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41
UNIVERSITY OF LONDON
GENERAL CERTIFICATE OF EDUCATION
EXAMINATION
SUMMER 1964
Ordinary Level
PURE MATHEMATICS I
Syllabus B
Two and a half hours
Answer all questions in Section A and any FOUR questions in Section B.
All necessary working must be shown.
Credit will be given for the orderly presentation of material; candidates who neglect this essential will be penalized.
Turn Over
Section A
1. (i) A car averages 32 miles per gallon using petrol which costs 4s. 7d. per gallon. Calculate the cost of the petrol for a journey of 864 miles.
(ii) Factorize 2x2 11xy 6y2.
(iii) If a = 4 and b = (1, calculate the value of 3a2 2ab 5b2.
2. Solve the equations
(a) 3x 2y = 8
y = 3x 7,
(b) EMBED Equation.3 .
3. Use tables to evaluate
(a) 0(3216 ( 41(03 0(741,
(b) EMBED Equation.3
4. The tangent at P to the circumcircle of a triangle PQR cuts RQ produced at A. If PQ = QA and the angle A = 37(, calculate the angle RPQ. If, also, QR = 3 in., calculate the length of PR.
5. (i) On January 1st 1960 the total amount of money invested in a Building Society was 20 per cent more than the corresponding amount on January 1st 1959. On January 1st 1961 the amount was 15 per cent less than that on January 1st 1960. Express the total amount invested at January 1st 1961 as a percentage of the amount invested at January 1st 1959.
(ii) Solve the equation 4x2 + 7x + 1 = 0, giving the roots correct to two places of decimals.
6. (i) If a = EMBED Equation.3 , express t in terms of a.
(ii) Without using tables, write down the value of EMBED Equation.3 .
(iii) Calculate the area of a triangle ABC in which AB = 1(6 in., BC = 2 in. and the angleB=64(.
Section B
Answer any FOUR questions in this section.
7. (i) In a parallelogram ABCD, the sides AB and AD are of length 5 in. and 8 in. respectively, and the angle DAB = 32(. Calculate the length of the longer diagonal.
(ii) An aeroplane at the position P (42( N, 58( W) flies due South to Q (24( S, 58( W) and then due East along the parallel of latitude to R (24( S, 22( E). Ignoring the altitude at which the aeroplane flies, calculate the total distance flown.
(Take ( as 3(142 and the earth as a sphere of radius 3,960 miles.)
8. The diagonal AB of a cyclic quadrilateral XAYB bisects the angle XAY and is a diameter of the circle through X, A, Y and B. The perpendicular BN to the tangent at Y to the circle meets this tangent at N. Prove that
(a) BN.AB=BX 2,
(b) BY bisects the angle XYN.
9. Measurements u, f and p taken in a certain experiment are related by the formula p = EMBED Equation.3 . If pis kept constant while u and f are varied, it is found that the value of f when u = 24 is 6 more than the value obtained when u = 8. Calculate the value of f when u = 8, given that f is positive for every value of u.
10. (i) The volume of a cone is given by the formula V = EMBED Equation.3 , where h is the height of the cone and r is the radius of the base. If h = 6 r, calculate the value of r for which the volume is a maximum.
(ii) Calculate the area between the curve y = 5x x3, the ordinates x = 1 and x = 2 and the xaxis.
11. A point A is 12 miles due West of a point B. A bird takes 45 minutes to fly in a straight line from B to A in a wind the velocity of which is 15 m.p.h. from 320( (N 40( W). Calculate
(a) the speed, taken to be constant, at which the bird is moving relative to the ground,
(b) the birds airspeed.
12. If a Channel tunnel could be constructed at a cost of 438,000,000 such that every day of the year (365 days) 120 trains could run through it each way, each train carrying on an average 250 tons of freight, calculate the cost per ton which should be charged to give a return of 5 per cent per annum on the cost of construction.
If the rate charged were 5 per ton, calculate how many years would elapse before the freight charges had cleared the construction costs.
PAGE
PAGE 4
TE&S 62/1603 9/2/2125/35300
1964 University of London
TE&S 62/1603 9/2/2125/35300
1964 University of London
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