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Examinations Council of Zambia — past paper
ECZ 2020 O-Level Paper 2 — Mathematics
12 questions · downloaded from g12titan.com, where this paper is marked free online
1.
(a)
Given that matrix A = 522x, find the
(a(i))
value of x for which A has no inverse, [2]
(a(ii))
inverse of A if x = 1. [2]
(b)
A bag contains 7 red and 3 white identical balls. Two balls are taken from the bag at random one after the other without replacement.
(b(i))
Draw a tree diagram to show all the possible outcomes. [2]
(b(ii))
Find the probability of taking at least one white ball. [3]
2.
(a)
Of the 115 students who attended an end of year party, 74 took fanta, 93 took sprite, 87 took coke, 61 took fanta and sprite, 71 took sprite and coke, 60 took fanta and coke and 50 took all the three drinks.
(a(i))
Illustrate this information on a Venn diagram. [2]
(a(ii))
How many students took
(a(ii)(a))
none of the drinks, [1]
(a(ii)(b))
fanta and sprite but not coke, [1]
(a(ii)(c))
at least two different drinks? [1]
(b)
In the diagram below, AVC is a straight line, AB = 8p, AC = 4p + 9q, BV = -6p + kq and AM = MB.
(b(i))
Express in terms of p, q and/or k
(b(i)(a))
AM, [1]
(b(i)(b))
AV. [1]
(b(ii))
Given that AV = hAC, by forming an equation involving p, q, h and k or otherwise, find the numerical values of h and k. [3]
3.
(a)
Solve the equation x2 - x - 19 = 0, giving your answers correct to 2 decimal places. [5]
(b)
Study the flow chart below.
Write a pseudo code corresponding to the flow chart programme above. [5]
4.
(a)
Construct a quadrilateral PQRS in which PQ = 8 cm, angle QPS = angle PQR = 120°, PS = 6 cm and QR = 5 cm. [1]
(b)
Measure and write the length of RS. [1]
(c)
Within the quadrilateral PQRS, draw the locus of points which are
(c(i))
4.5 cm from Q, [1]
(c(ii))
equidistant from P and Q, [1]
(c(iii))
equidistant from QR and RS. [2]
(d)
A point T, within the quadrilateral PQRS, is such that it is less than or equal to 4.5 cm from Q, nearer to Q than P and nearer to RS than QR. Indicate, by shading, the region in which T must lie. [2]
5.
(a)
Simplify 2 - 18p23p + 1. [2]
(b)
The first three terms of a geometric progression are: m - 2, m + 1 and m + 7. Find
(b(i))
the value of m, [3]
(b(ii))
the common ratio, [2]
(b(iii))
the sum of the first 6 terms. [2]
6.
The figure below is a frustum of a cone. The base radius and top radius are 10 cm and 5 cm respectively, while the height is 12 cm. (Take π as 3.142)
(a)
Calculate its volume. [6]
7.
(a)
The values of x and y are connected by the equation y = x3 + 4x2 + x - 5 as shown in the table below. x-4-3-2-1012y-911-3-51p
(a(i))
Calculate the value of p. [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 ≤ 2 and -10 ≤ y ≤ 25, draw the graph of y = x3 + 4x2 + x - 5. [3]
(a(iii))
Use your graph to find the solutions of the equations
(a(iii)(a))
x3 + 4x2 + x - 5 = 0, [2]
(a(iii)(b))
x3 + 4x2 + x - 5 = 2x + 4. [3]
(b)
Express 42x - 1 - 13x + 2 as a single fraction in its lowest terms. [3]
8.
(a)
P, Q, R and T are points on the surface of the earth as shown in the diagram below.
(Take π as 3.142 and R = 3437 nm)
(a(i))
Find the difference in longitude between points Q and R. [2]
(a(ii))
Calculate the distance, in nautical miles, of
(a(ii)(a))
PQ, [2]
(a(ii)(b))
RT. [2]
(b)
A curve has gradient x2 - 4x + 3. Find the equation of this curve if it passes through the point (3, -1). [3]
(c)
Find the equation of the normal to the curve y = x3 - 2x2 - 4x + 1 at the point (-1, 2). [3]
9.
The table below shows the distribution of lengths of plots in a certain locality. Length of plot in m1≤x≤56≤x≤1011≤x≤1516≤x≤2021≤x<2526≤x<3031≤x≤35Frequency0211443184
(a)
Calculate the standard deviation. [6]
(b(i))
Using the table above, copy and complete the cumulative frequency table below. Length of plot in m≤5≤10≤15≤20≤25≤30≤35Frequency0213100 [1]
(b(ii))
Using a scale of 2 cm to represent 5 units on the horizontal axis and 2 cm to represent 10 units on the vertical axis, draw a smooth cumulative frequency curve. [3]
(b(iii))
Showing your method clearly, use your graph to estimate the interquartile range. [2]
10.
(a)
Three villages A, B and C are joined by straight roads as shown in the diagram below.
Given that AB = 10.2 km, BC = 15.6 km and AC = 9.4 km, calculate
(a(i))
angle BAC to the nearest whole number, [5]
(a(ii))
the area of triangle ABC, [2]
(a(iii))
the shortest distance from A to BC. [2]
(b)
Solve the equation 2 cos θ = 1 for 0° ≤ θ ≤ 180°. [1]
(c)
Simplify 3x2 y8xy3 ÷ 9x34y4. [2]
11.
Study the diagram below to answer the questions that follow.
(a)
Triangle ABC is mapped onto triangle A1B1C1 by an enlargement. Find its centre and scale factor. [2]
(b)
Triangle ABC is mapped onto triangle A2B2C2 by a single transformation. Find
(b(i))
the matrix representing this transformation, [2]
(b(ii))
the area scale factor of the transformation. [2]
(c)
Triangle ABC is mapped onto triangle A3B3C3 by a single transformation. Describe fully this transformation. [3]
(d)
The 1301 maps triangle ABC onto triangle A4B4C4 (not on the diagram). Find the coordinates of A4, B4 and C4. [3]
12.
A carpenter intends to manufacture at least 10 tables and at least 20 chairs. Each table requires 4 hours of assembling and 2 hours of varnishing. Each chair requires 3 hours of assembling and 1 hour of varnishing. There are 240 hours available for assembling and 100 hours for varnishing.
(a)
Given that x represents the number of tables and y the number of chairs, write four inequalities which represent these conditions. [4]
(b)
Using a scale of 2 cm to represent 10 pieces of furniture on each axis, draw the x and y axes for 0 ≤ x ≤ 70 and 0 ≤ y ≤ 100 respectively and shade the unwanted region to show clearly the region where the solution of the inequalities lie. [4]
(c)
Each table sold yields a profit of K300.00 while each chair sold yields a profit of K250.00. Find the best combination of the number of tables and chairs to gain the maximum profit. [2]
(d)
Calculate this estimate of the maximum profit. [2]