCDE which was made by cutting off the smaller cone ADE from the cone ABC. [Take π = 3.142]
Given that AF = 9 cm, BG = 8 cm, EF = 3 cm and GF = h cm, calculate the
(b(i))
value of h, [2]
(b(ii))
volume of the bin BCDE. [4]
12.
Study the following diagram 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
(b(i))
centre of enlargement, [1]
(b(ii))
scale factor. [2]
(c)
Triangle ABC is mapped onto triangle A3B3C3 by a single transformation. Find the matrix representing this transformation. [3]
(d)
The transformation with matrix 100-2 maps triangle ABC onto triangle A4B4C4 not shown on the diagram. Find the coordinates of A4, B4 and C4. [3]