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361
UNIVERSITY OF LONDON
SCHOOL EXAMINATIONS BOARD
General Certificate of Education Examination
JUNE 1988 ORDINARY LEVEL
Subject TitleMathematicsSyllabusSyllabus BPaper No./TitlePaper 2Subject Code No361
Two and a half hours
Answer ALL questions in Section A. In Section B, full marks may be obtained for answers to SIX questions.
(If you attempt more than six questions in Section B, only the best six answers will be taken into account.)
All necessary working must be shown.
You are reminded of the necessity for good English and orderly presentation in your answers.
In calculations you are advised to show all the steps in your working, giving your answer at each stage.
University of London 1988
Section A
Answer ALL questions in this section.
1. A circular fish pond has a radius of 27 m. A sector, of angle 75(, is used for fish breeding.
Calculate, in m2 to one decimal place,
(a) the area of the pond,
(b) the area of the breeding sector.
[The area of a circle is (r 2.]
(4 marks)
2. (i) The interior angles of a quadrilateral are (3x 12)(, (2x + 42)(, (x + 88)( and (2x 30)(. Form an equation in x and solve it.
Hence find the angles of the quadrilateral.
A
20(
72(
B C D
Fig. 1
(ii) In Fig. 1, AC = CD and BCD is a straight line. Given that (ADC = 72( and (BAC = 20(, calculate the size of (ACB and (ABC.
(6 marks)
3. Given that X = EMBED Equation.3 and Y = EMBED Equation.3 ,
find (a) X + Y, (b) Y2, (c) the inverse of X.
[The inverse of matrix is EMBED Equation.3 is EMBED Equation.3 .]
(4 marks)
4. (i) Calculate the probability that when a die is thrown, the number obtained will not be divisible by 3.
(ii) A coin is tossed and a die is thrown. Calculate the probability of obtaining a tail on the coin and a 5 on the die.
(4 marks)
5. C
A
a
O
b B D
Fig. 2
In Fig. 2, OB is produced to D so that BD = 2OB and OA is produced to C so that AC = 2OA.
Given that EMBED Equation.3 = a and EMBED Equation.3 = b,
(a) express in terms of a and b
(i) EMBED Equation.3 , (ii) EMBED Equation.3 , (iii) EMBED Equation.3 , (iv) EMBED Equation.3 ,
(b) state two geometrical relationships between AB and CD.
(6 marks)
6. The operation * is defined for the rational numbers x and y by
x * y = (x y)2.
(a) Calculate 9 * 5.
(b) Calculate 7 * (5 * 3).
(c) Solve for x, the equation, x * 2 = 36.
(4 marks)
Section B
Answer SIX questions in this section.
7. A survey was made of the form of heating in the 540 homes in a small town. The results of the survey were as follows:
Gas 180
Oil 150
Electricity 66
Solid fuel 120
Underfloor 24
(a) On graph paper, using 2 cm to represent each form of heating and 1 cm to represent 20 homes, illustrate this information by means of a bar chart.
(b) Draw a circle of radius 6 cm. Use this circle to illustrate the given information on a pie chart. Label, and show clearly the angle of, each sector.
(c) Calculate the percentage of homes in this town which have underfloor heating.
(12 marks)
8. Given that f(x) = 3x 2 and g(x) = x2,
(a) find f(1) and g((3).
(b) Express f(1 (x) and ff(x) in terms of x.
(c) Solve, for x, the equation f(x) = 10.
(d) Express gf(x) in terms of x, simplifying your answer.
(e) Solve, for x, the equation gf(x) g(x) = 25, giving your answers to 2 decimal places.
[The solutions of ax2 + bx + c = 0 are x = EMBED Equation.3 .]
(12 marks)
9. (a) On graph paper, taking (8 ( x ( 10, (8 ( y ( 10 and using a scale of 1 cm to represent 1 unit on each axis, plot the points A (1, 4), B (5, 5) and C (6, 7). Draw and label the ABC.
ABC is rotated through 180( about the point (1, 4) to give A1B1C1.
(b) Draw and label the A1B1C1 and write down the coordinates of A1, B1 and C1.
A1B1C1 is reflected in the xaxis to give A2B2C2.
(c) Draw and label the A2B2C2.
A2B2C2 is now reflected in the line y = x to give A3B3C3.
(d) Draw and label the A3B3C3.
ABC can be mapped on to A3B3C3 by rotation.
(e) By construction, determine the centre of rotation and write down the angle of this rotation.
