(O 25, B,I)
UNIVERSITIES OF MANCHESTER LIVERPOOL
LEEDS SHEFFIELD AND BIRMINGHAM
JOINT MATRICULATION BOARD
GENERAL CERTIFICATE OF EDUCATION
MATHEMATICS (O 25)
SYLLABUS B, PAPER I
Monday 17 June 1963 9-30—12
Negligently presented or slovenly work will be penalized.
Mathematical tables and one sheet of graph paper will be provided.
Answer all questions in Section A and any three questions from Section B.
In each question necessary details of working, including rough work, must be shown with the answer
A 1. (a) To build a motorway costs half a million pounds per mile. Find the cost per yard, in pounds, correct to the nearest pound.
(b) Solve the equation
(c) In a triangle ABC, AB = AC = 20 cm.
BC = 32 cm. Calculate C.
A 2. (a) Find the angle whose sine is
(b) Calculate the side of a regular polygon of 40 sides which is inscribed in a circle of radius 8 cm.
(c) Find x, where
A 3. (a) The angle A is obtuse and
Without using tables, find tan A.
(c) A tangent from a point P to a circle with centre O touches the circle at T. The line joining O to P cuts the circle at S and TPS = 20°. Calculate the obtuse angle of triangle TPS.
A 4. (a) Through how many degrees does the hour hand of a clock rotate in x minutes?
(b) Four angles of a pentagon are 30°,. 88°, 112° and 145°. Find the fifth angle.
(c) If , and y = 0 when x = -1, find y in terms of x.
A 5. (a) An isosceles right-angled triangle is equal in area to a circle whose radius is in. Calculate the length, of one of the equal sides of the triangle, taking as .
(b) A man is allowed two-ninths of a sum of money tax-free and pays tax at 7s. 9d. in the £ on the remainder. Find the sum if the tax payable is £217.
A 6., (a) . Calculate the largest angle of a triangle with sides 4, 5 and 6 cm.
. (b) A quantity h is equal to Find the. percentage increase in h when a is increased by 25 per cent, b by 8 per cent and c by 20 per cent..
Answer three questions from this section.
B 7. In a triangle ABC, AB =8 cm., A = 130° and B = 40°. Calculate BC and the radius of’the circumcircle.
The inscribed circle touches AB at D. Write, down the value of each of AD and DB in terms of the radius of this circle and hence calculate the radius.
B 8. ABCD is a parallelogram and X, Y are the mid-points of AD, BC respectively. AC meets BX at P and XY at Q.
(i) Prove that P is a point of trisection of AC.
(ii) Find the areas of QYC, ABP and PBYQ as fractions of the area of ABC.
B 9. Draw the graph of from x = 1 to
x= 5, taking 1 in. as the unit of both x and y.
From your graph estimate two roots of the equation . Calculate the gradient of the curve at the point where x =2, and hence draw the tangent to the curve at this point.
B 10. A factory makes cylindrical pencils 9 in. long and of radius in. The graphite core is cylindrical, of radius in., and it is surrounded by wood. During manufacture there is a wastage of 12 per. cent of graphite and 20 per cent of wood. Taking as , find how many pencils may be made from 50 cu. ft. of graphite and calculate the volume of wood required.
B 11. A 100 h.p. racing car travels at a steady speed of 40 metres per sec. in a race of 300 miles. Taking 8 km. as 5 miles, calculate the time in hours and minutes to complete the course.
Calculate the time taken for a 200 h.p. car to complete the course, assuming that the speed is proportional to the square root of the horse-power. Give your answer to the nearest minute.