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Examinations Council of Zambia — past paper
ECZ 2016 O-Level Paper 2 — Mathematics
12 questions · downloaded from g12titan.com, where this paper is marked free online
1.
(a)
Given that matrix Q = 3-2x4, find the
(a(i))
value of x, given that the determinant of Q is 2, [2]
(a(ii))
inverse of Q. [2]
(b)
Solve the equation x2 + 2x = 7, giving your answers correct to 2 decimal places. [5]
2.
(a)
Of the 50 villagers who can tune in to Kambani Radio Station, 29 listen to news, 25 listen to sports, 22 listen to music, 11 listen to both news and sports, 9 listen to both sports and music, 12 listen to both news and music, 4 listen to all the three programmes, and 2 do not listen to any programme.
(a(i))
Draw a Venn diagram to illustrate this information. [2]
(a(ii))
How many villagers
(a(ii)(a))
listen to music only, [1]
(a(ii)(b))
listen to one type of programme only, [1]
(a(ii)(c))
listen to two types of programmes only? [1]
(b)
A survey carried out at a certain hospital indicates that the probability that a patient tested positive for malaria is 0.6. What is the probability that two patients selected at random
(b(i))
one tested negative while the other positive, [3]
(b(ii))
both patients tested negative. [2]
3.
(a)
The program below is given in the form of a pseudocode.
Start
Enter radius
If radius < 0
Then display "error message" and re-enter positive radius
Else enter height
If height < 0
Then display "error message" and re-enter positive height
Else Volume = 13 * π * square radius * height
End if
Display volume
Stop
Draw the corresponding flowchart for the information given above. [5]
(b)
In the diagram below, OAB is a triangle in which OA = 3a and OB = 6b. OC : CA = 2 : 3 and AD : DB = 1 : 2. OD meets CB at E.
(b(i))
Express each of the following in terms of a and/or b
(b(i)(a))
AB, [1]
(b(i)(b))
OD, [1]
(b(i)(c))
BC. [2]
(b(ii))
Given that BE = hBC, express BE in terms of h, a and b. [1]
4.
(a(i))
Construct a triangle ABC where AB = BC = CA = 7 cm. [1]
(a(ii))
Measure and write the size of angle CAB. [1]
(b)
Within the triangle ABC, construct the locus of points
(b(i))
equidistant from AB and BC, [1]
(b(ii))
4 cm from B, [1]
(b(iii))
3 cm from AB. [2]
(c)
A point R, within triangle ABC, is such that it is nearer to BC than AB, less than 3 cm from AB and less than 4 cm from B. Shade the region in which R must lie. [2]
5.
(a)
Simplify x - 1x2 - 1. [2]
(b)
The first three terms of a geometric progression are x + 1, x - 3 and x - 1. Find
(b(i))
the value of x, [3]
(b(ii))
the first term, [1]
(b(iii))
the sum to infinity. [3]
6.
The equation of a curve is y = x3 - (32)x2. Find
(a)
the equation of the normal where x = 2, [3]
(b)
the coordinates of the stationary points. [3]
7.
The ages of people living at Pamodzi Village are recorded in the frequency table below. Ages0<x≤1010<x≤2020<x≤3030<x≤4040<x≤5050<x≤60Number of people7222823155
(a)
Calculate the standard deviation. [6]
(b(i))
Using the information in the table above, copy and complete the cumulative frequency table below. Age≤10≤20≤30≤40≤50≤60Number of people729100 [1]
(b(ii))
Using a scale of 2 cm to represent 10 units on both axes, draw a smooth cumulative frequency curve where 0 ≤ x ≤ 60 and 0 ≤ y ≤ 100. [3]
(b(iii))
Showing your method clearly, use your graph to estimate the semi-interquartile range. [2]
8.
Study the diagram below to answer the questions that follow.
(a)
An enlargement maps triangle ABC onto triangle A1B1C1. Find
(a(i))
the centre of enlargement, [1]
(a(ii))
the scale factor. [1]
(b)
Triangle ABC is mapped onto triangle A2B2C2 by a shear. Find the matrix which represents this transformation. [3]
(c)
Triangle ABC is mapped onto triangle A3B3C3 by a single transformation. Describe this transformation fully. [3]
(d)
A transformation with matrix -3001 maps triangle ABC onto triangle A4B4C4 not drawn on the diagram. Find
(d(i))
the scale factor of this transformation, [1]
(d(ii))
the coordinates of A4, B4 and C4. [3]
9.
(a)
The points A, B, C and D are on the surface of the earth.
(Take π = 3.142 and R = 3437 nm)
(a(i))
Find the difference in latitude between points C and B. [1]
(a(ii))
Calculate the length of the circle of latitude 50° N in nautical miles. [2]
(a(iii))
Find the distance AD in nautical miles. [3]
(b)
The cross section of a rectangular tank measures 1.2 m by 0.9 m. If it contains fuel to a depth of 10 m, find the number of litres of fuel in the tank. (1 m3 = 1000 litres.) [3]
(c)
A cone has a perpendicular height of 12 cm and slant height of 13 cm. Calculate its total surface area. (Take π = 3.142) [3]
10.
(a)
The diagram below shows the location of three secondary schools, namely Kamubala (K), Belengani (B) and Pendeni (P) in a district. P is 5 km from K, B is 3 km from K and angle PKB = 110°.
Calculate
(a(i))
BP, [5]
(a(ii))
the area of triangle BKP, [2]
(a(iii))
the shortest distance from K to BP. [2]
(b)
Solve the equation tan θ = 0.7 for 0° ≤ θ ≤ 180°. [1]
(c)
Simplify 17k220a2 ÷ 51k25a. [2]
11.
A Health Lobby Group produced a guide to encourage healthy living among the local community. The group produced the guide in two formats: a short video and a printed book. The group needed to decide the number of each format to produce for sale to maximise profit. Let x represent the number of videos produced and y the number of printed books produced.
(a(i))
Write the inequalities which represent each of the following conditions:
(a(i)(a))
the total number of copies produced should not be more than 800, [1]
(a(i)(b))
the number of video copies to be at least 100, [1]
(a(i)(c))
the number of printed books to be at least 200. [1]
(a(ii))
Using a scale of 2 cm to represent 100 copies on both axes, draw the x and y axes for 0 ≤ x ≤ 800 and 0 ≤ y ≤ 800 respectively and shade the unwanted region to indicate clearly the region where the solution of the inequalities lie. [4]
(a(iii))
The profit on the sale of each video copy is K15.00 while profit on each printed book is K8.00. How many of each type were produced to make maximum profit? [2]
(b)
Maphone manufacturing company paid a total dividend of K12 600 000.00 at the end of 2015 on 600 shares.
(b(i))
Calculate the dividend per share. [2]
(b(ii))
Magula owned 200 shares in the company, how much was paid out in dividends to her? [1]
12.
(a)
The values of x and y are connected by the equation y = x(x - 2)(x + 2). Some corresponding values of x and y are given in the table below. x-3-2-10123y-15030-30k
(a(i))
Calculate the value of k. [1]
(a(ii))
Using a scale of 2 cm to 1 unit on the x-axis for -3 ≤ x ≤ 3 and 2 cm to represent 5 units on the y-axis for -16 ≤ y ≤ 16, draw the graph of y = x(x - 2)(x + 2). [3]
(a(iii))
Use your graph to solve the equations
(a(iii)(a))
x(x - 2)(x + 2) = 0, [2]
(a(iii)(b))
x(x - 2)(x + 2) = x + 2. [3]
(b)
Express 22x - 1 - 13x + 1 as a single fraction in its simplest form. [3]