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UNIVERSITY OF LONDON
GENERAL CERTIFICATE OF EDUCATION
EXAMINATION
JANUARY 1968
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.
Credit will be given for the orderly presentation of material; candidates who neglect this essential will be penalised.
All necessary working must be shown.
Section A
1. (i) Multiply 2a + 3b by 3a 2b.
(ii) A motorist travels for 20 miles at an average speed of 30 m.p.h. and for a further 30 miles at an average speed of 20 m.p.h. Calculate his average speed for the whole journey of 50miles, giving your answer correct to the nearest m.p.h.
(iii) A quadrilateral ABCD is inscribed in a circle, the tangent at A to the circle making an angle of 42( with AB. The angles BDC and CAD are 15( and 67( respectively. Calculate the angle ABD.
TE&S 65/1319 11/2/100/17800
( 1968 University of London
2. (i) Find the value of t (1 t + t 2) when t = 1.
(ii) The size of an interior angle of a regular polygon is 5 times that of an exterior angle. Calculate the number of sides of the polygon.
(iii) In a triangle ABC the angle C = 90( and sin B = EMBED Equation.3 .
Calculate the value of sin A.
If the length of AC is 4(8 cm, calculate the length of AB.
3. (i) If R = EMBED Equation.3 , express R1 in terms of R and R2.
(ii) In a triangle PQR the sides PQ and PR are equal and S is a point on PR such that QS=QR. If PQ = 3QR show that PR = 9RS.
4. (i) Express as a single fraction in its simplest form
EMBED Equation.3 EMBED Equation.3 .
(ii) The lengths of the diagonals of a rhombus ABCD are AC = 6(4 cm and 5(6 cm. Calculate
(a) the angle BCD,
(b) the area of the rhombus,
(a) the length of AB.
5. (i) From a point T a tangent is drawn to a circle to touch the circle at P. A secant from T intersects the circle at Q and S. If TQ = x cm, QS = x + 9 cm and TP = 2x cm, form an equation for x and hence calculate the length of TP.
(ii) A cleaner is paid 4s.6d. per hour for normal work up to 35 hours per week and 6s.9d. per hour for overtime in excess of 35 hours. In one week her total pay was 911s.3d. Calculate the number of hours of overtime she worked.
6. In Fig. 1, AR = DR and AB = DC. The lines QS and AD are parallel. Prove that
(a) QR = SR,
(b) QD = SA,
(c) the triangles ASG and DQB are congruent.
Fig. 1
Section B
Answer FOUR questions in the section.
7. The average yearly rainfall for a certain area is 24 in. During a heavy storm in that area the rainfall was 0(6 in. Calculate the percentage of the yearly rainfall which fell during the storm.
Taking 1 cu ft of water as 6(23 gallons and the weight of 1 gallon of water as 10 lb, calculate also, giving your answer to the nearest ton, the weight of water which fell during the storm on a schools playing fields of area 18 acres. [1 acre = 4,840 sq yd.]
8. Three points, A, B and C on a circle with centre O are such that AB = BC with B on the minor arc AC. The tangents to the circle at A and C intersect at T. Prove that
(a) AB bisects the angle CAT,
(b) O lies on the circumcircle of the triangle ATC,
(c) if B is the centre of the circumcircle of the triangle ATC then the angle ATC = 60(.
9. Draw the graph of y = 5x x2 for values of x from 0 to 5 inclusive, taking a scale of 1 in. to represent 1 unit on each axis.
With the same axes and the same scale draw the graph y = EMBED Equation.3 for values of x from 1 to 5 inclusive.
Use your graph to find
(a) the range of values of x for which 5x x2 is greater than EMBED Equation.3 ,
(b) the minimum value of 5x x2,
(c) the gradient of y = EMBED Equation.3 at the point where x = 3.
10. From a boat at a point X the bearing of a lighthouse at a point H is 112( (S 68( E). The boat sails due North to a point Y, 10 miles from the lighthouse. The bearing of the lighthouse fromY is 150( (S 30( E). The boat then sails North-East from Y for 8 miles to a point Z.
Calculate
(a) the distance XY,
(b) the distance ZH,
(c) the bearing of Z from X.
11. In Fig. 2, the diagonals AC and BD of the quadrilateral ABCD intersect at R, and AR = RC. The lines PR and DC are parallel and QR is parallel to BC.
Prove that
(a) PQ is parallel to DB,
(b) the area of the triangle PCQ is 3 times the area of the triangle PQR,
(c) the area of the triangle PCQ is equal to the area of the quadrilateral DPQB.
Fig. 2
12. (i) Show by calculation that the line y = 5 intersects the curve y = x2 + 1 at the points (2, 5) and(2, 5). Calculate the area enclosed between the line y = 5 and the curve y = x2 + 1.
(ii) The area bounded by the x-axis, the lines x = 1 and x = 2 and the curve y = x2 + 2 is rotated about the x-axis. Calculate the volume of the solid of revolution so formed. (Leave ( as a factor of your answer.)
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