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Examinations Council of Zambia — past paper
ECZ 2014 O-Level Paper 2 — Mathematics
12 questions · downloaded from g12titan.com, where this paper is marked free online
1.
(a)
Evaluate 1 23 - 1 23 × 14. [2]
(b)
Solve the equation 5x - 8 - 3(x + 1) = -7. [2]
(c)
Simplify 5(2y - 3) - 2(5 - 2y). [2]
(d)
Mataya's digital camera stores images on a memory card of capacity 512 units. Given that two thirds of the memory card is used, calculate the number of unused units, giving your answer correct to 2 decimal places. [2]
2.
(a)
Given that A = 5210 and B = -100-1, find
(a(i))
the inverse of matrix A, [2]
(a(ii))
3A - B, [2]
(a(iii))
AB. [2]
(b)
Express 52y - 1 - 63y - 1 as a single fraction in its simplest form. [3]
(c)
Solve the inequation 9t - 4 < 12t - 10. [2]
3.
(a)
In the diagram below, P, Q, R and S are points on the circumference of a circle with centre O. AQB is a tangent to the circle at Q.
Given that PS = PQ, QR = RS and angle PSQ = 70°, calculate
(a(i))
angle QPS, [1]
(a(ii))
angle QRS, [1]
(a(iii))
angle AQP, [1]
(a(iv))
angle SOQ. [1]
(b)
Solve the equation 1 - 2m - 5m2 = 0, giving your answers correct to 2 decimal places. [5]
4.
(a(i))
Construct triangle ABC in which AC = 11.5 cm, BC = 7 cm and AB = 9.5 cm. [1]
(a(ii))
Measure and write the size of angle ABC. [1]
(b)
On the same diagram and within triangle ABC, construct the locus of points
(b(i))
equidistant from AC and BC, [1]
(b(ii))
7.5 cm from A, [1]
(b(iii))
2.5 cm from AC. [2]
(c)
Given that N is a point within triangle ABC such that it is greater than or equal to 7.5 cm from A, nearer to AC than BC and less than or equal to 2.5 cm from AC, shade clearly the region in which N must lie. [2]
5.
(a)
Tokhozani Sports Club offers Squash (S), Badminton (B) and Tennis (T). The Venn diagram below shows choices of the 73 members of the club.
(a(i))
Calculate the value of y. [2]
(a(ii))
Find the number of members who played Squash or Tennis but not Badminton. [1]
(a(iii))
How many members played two different sports only? [1]
(a(iv))
Find the number of members who played one sport only. [1]
(b)
To write an examination, a candidate is required to pay an entry fee of K38.50 and K25.00 for each subject.
(b(i))
If Mary intends to write 4 subjects, how much is she expected to pay? [1]
(b(ii))
Given that Martin paid a total of K188.50, how many subjects did he write? [2]
6.
(a)
In the diagram below, OP = 2p and OQ = 2q. X is a point on PQ such that PXPQ = 23.
(a(i))
Express in terms of p and/or q
(a(i)(a))
PQ, [1]
(a(i)(b))
QX. [1]
(a(ii))
Given also that OY = hOX, express OY in terms of p, q and h. [2]
(b)
The diagonals of a rhombus are 20 cm and 16 cm long. Calculate the length of the side of the rhombus, giving your answer correct to 2 decimal places. [2]
(c)
Factorise completely 2xy + 8x - 3y - 12. [2]
7.
(a)
Positions of Kabwela (K), Chapa (C) and Muzi (M) are as shown in the diagram below. Chapa is 78 km from Kabwela and Muzi is 123 km from Kabwela.
(a(i))
Given that angle CKM = 81°, calculate the area of triangle CMK. [3]
(a(ii))
Calculate the distance CM. [5]
(a(iii))
A company has been contracted to construct a road from Muzi (M) to Chapa (C). Find the total cost of constructing this road, if the company charges K215 000.00 per kilometre. [2]
(b)
Mrs Mwasona invested K800 000.00 for 4 years at 5% per annum simple interest with a certain bank. How much money will be in her account at the end of the period? [2]
8.
