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Examinations Council of Zambia — past paper
ECZ 2010 O-Level Paper 2 — Mathematics
12 questions · downloaded from g12titan.com, where this paper is marked free online
1.
(a)
Factorise completely 5 - 20x2. [2]
(b)
Solve the equation 2(2x - 5) + 2 = x + 7. [3]
(c)
Express 21 - 3x + 45 + x as a single fraction in its simplest form. [3]
2.
(a)
Solve the equation 7 - 5x - x2 = 0, giving your answers correct to 2 decimal places. [5]
(b)
In the diagram below, ABC is a tangent and O is the centre of the circle. BE is the diameter, angle DEO = 25° and angle FEO = 22°.
Calculate
(b(i))
angle BOD, [1]
(b(ii))
angle CBD, [1]
(b(iii))
angle DBF. [1]
3.
(a)
Given that d = p + y5y, express y in terms of d and p. [3]
(b)
Given that 2x - 7y - 1 = 0, find the gradient of this equation. [1]
(c)
If P = 2061 and Q = a01b, find
(c(i))
PQ, [2]
(c(ii))
the values of a and b, given that PQ = P - Q. [3]
4.
(a)
Construct triangle DEF in which EF = 10 cm, angle DEF = 38° and angle DFE = 95°. [1]
(b)
On your diagram, draw the locus of points which are:
(b(i))
4.5 cm from F, [1]
(b(ii))
equidistant from DE and EF. [2]
(c)
P is a point inside the triangle DEF such that it is 4.5 cm from F and is equidistant from DE and EF. Label the point P. [1]
(d)
Measure and write down the length of EP. [1]
5.
In the diagram below, triangles ABC and ACD are right angled. Angle ABC = angle CAD = 90°, AC = 20.2 cm, BC = 9.5 cm and AD = 12 cm.
Calculate
(a)
AB, [2]
(b)
angle BAC, [2]
(c)
the area of quadrilateral ABCD, [3]
(d)
the perimeter of quadrilateral ABCD. [3]
6.
(a)
The diagram below shows a trapezium in which OA = a, OC = c and CB is parallel to OA. CB is twice OA, the points D and E are midpoints of AB and CB respectively.
Express in terms of a and/or c the following vectors
(a(i))
CA, [1]
(a(ii))
AB, [1]
(a(iii))
EB, [2]
(a(iv))
OD, [2]
(a(v))
BO. [1]
(b)
At one college, a group of 25 students were asked which Cell Phone service providers they subscribed to. The results are shown in the Venn diagram below.
(b(i))
Calculate the value of x. [1]
(b(ii))
Given that G = {Glo}, R = {Rodgers} and D = {Du}, find,
(b(ii)(a))
n(G ∩ R) [1]
(b(ii)(b))
n(D ∪ G'). [2]
7.
(a)
The figure below shows the net of a pyramid with a square base ABCD of side 8 cm.
Given that each of the triangles is an equilateral triangle, calculate
(a(i))
the perimeter of the net, [1]
(a(ii))
the area of the figure in square centimeters, correct to 3 significant figures. [3]
(a(iii))
the volume of the pyramid when the shape is folded along the dotted lines.
[volume of pyramid = 13 base area x perpendicular height] [3]
(b)
Given that x = 7 and y = -9,
(b(i))
find the value of 3x - y. [2]
(b(ii))
Simplify 3y2 - 5y - 12y2 - 9. [3]
8.
P, Q and R are fishing camps along the banks of Lake Kariba joined by straight paths PQ, QR and RP. P is 7.6 km from Q, and Q is 13.2 km from R and angle PQR = 120°.
(a)
Calculate
(a(i))
the distance PR, [5]
(a(ii))
the area of triangle PQR. [3]
(b)
Find the shortest distance from Q to PR. [2]
(c)
A fisherman takes 30 minutes to move from R to P. Calculate his average speed in km/h. [2]
9.
The variables x and y are connected by the equation y = (15)x(10 - x2).
The table below shows some corresponding values of x and y. The values of y are given correct to one decimal place where necessary. x-101233.54y-1.801.82.40.6-1.6p
(a)
Calculate the value of p. [1]
(b)
Using a scale of 2 cm to represent 1 unit on both axes for -1 ≤ x ≤ 5 and -5 ≤ y ≤ 3, draw the graph of y = (15)x(10 - x2). [3]
(c)
By drawing a tangent to the curve, estimate the gradient of the curve at the point (1, 1.8). [2]
(d)
On the same graph, draw the line whose equation is 5y + 4x = 4. [2]
(e)
Use your graphs to find the solutions of (15)x(10 - x2) = -(45)x + 45. [2]
(f)
Estimate the area under the curve between x = 1, x = 3 and the line y = 0. [2]
10.
The following table shows the total number of points scored by 110 candidates at Mingule High School, in the 2007 Grade 12 Final Examinations. Points scored6 < x ≤ 88 < x ≤ 1010 < x ≤ 1212 < x ≤ 1414 < x ≤ 1616 < x ≤ 20Frequency13169152235
(a)
Calculate the estimated mean points scored. [2]
(b)
Copy and complete the table showing the cumulative frequency distribution. Points scored≤6≤8≤10≤12≤14≤16≤20Frequency013110 [2]
(c)
Using a horizontal scale of 2 cm to represent 2 points and a vertical scale of 2 cm to represent 10 candidates, draw a smooth cumulative frequency curve. [3]
(d)
Use your graph to find the median points scored. [2]
(e)
A scholarship was offered to two candidates. Find the probability that the two candidates chosen scored less than 14 points. [2]
(f)
Selection to the University of Zambia was done and only those who scored 14 points or less were selected. How many candidates were not selected from this group? [1]
11.
Mr. Simukadi intends to buy a total of 500 scratch cards from Unitel and Bitel. He decides to buy Bitel scratch cards which must be at least one third of Unitel scratch cards. He wants to buy at least 150 Bitel scratch cards and not more than 300 Unitel scratch cards. Let x be the number of Unitel scratch cards and y be the number of Bitel scratch cards.
(a)
Write down four inequalities which satisfy the above conditions. [4]
(b)
Show these inequalities on graph paper using 2 cm to represent 100 scratch cards on both axes. [4]
(c)
The dealer makes a profit of K500 on each Unitel scratch card and K150 on each Bitel scratch card sold.
(c(i))
Find the number of scratch cards of each type he must buy in order to maximize his profit. [2]
(c(ii))
Calculate the maximum profit. [2]
12.
(a)
Using a scale of 1 cm to represent 1 unit on each axis, draw x and y axes for -5 ≤ x ≤ 8 and 2 ≤ y ≤ 12. Draw and label triangle PQR whose vertices are P(5, 2), Q(1, 1) and R(3, 6). [1]
(b)
An enlargement maps triangle PQR onto triangle PAB. Given that the co-ordinates of A are (-3, 0), find
(b(i))
the centre of enlargement, [1]
(b(ii))
the scale factor, [1]
(b(iii))
the co-ordinates of the point B, [1]
(b(iv))
the ratio of the area of triangle PAB to triangle PQR. [1]
(c)
Given that C is the point (2, 5), D is the point (6, 3) and that a single transformation maps triangle PQR onto triangle CQD,
(c(i))
describe fully the single transformation, [2]
(c(ii))
write down the matrix of this transformation. [1]
(d)
Given that PQRS is a parallelogram,
(d(i))
write down the co-ordinates of S, [1]
(d(ii))
describe the transformation which maps triangle PQR onto triangle RSP. [3]