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Examinations Council of Zambia — past paper
ECZ 2012 O-Level Paper 2 — Mathematics
12 questions · downloaded from g12titan.com, where this paper is marked free online
1.
(a)
Evaluate 2 13 - 2 14 ÷ 1 12. [2]
(b)
Solve the equation 6x + 2 = 23. [2]
(c)
Simplify x + 2x2 - 4. [2]
(d)
A sugar cane stick has eleven equal segments. Given that one third of it is spoiled by stalk borers, calculate the number of the remaining segments giving your answer correct to 2 decimal places. [2]
2.
(a)
Given that A = 23-15 and B = 23, find the
(a(i))
determinant of A, [1]
(a(ii))
inverse of A, [1]
(a(iii))
value of AB. [2]
(b)
Express 42x - 1 - 3x - 1 as a single fraction in its simplest form. [3]
(c)
Solve the inequation 4b - 3 < 6b + 4. [2]
3.
(a)
In the diagram below, A, B, C and D are points on the circumference of a circle. TCM and MDE are tangents to the circle at C and D respectively.
Given that AB = AD, angle CBD = 47° and angle BDC = 23°, calculate
(a(i))
angle BCT, [1]
(a(ii))
angle BAD, [1]
(a(iii))
angle ABD, [1]
(a(iv))
angle DMC. [2]
(b)
Solve the equation 2x2 + 5x - 8 = 0, giving your answers correct to 2 decimal places. [5]
4.
(a)
Construct triangle PQR in which PQ = 10 cm, PR = 7 cm and QR = 8 cm. [1]
(b)
Measure and write the size of angle PRQ. [1]
(c)
On your diagram, draw the locus of points within the triangle which are
(c(i))
equidistant from Q and R, [1]
(c(ii))
5 cm from R. [1]
(d)
T is a point inside the triangle PQR such that it is 5 cm from R and equidistant from Q and R. Label the point T. [2]
(e)
Another point B within the triangle PQR is nearer to Q than R and greater than or equal to 5 cm from R. Indicate clearly, by shading, the region in which B must lie. [2]
5.
(a)
A class of 41 girls takes History (H), Commerce (C) and Geography (G) as optional subjects. The Venn diagram below shows their choice distribution.
(a(i))
Calculate the value of x. [2]
(a(ii))
Find
(a(ii)(a))
n(H ∪ G), [1]
(a(ii)(b))
n(G' ∩ H'). [1]
(b)
Express 3 25 % as a decimal. [1]
(c)
The ratio of adults to children that bought tickets for a video show was 17:15 respectively.
The total number of tickets sold for the video show was 4 704.
(c(i))
How many more adults than children attended the video show? [2]
(c(ii))
If the tickets for adults were sold at K12 500 each and the tickets for children were sold at K8 500 each, calculate the total amount realised from the sell of tickets. [2]
6.
(a)
In the diagram below, OABC is a parallelogram in which OA = 2a and OC = 2b. P is a point on OA such that OP = (14)OA and Q is a point on CB such that CQ : QB = 3 : 1.
(a(i))
Express in terms of a and/or b
(a(i)(a))
OB, [1]
(a(i)(b))
OP, [1]
(a(i)(c))
QC. [1]
(a(ii))
Given that OX = hOB, express OX in terms of a, b and h. [1]
(b)
Factorise completely 2xy + x - 10y - 5. [2]
(c)
The width of a rectangle is 3 m. If its diagonal is 8.5 m long, calculate the length giving your answer correct to 2 decimal places. [2]
7.
(a)
The diagram below shows Kapenta (K), Bream (B) and Chisense (C) fishing camps on lake Manzi. B is 20 km due east of K, BC = 16 km and KC = 30.4 km.
(a(i))
Calculate
(a(i)(a))
angle KBC to the nearest degree, [5]
(a(i)(b))
the area of triangle KBC. [3]
(a(ii))
Another fishing camp Ndombe (N) is on KB produced, such that angle BNC = 90°. Calculate the distance between C and N. [2]
(b)
Mrs Kongolani obtained a seasonal loan of K5 000 000 from a bank, payable over 1 12 years at the rate of 20% per annum. How much did she pay to the bank at the end of 1 12 years? [2]
8.
