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Examinations Council of Zambia — past paper
ECZ 2017 GCE Paper 2 — Mathematics
12 questions · downloaded from g12titan.com, where this paper is marked free online
1.
(a)
Given that matrix K = 10-211-2, find
(a(i))
the determinant of K, [2]
(a(ii))
the inverse of K. [2]
(b)
Solve the equation 3z2 = 7z - 1, giving your answers correct to 2 decimal places. [5]
2.
(a)
Simplify m2 - 1m2 - m. [2]
(b)
The first three terms of a geometric progression are 6 + n, 10 + n and 15 + n. Find
(b(i))
the value of n, [2]
(b(ii))
the common ratio, [2]
(b(iii))
the sum of the first 6 terms of this sequence. [3]
3.
(a)
In a box of 10 bulbs, 3 are faulty. If two bulbs are drawn at random one after the other, find the probability that
(a(i))
both are good, [2]
(a(ii))
one is faulty and the other is good. [3]
(b)
The Venn diagram below shows tourist attractions visited by students in a certain week.
(b(i))
Find the value of y, if 7 students visited Mambilima Falls only. [2]
(b(ii))
How many students visited
(b(ii)(a))
Victoria Falls but not Gonya Falls, [1]
(b(ii)(b))
two tourist attractions only, [1]
(b(ii)(c))
one tourist attraction only? [1]
4.
(a)
Express 35x - 2 - 2x + 3 as a single fraction in its simplest form. [3]
(b)
Evaluate ∫52 (3x2 + 2) dx. [3]
5.
(a(i))
Construct triangle PQR in which PQ is 9 cm, angle PQR = 60° and QR = 10 cm. [1]
(a(ii))
Measure and write the length of PR. [1]
(b)
On your diagram, draw the locus of points within triangle PQR which are
(b(i))
3 cm from PQ, [1]
(b(ii))
7 cm from R, [1]
(b(iii))
equidistant from P and R. [2]
(c)
A point M, within triangle PQR, is such that it is nearer to R than P, less than or equal to 7 cm from R and less than or equal to 3 cm from PQ. Shade the region in which M must lie. [2]
6.
(a)
In the diagram below, OABC is a parallelogram in which OA = a and AB = 2b. OB and AC intersect at D. E is the midpoint of CD.
Express in terms of a and/or b.
(a(i))
OB, [1]
(a(ii))
OE, [2]
(a(iii))
CD. [2]
(b)
The program below is given in the form of a flow chart.
Write a pseudo code corresponding to the flow chart program above. [5]
7.
Answer the whole of this question on a sheet of graph paper.
Using a scale of 1 cm to represent 1 unit on each axis, draw x and y axes for -6 ≤ x ≤ 10 and -6 ≤ y ≤ 12.
(a)
Quadrilateral ABCD has vertices A(1, 2), B(2, 1), C(3, 2) and D(2, 3).
Quadrilateral A1B1C1D1 has vertices A1(3, 2), B1(6, 1), C1(9, 2) and D1(6, 3)
(a(i))
Draw and label quadrilaterals ABCD and A1B1C1D1. [2]
(a(ii))
Describe fully a single transformation which maps quadrilateral ABCD onto quadrilateral A1B1C1D1. [3]
(b)
The 1031, maps quadrilateral ABCD onto quadrilateral A2B2C2D2.
(b(i))
Find the coordinates of quadrilateral A2B2C2D2. [3]
(b(ii))
Draw and label quadrilateral A2B2C2D2. [1]
(c)
Quadrilateral A3B3C3D3 has vertices A3(-2, -4), B3(-4, -2), C3(-6, -4) and D3(-4, -6). Describe fully the transformation which maps quadrilateral ABCD onto quadrilateral A3B3C3D3. [3]
8.
The frequency table below shows the number of copies of newspapers allocated to 48 newspaper vendors.
Copies of newspapers25 < x ≤ 3030 < x ≤ 3535 < x ≤ 4040 < x ≤ 4545 < x ≤ 5050 < x ≤ 5555 < x ≤ 60Number of vendors547111281
(a)
Calculate the standard deviation. [6]
(b)
Answer this part of the question on a sheet of graph paper.
(b(i))
Using the information in the table above, copy and complete the cumulative frequency table below.
Copies of newspapers≤ 25≤ 30≤ 35≤ 40≤ 45≤ 50≤ 55≤ 60Number of vendors0591627 [1]
(b(ii))
Using a horizontal scale of 2 cm to represent 10 newspapers on the x-axis for 0 ≤ x ≤ 60 and a vertical scale of 4 cm to represent 10 vendors on the y-axis for 0 ≤ y ≤ 50, draw a smooth cumulative frequency curve. [3]
(b(iii))
Showing your method clearly, use your graph to estimate the 50th percentile. [2]
9.
(a)
The diagram below shows the graph of y = x3 + x2 - 5x + 3.
Use the graph
(a(i))
to calculate an estimate of the gradient of the curve at the point (2, 5). [2]
(a(ii))
to solve the equations
(a(ii)(a))
x3 + x2 - 5x + 3 = 0, [2]
(a(ii)(b))
x3 + x2 - 5x + 3 = 5x. [3]
(a(iii))
to calculate an estimate of the area bounded by the curve, x = 0, y = 0 and x = -2. [2]
(b)
Find the equation of the tangent to the curve y = x2 - 3x - 4 at the point where x = 2. [3]
10.
(a)
In triangle PQR below, QR = 36.5 m, angle PQR = 36° and angle QPR = 46°.
Calculate
(a(i))
PQ, [4]
(a(ii))
the area of triangle PQR, [2]
(a(iii))
the shortest distance from R to PQ. [2]
(b)
Solve the equation sin θ = 0.6792 for 0° ≤ θ ≤ 360°. [2]
(c)
Simplify p2 q34 × 8pq ÷ 2p2 q. [2]
11.
Answer this question on a sheet of graph paper.
Makwebo prepares two types of sausages, hungarian and beef, daily for sale. She prepares at least 40 hungarian and at least 10 beef sausages. She prepares not more than 160 sausages altogether. The number of beef sausages prepared are not more than the number of hungarian sausages.
(a)
Given that x represents the number of hungarian sausages and y the number of beef sausages, write four inequalities which represent these conditions. [4]
(b)
Using a scale of 2 cm to represent 20 sausages on both axes, draw the x and y axes for 0 ≤ x ≤ 160 and 0 ≤ y ≤ 160 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 each hungarian sausage is K3.00 and on each beef sausage is K2.00. How many of each type of sausages are required to be prepared to make maximum profit? [2]
(d)
Calculate this maximum profit. [2]
12.
(a)
The figure below is a cone ABC from which BCXY remained after the small cone AXY was cut off. [Take π as 3.142]
Given that EX = 4 cm, DB = 12 cm and DE = 15 cm, calculate
(a(i))
the height AE, of the smaller cone AXY. [2]
(a(ii))
the volume of XBCY, the shape that remained. [4]
(b)
P(80° N, 10° E), Q(80° N, 70° E), R(85° S, 70° E) and S(85° S, 10° E) are four points on the surface of the earth.
(b(i))
Show these points on a clearly labelled sketch of the surface of the earth. [2]
(b(ii))
Find in nautical miles
(b(ii)(a))
the distance QR along the longitude, [2]
(b(ii)(b))
the circumference of the circle of latitude 85° S. [Take π as 3.142 and R = 3 437 nm] [2]