A bag containing 250 potatoes was supplied to a supermarket. Each of the potatoes was weighed to the nearest gram and the results were as shown in the table below. Mass in g50 < x ≤ 100100 < x ≤ 150150 < x ≤ 200200 < x ≤ 250250 < x ≤ 300300 < x ≤ 350Number of potatoes53065105405
(a)
Copy and complete the cumulative frequency table below. Mass in g≤50≤100≤150≤200≤250≤300≤350Number of potatoes0535250 [1]
(b)
Using a scale of 2 cm to represent 50 grams on the x-axis and 4 cm to represent 50 potatoes on the y-axis, draw a smooth cumulative frequency curve. [3]
(c)
Showing your method clearly, use your graph to estimate
(c(i))
the median mass, [1]
(c(ii))
the interquartile range, [2]
(c(iii))
the 60th percentile. [2]
(d)
Potatoes weighing between 180 g and 290 g are graded 'B'. Use your graph to estimate the number of grade 'B' potatoes. [3]
9.
(a)
A pattern on a chitenge material consists of three circles as shown in the diagram below. AC is the diameter of the bigger circle.
Given that the diameter of each of the small circles is 20 cm and taking π = 3.142, calculate the
(a(i))
total perimeter of the two small circles, [3]
(a(ii))
area of the shaded part. [3]
(b)
The diagram below shows a model of the earth. The points C and D are on the same longitude. The latitudes of C and D are 45° N and 55° S respectively.
(Take π = 3.142 and R = 3 437 nm)
(b(i))
Write the position of the point C. [1]
(b(ii))
Calculate the difference in latitude between C and D. [1]
(b(iii))
Find the distance CD in nautical miles. [2]
(b(iv))
Calculate the circumference of the latitude 45° N in nautical miles. [2]
10.
(a)
The graphs of y = 7 - 3x - x2 and y = x + 4 are shown in the diagram below.
(a(i))
Use the graph above to solve the following equations;
(a(i)(a))
9 - 3x - x2 = 0, [3]
(a(i)(b))
7 - 3x - x2 = x + 4. [2]
(a(ii))
Estimate the area bounded by the curve, x = -3, x = -1 and y = x + 4. [3]
(a(iii))
Find the maximum value of y = 7 - 3x - x2. [1]
(b)
In a certain school, the ratio of boys to girls is 3 : 5. Given that there are 600 girls, calculate the total number of pupils in the school. [3]
11.
(a)
The graph below shows four inequalities that satisfy the intentions of a businessman to purchase two types of soft drinks, A and B.
(a(i))
Given that x represents the number of type A soft drinks and y the number of type B, write the four inequalities that represent the unshaded region R. [4]
(a(ii))
If the profit on type A soft drinks is K2.00 per bottle and profit on type B is K3.00 per bottle, find the number of each type of soft drinks he must buy in order to make maximum profit. [1]
(a(iii))
Find this maximum profit. [2]
(b)
Two pupils are to represent a school at a Human Rights Conference. If the two are chosen at random from a group of 8 girls and 6 boys, calculate the probability that the two pupils picked
(b(i))
are both girls, [2]
(b(ii))
at least one is a boy. [3]
12.
Using a scale of 2 cm to represent 1 unit on each axis, draw x and y axes for -3 ≤ x ≤ 7 and -3 ≤ y ≤ 7.
(a)
Triangle ABC has vertices A(3, 6), B(3, 7) and C(1, 7). Triangle A1B1C1 has vertices A1(-1, 4), B1(-2, 4) and C1(-2, 2).
(a(i))
Draw and label triangle ABC and triangle A1B1C1. [2]
(a(ii))
Describe fully a single transformation which maps triangle ABC onto triangle A1B1C1. [2]
(b)
The matrix 0110 maps triangle ABC onto triangle A2B2C2.
(b(i))
Find the coordinates of triangle A2B2C2. [2]
(b(ii))
Draw and label triangle A2B2C2. [1]
(c)
Triangle A1B1C1 can be mapped onto triangle A3B3C3 by a translation. Given that A3 is the point (3, -1),
(c(i))
find the column vector representing this translation, [1]
(c(ii))
draw and label triangle A3B3C3. [1]
(d)
Triangle A4B4C4 has vertices A4(0, 3), B4(0, 1) and C4(4, 1).
(d(i))
Draw and label triangle A4B4C4. [1]
(d(ii))
Describe fully the transformation which maps triangle ABC onto triangle A4B4C4. [2]