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Examinations Council of Zambia — past paper
ECZ 2015 O-Level Paper 2 — Mathematics
12 questions · downloaded from g12titan.com, where this paper is marked free online
1.
(a)
Evaluate (1.3)2 + 1.3 × 0.3. [2]
(b)
Factorise completely 5px - 5py + 3qx - 3qy. [2]
(c)
Bana Makwebo bought a display cabinet at K2 500.00 and later sold it at K2 000.00. What was the percentage loss? [2]
(d)
Simplify 2y2 - 3y - 5y2 - 1. [3]
2.
(a)
Solve the equation 2y2 + 6y - 1 = 0, giving your answers correct to 2 decimal places. [5]
(b)
Given that matrix Q = a23-2,
(b(i))
write an expression in terms of a for the determinant of Q, [1]
(b(ii))
find the value of a, given that the determinant of Q is 2, [2]
(b(iii))
write Q-1. [1]
3.
(a)
In the diagram below, a circle with centre O passes through the points Q, U, S and V. TP and TR are tangents to the circle at the points Q and S respectively. Angle VQO = 20° and angle QVS = 60°.
Find
(a(i))
angle QUS, [1]
(a(ii))
angle VQP, [1]
(a(iii))
angle PTR, [2]
(a(iv))
reflex angle SOQ, [1]
(a(v))
the length ST, given that OS = 3 cm and OT = 15 cm. [2]
(b)
Express 4x - 2 - 2x + 3 as a single fraction in its simplest form. [3]
4.
The Venn diagram shows the number of students who took Business Studies (B), Human Resources (H) and Community Development (C) at Mafundisho College. 100 students took these three courses.
(a)
Find
(a(i))
the value of x, [2]
(a(ii))
the number of students who took Human Resources, [1]
(a(iii))
n(B ∩ C) ∩ H', [1]
(a(iv))
n(B ∪ C) ∩ H'. [1]
(b)
If a student is chosen at random, what is the probability that the student took
(b(i))
one course, [2]
(b(ii))
at least two courses? [3]
5.
(a)
Construct triangle ABC in which AB = 11 cm, angle BAC = 60° and AC = 6 cm. [1]
(b)
Measure and write the length BC. [1]
(c)
On your diagram, draw the locus of points within triangle ABC which are
(c(i))
equidistant from AB and AC, [1]
(c(ii))
equidistant from A and B. [1]
(d)
R is a point inside the triangle ABC such that it is equidistant from AB and AC and equidistant from A and B. Label point R. [2]
(e)
Another point S within triangle ABC is such that it is nearer to AB than AC and nearer to A than B. Indicate clearly, by shading, the region in which S must lie. [2]
6.
(a)
Solve the inequation -2(x - 4) ≤ 2 - 4x. [2]
(b)
In the diagram below, OABC is a parallelogram in which OA = a and AB = b. AC and OB meet at D such that OD = DB. OC is produced to E, such that CE = (12)OC.
Express each of the following in terms of a and/or b
(b(i))
OB, [1]
(b(ii))
AD, [2]
(b(iii))
BE. [1]
7.
(a)
The variables x and y are connected by the equation y = x2 - 2x + 1. Some of the corresponding values of x and y correct to 1 decimal place are given in the table below. x-1.5-1-0.500.511.522.53yp42.310.300.312.34
(a(i))
Calculate the value of p, [1]
(a(ii))
Using a scale of 2 cm to represent 1 unit on both axes, draw the graph of y = x2 - 2x + 1 for -2 ≤ x ≤ 3 and 0 ≤ y ≤ 10. [3]
(a(iii))
Calculate an estimate of the gradient of the curve at the point (0, 1). [2]
(a(iv))
Showing your method clearly, use your graph to solve the equation x2 - 2x + 1 = 1.5. [3]
(b)
Given that Kachinja uses K46.90 to buy $7,
(b(i))
calculate the cost of buying $1, [1]
(b(ii))
how much Kwacha will be required to buy $3? [2]
8.
Triangle P has vertices (2, 2), (3, 1) and (3, 2). Triangle Q has vertices (-2, 2), (-3, 1) and (-3, 2).
(a)
Using a scale of 2 cm to represent 1 unit on each axis, draw axes for values of x and y in the range -4 ≤ x ≤ 4 and -4 ≤ y ≤ 6. Draw and label triangles P and Q. [2]
(b)
Describe fully a single transformation that maps triangle P onto triangle Q. [2]
(c)
Triangle R is the image of triangle P after a rotation of 180° about the origin. Draw triangle R. [2]
(d)
Triangle P is mapped onto triangle S with coordinates (1, -2), (2, -2) and (2, -3). Describe fully a single transformation that maps triangle P onto triangle S. [3]
(e)
A transformation with matrix 1002.5 maps triangle P onto triangle T. Draw and label triangle T and name this transformation. [3]
9.
On a particular day, a tuck shop owner recorded the expenditure of 350 boys and the results were as shown in the table below. Amount K10<x≤2020<x≤3030<x≤4040<x≤5050<x≤6060<x≤7070<x≤8080<x≤9090<x≤100Number of boys20505570604535105
(a)
Calculate the mean amount of money spent. [3]
(b)
Copy and complete the cumulative frequency distribution. Amount K≤10≤20≤30≤40≤50≤60≤70≤80≤90≤100Number of boys02070125195255300350 [1]
(c)
Using a horizontal scale of 2 cm to represent K10.00 and a vertical scale of 2 cm to represent 50 boys, draw a smooth cumulative frequency curve. [3]
(d)
Showing your method clearly, use your graph to estimate
(d(i))
the median, [1]
(d(ii))
the semi-interquartile range. [2]
(e)
Given that those who spent K75.00 or more qualified for a draw in a competition to win a prize, find the number of boys who qualified for the draw. [2]
10.
(a)
In Votani Constituency, A, B and C are polling stations as shown on the diagram below.
Calculate
(a(i))
angle BAC, [5]
(a(ii))
angle ACB, [1]
(a(iii))
the area of triangle ABC correct to 1 decimal place, [2]
(a(iv))
the shortest distance from C to AB. [2]
(b)
Solve the equation 3(t - 5) - 2 = -1 + t. [2]
11.
(a)
A, B, C and D are points on the surface of the earth as shown in the diagram below.
(a(i))
Using latitudes and longitudes, write the positions of the points A and B. [2]
(a(ii))
Find the difference in longitude between points C and D. [1]
(a(iii))
Calculate the distance CD in nautical miles [π = 3.142 and R = 3437 nm]. [2]
(a(iv))
Given that the local time at D is 13 05 hours, find the time at C. [1]
(b)
A cylindrical water tank at Mwaiseni Lodge has diameter of 200 cm and height 250 cm as shown below.
Taking π to be 3.142, find
(b(i))
the total surface area of the tank if it is closed, [3]
(b(ii))
the number of litres of water the tank can hold. [3]
12.
(a)
The graph below shows three inequalities that satisfy region R.
(a(i))
Write the three inequalities that define the unshaded region R. [6]
(a(ii))
Find the largest value of 4x - 5y within region R. [1]
(b)
Before the ZESCO prepaid meters were installed in Mr Malaiti's house, his electricity bill used to consist of a fixed charge of K20.00 and 20 ngwee for every unit of electricity used.
(b(i))
Given that in a particular month, he used 387.3 units, calculate the amount he paid to ZESCO. [2]
(b(ii))
In another month, he was given a bill of K120.40. Find the number of units he used in this month. [3]