In a Survey, 200 shoppers were asked how much they had spent at Gulani Super Market on a particular day. The results are shown in the table below. Amount in '000 of Kwacha0 < x ≤ 2020 < x ≤ 4040 < x ≤ 6060 < x ≤ 8080 < x ≤ 100100 < x ≤ 140Number of shoppers103248543620
(a)
Write the modal class. [1]
(b)
Estimate the mean amount spent. [3]
(c)
Copy and complete the following cumulative frequency table. Amount in '000 of Kwacha≤0≤20≤40≤60≤80≤100≤140Number of shoppers0104290 [1]
(d)
Using a scale of 2 cm to represent K20 000 on the horizontal axis and 2 cm to represent 20 shoppers on the vertical axis, draw a smooth cumulative frequency curve. [3]
(e)
Showing your method clearly, use your graph to estimate the interquartile range. [2]
(f)
Given that shoppers who spent at least K110 000 qualified for entry into a raffle draw, estimate the number of those who qualified. [2]
9.
(a)
The diagram below shows a wire model of the earth. The circle of latitude in the north is 50° N and the circle of latitude in the south is 60° S. A and C are on longitude 55° W while B and D are on longitude 50° E.
(Take π = 3.142 and R = 3437 nm)
(a(i))
Write the positions, using longitudes and latitudes, of the points A and D. [2]
(a(ii))
Calculate the difference in longitudes between A and B. [1]
(a(iii))
Given that the time at town D is 09 20 hours, what would be the time at town C? [1]
(a(iv))
Calculate the distance BD along the longitude 50° E in nautical miles. [2]
(b)
A cylindrical geyser of radius 21 cm and length 50 cm is placed with its curved surface on a horizontal ground. It is filled partially with water and the segment ABY in the diagram shows the cross section of water in the geyser. O is the centre of the circular end of the geyser, Y is vertically below O and angle AOB = 120°. (Take π = 227).
Calculate
(b(i))
the curved surface area of the geyser, [2]
(b(ii))
the volume of water in the geyser correct to the nearest whole number. [4]
10.
(a)
The variables x and y are connected by the equation y = 42 - x - x2. The table below shows some corresponding values of x and y. x-8-6-4-2-1035yq12304042423012
(a(i))
Calculate the value of q. [1]
(a(ii))
Using a scale of 1 cm to represent 1 unit on the x-axis for -8 ≤ x ≤ 6 and a scale of 2 cm to represent 10 units on the y-axis for -20 ≤ y ≤ 50, draw the graph of y = 42 - x - x2. [3]
(a(iii))
By drawing the line 7y - 20x = 140 on the same graph, solve the equation 42 - x - x2 = 20x7 + 20. [3]
(a(iv))
Estimate the area under the curve between x = 0, y = 0 and x = 4. [2]
(b)
Given that x = 2 and y = -3, find the value of x2 - 6xy + 3y - 3x. [3]
11.
(a)
The graph below shows three inequalities that satisfy Kashita's intentions to purchase Science and Mathematics text books.
(a(i))
Given that x represents the number of Science text books and y represents Mathematics text books, write the three inequalities that represent the unshaded region R. [4]
(a(ii))
Given that a Science textbook costs K25 000 and a Mathematics textbook K10 000, find the largest number of Science and Mathematics textbooks that can be bought. Hence calculate the total cost of the textbooks. [3]
(b)
A box has 7 identical sweets. 3 of these are green and the rest are red. Kamwanga picks one sweet at random and eats it. After sometime, he picks another one and eats it.
(b(i))
Construct a tree diagram to illustrate the outcomes of the two sweets taken. [3]
(b(ii))
Calculate the probability that the first sweet was red and the second was green. [2]
12.
Using a scale of 2 cm to represent 10 units on each axis, draw x and y axes for -30 ≤ x ≤ 40 and -30 ≤ y ≤ 30.
(a)
Triangle ABC has vertices A(-30, 10), B(-30, 20) and C(-10, 20). Triangle A1B1C1 has vertices A1(10, -30), B1(20, -30) and C1(20, -10).
(a(i))
Draw and label triangle ABC and triangle A1B1C1. [1]
(a(ii))
Describe fully a single transformation that maps triangle ABC onto triangle A1B1C1. [2]
(b)
An enlargement centre (10, -30) and scale factor 2 maps triangle A1B1C1 onto triangle A2B2C2. Draw and label triangle A2B2C2. [1]
(c)
A reflection in the line y = 0 maps triangle ABC onto triangle A3B3C3. Draw and label triangle A3B3C3. [1]
(d(i))
Triangle A4B4C4 is the image of triangle A1B1C1 under a translation. Given that A4 is the point (10, 10), write the column vector representing this translation. [2]
(d(ii))
draw and label triangle A4B4C4. [1]
(e)
The matrix 11201 maps triangle A4B4C4 onto triangle A5B5C5.
(e(i))
Draw and label triangle A5B5C5. [2]
(e(ii))
Describe fully the single transformation represented by this matrix. [2]