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Examinations Council of Zambia — past paper
ECZ 2018 O-Level Paper 2 — Mathematics
12 questions · downloaded from g12titan.com, where this paper is marked free online
1.
(a)
Given that matrix A = 4-512 and B = 8y35,
(a(i))
find the value of y for which the determinants of A and B are equal, [2]
(a(ii))
hence find the inverse of B. [2]
(b)
A small bag contains 6 black and 9 green pens of the same type. Two pens are taken at random one after the other from the bag without replacement. Calculate the probability that both pens
(b(i))
are black, [2]
(b(ii))
are of different colours. [3]
2.
(a)
Solve the equation 3x2 - x - 5 = 0, giving your answers correct to 2 decimal places. [5]
(b)
At Sambilileni College, 20 students study at least one of the three subjects; Mathematics (M), Chemistry (C) and Physics (P). All those who study Chemistry also study Mathematics. 3 students study all the three subjects. 4 students study Mathematics only, 8 students study Chemistry and 14 students study Mathematics.
(b(i))
Draw a Venn diagram to illustrate this information. [2]
(b(ii))
How many students study
(b(ii)(a))
Physics only, [1]
(b(ii)(b))
two subjects only, [1]
(b(ii)(c))
Mathematics and Physics but not Chemistry? [1]
3.
(a)
Evaluate ∫2-1 (2 + x - x2) dx. [3]
(b)
Find the equation of the normal to the curve y = x + 4x at the point where x = 4. [3]
4.
(a(i))
Construct a triangle XYZ in which XY = 9 cm, YZ = 7 cm and angle XYZ = 38°. [1]
(a(ii))
Measure and write the length of XZ. [1]
(b)
On your diagram, within triangle XYZ, construct the locus of points which are
(b(i))
6 cm from Y, [1]
(b(ii))
equidistant from XZ and XY. [2]
(c)
Mark clearly with the letter P, within triangle XYZ, a point which is 6 cm from Y and equidistant from XZ and XY. [1]
(d)
A point T, within triangle XYZ, is such that its distance from Y is less than or equal to 6 cm and it is nearer to XZ than XY. Indicate clearly, by shading, the region in which T must lie. [2]
5.
(a)
Simplify b - aa2 - b2. [2]
(b)
The first three terms of a geometric progression are k + 4, k and 2k - 15 where k is a positive constant.
(b(i))
Find k. [3]
(b(ii))
List the first three terms of the progression. [2]
(b(iii))
Calculate the sum to infinity. [2]
6.
(a)
In the quadrilateral ABCD below, AB = a, AD = b, BC = 2b and AE : AC = 1 : 3.
(a(i))
Find in terms of a and/or b
(a(i)(a))
AE, [1]
(a(i)(b))
BE, [1]
(a(i)(c))
BD. [1]
(a(ii))
Hence or otherwise, show that the points B, E and D are collinear. [2]
(b)
The program below is given in pseudo code.
Start
Enter x, y
Let M = square root (x squared + y squared)
IF M < 0
THEN display error message "M must be positive"
ELSE
END IF
Display M
Stop
Draw the corresponding flow chart for the information given above. [5]
7.
(a)
The values of x and y are connected by the equation y = 2x3 - 3x2 + 5. Some corresponding values of x and y are given in the table below. x-2-1.5-1-0.500.511.52yp-8.50454.5459
(a(i))
Calculate the value of p. [1]
(a(ii))
Using a scale of 4 cm to represent 1 unit on the x-axis for -2 ≤ x ≤ 2 and 2 cm to represent 5 units on the y-axis for -25 ≤ y ≤ 10, draw the graph of y = 2x3 - 3x2 + 5. [3]
(a(iii))
Use your graph to solve the equation 2x3 - 3x2 + 5 = x. [3]
(a(iv))
Calculate an estimate of the gradient of the curve at the point where x = 1.5. [2]
(b)
Express 3x + 1 - 2x - 1 as a single fraction in its lowest terms. [3]
8.
(a)
In the diagram below, K, N, B and R are places on horizontal surface. KN = 80 m, NB = 50 m, angle KNR = 60° and angle KRN = 52°.
(a(i))
Calculate
(a(i)(a))
KR, [4]
(a(i)(b))
the area of triangle KNB. [2]
(a(ii))
Given that the area of triangle KNR is equal to 3 260 m2, calculate the shortest distance from R to KN. [2]
(b)
Sketch the graph of y = cos θ for 0° ≤ θ ≤ 360°. [2]
(c)
Simplify 12dn315cd3 ÷ 9c3n10c2d2. [2]
9.
The frequency table below shows the distribution of marks obtained by 90 learners on a test. Marks10<x≤2020<x≤3030<x≤4040<x≤5050<x≤6060<x≤70Frequency21015233010
(a)
Calculate the standard deviation. [6]
(b(i))
Copy and complete the relative cumulative frequency table below. Marks≤10≤20≤30≤40≤50≤60≤70Cumulative frequency021227508090Relative cumulative frequency00.020.130.3 [1]
(b(ii))
Using a scale of 2 cm to represent 10 units on the x-axis for 0 ≤ x ≤ 70 and 2 cm to represent 0.1 units 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 65th percentile. [2]
10.
Study the diagram below and answer the questions that follow.
(a)
Triangle R is the image of triangle P under a rotation. Find the coordinates of the centre, angle and direction of the rotation. [3]
(b)
A single transformation maps triangle P onto triangle M. Describe fully this transformation. [3]
(c)
Triangle P maps onto triangle V by a stretch. Find the matrix of this transformation. [3]
(d)
If triangle P is mapped onto triangle S by a shear represented by the 10-21, find the coordinates of the triangle S. [3]
11.
A hired bus is used to take learners and teachers on a trip. The number of learners and teachers must not be more than 60. There must be at least 35 people on the trip. There must be at least 6 teachers on the trip. The number of teachers on the trip should not be more than 14. Let x be the number of learners and y the number of teachers.
(a)
Write four inequalities which represent the information above. [4]
(b)
Using a scale of 2 cm to represent 10 units on both axes, draw the x and y axes for 0 ≤ x ≤ 70 and 0 ≤ y ≤ 70 respectively and shade the unwanted region to indicate clearly the region where the solution of the inequalities lie. [4]
(c(i))
If the group has 25 learners, what is the minimum number of teachers that must accompany them? [1]
(c(ii))
If 8 teachers go on this trip, what is the maximum number of learners that can be accommodated on the bus? [1]
(d)
If T is the amount in Kwacha paid by the whole group, what is the cost per learner if T = 30x + 50y? [2]
12.
(a)
The diagram below is a frustum of a rectangular pyramid with a base 14 cm long and 10 cm wide. The top of the frustum is 8 cm long and 4 cm wide.
Given that the height of the frustum is 11.4 cm, calculate its volume. [6]
(b)
The points A(15° N, 40° E), B(35° S, 70° E) and C(35° S, 40° E) are on the surface of the earth. [Use π = 3.142 and R = 6370 km]
(b(i))
Calculate the distance AC in kilometres. [2]
(b(ii))
An aeroplane takes off from point B and flies due west on the same latitude covering a distance of 900 km to a point Q.
(b(ii)(a))
Calculate the difference in longitudes between B and Q. [3]
(b(ii)(b))
Find the position of Q. [1]