(12 marks)
10. A ball is thrown vertically downwards at 2 m/s from the top of a high building. Given that the distance, s metres, travelled by the ball in the first t seconds after being thrown is
s = 2t + 5t2,
(a) calculate
(i) the distance, in metres, travelled by the ball in the first 3 seconds,
(ii) the distance, in metres, travelled by the ball in the third second,
(iii) the time, in seconds, taken to travel the first 7 metres.
(b) (i) Obtain an expression, in terms of t, for the speed, in m/s, at which the ball is travelling tseconds after being thrown.
(ii) Calculate the speed, in m/s, of the ball 2 EMBED Equation.3 seconds after being thrown.
(iii) Find the time, in seconds, taken to reach a speed of 17 m/s.
(c) Find the acceleration, in m/s2, of the ball after t seconds.
(12 marks)
11. A coarse mix of concrete is made by using 2 parts of water with 3 parts of sand, 4 parts of chippings and 2 parts of cement by volume. Assuming that there is no loss in volume on mixing, calculate
(a) the number of spadefulls of chippings needed to make 22 spadefulls of concrete,
(b) the amount of concrete, in cm3 to 2 significant figures, which would be obtained by using 2850 cm3 of clippings.
A rectangular base is to be made for a garage using this coarse mix of concrete. The base measures 6 m by 3 EMBED Equation.3 m and it is to be 15 cm thick. Calculate
(c) the volume of the base in cm3,
(d) the minimum quantity of sand, in cm3 to 2 significant figures, which will be needed to make this base.
The cost of a bag of cement is 2.90 and it contains 40 000 cm3; the cost of a sack of sand is 1.50 and it contains 80 000 cm3; the cost of a barrowfull of chippings is 1.25 and it contains 150 000 cm3. Assuming that cement, sand and chippings are only for sale in multiples of these quantities,
(e) find the total cost of buying sufficient materials from which to make the base.
(12 marks)
12.
B C
D
Fig. 3
Sixty pupils in a school answered questions about their pets. 16 pupils said that they had only a cat, 13 said that they had only a dog and 5 had only a budgerigar.
8 pupils said that they had a cat, a dog and a budgerigar but 4 pupils said that they had no pets.
10 pupils said that they had a budgerigar and a cat.
The number of pupils who had a cat and a dog, but not a budgerigar, was three times the number who had a budgerigar and a dog, but not a cat.
(a) Copy the Venn diagram shown in Fig. 3, where B denotes the set of pupils having a budgerigar, C the set of pupils having a cat and D the set of pupils having a dog. Denoting the number of pupils who have a budgerigar and a dog, but not a cat, by x, show the above information on your Venn diagram.
(b) Form an equation in x and solve it to find x.
(c) Calculate n(C ( D).
(d) Calculate n(C ( D)(.
(e) Calculate the number of pupils who did not have a cat.
(12 marks)
13.
N
X
C
A B
Fig. 4
In Fig. 4, AB is a diameter of the circle ABC and (ABC = 70(. The tangent to the circle at C meets, at N, the line drawn through A at right angles to this tangent. The line AN cuts the circle atX.
Giving your reasons
(a) find the size of (CXA,
(b) state the sizes of the angles ACB and BAC,
(c) prove that CA bisects (NAB,
(d) state why ABC and ACN are similar triangles,
(e) prove that EMBED Equation.3 .
(12 marks)
14. (a) Given that y = 4 + 15x 2x2, copy and complete the table:
x(1012345678y(134292211(4
(b) On graph paper, using a scale of 2 cm for one unit on the xaxis, and 2 cm for 5 units on the yaxis, plot the points from your completed table and join them to form a smooth curve.
(c) On the same axes, draw the graph of the line whose equation is y = 2x + 3.
(d) Use your graphs to solve the equation
2x2 13x 1 = 0,
giving your answers to one decimal place.
(e) From your graph find, to one decimal place, the range of values of x for which
14 + 15x 2x2 > 8.
(12 marks)
15. From an observation post, P, on the top of a vertical cliff 65 m above sea level, a small boat is seen to be at the point B1 due east of P. The angle of depression of the boat, when at the point B1, is 28( from the observation post P.
(a) Calculate the distance, in metres to the nearest metre, of the boat from the foot of the cliff when it is at B1.
During the next five minutes the boat travels eastwards, directly away from the observer at P, and it travels 800 m to a new position B2.
(b) Calculate the speed of the boat in km/h.
(c) Calculate, when the boat is at B2,
(i) the distance, in metres to the nearest metre, of the boat from the foot of the cliff,
(ii) the angle of depression, to the nearest degree, of the boat from P.
The boat now changes course and travels 300 m due south to a new position B3, without changing speed.
(d) Calculate the distance, in metres to the nearest metre, of B3 from the foot of the cliff immediately below P.
(12 marks)
PAGE
PAGE 3
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