w. MECHS. I
(0 26 1)
UNIVERSITIES OF MANCHESTER LIVERPOOL
LEEDS SHEFFIELD AND BIRMINGHAM
JOINT MATRICULATION BOARD
GENERAL CERTIFICATE OF EDUCATION
PURE MATHEMATICS WITH MECHANICS (0 26), PAPER 1
Wednesday 26 June 1963 9-30—12
Negligently presented or slovenly work will be penalized.
Mathematical tables (green covers) will be provided.
Answer seven questions.
1. Three motorists A, B and C start at the same time from a town X and travel with constant speeds V1 m.p.h., V2 m.p.h., V3 m.p.h. respectively towards a town Y, distant a miles from X. A fourth motorist D starts at the same time from Y and travels with a constant speed V2 m.p.h. towards X. Show that A and D will meet a/(V1 + V2) hr. after starting.
Show that, if D meets A, B and C at times that are in arithmetic progression, then V1, V2 and V3 will be in geometric progression.
Calculate the speeds of all four motorists if the distance between X and Y is 70 miles and if D meets A, B and C 50, 60 and 70 min. respectively after leaving Y.
2. (a) Show that the term independent of x in the
expansion of is
(b) Show that, if where and x is not equal to 1, then
3. Prove that, if then
Show that the equation has two unequal roots between 0 and 2if
Show that in this case, if and are the roots of this equation, then
4. Show that the straight lines whose equations are , and form a right-angled triangle and find the coordinates of the vertices of this triangle.
Show also that if the triangle has an area of 5 sq. units then
5. Find the equation of the straight line which intersects the coordinate axes at (a, 0) and (0, b).
The straight line meets the axes of x and y in A and B respectively, while the straight line meets these axes in C and B respectively. The line joining 0, the origin, to the mid-point of CB meets AB in P, while the line drawn from 0 perpendicular to CB meets AB in Q. Find the equations of the lines OP, OQ and prove that PQ is of length units.
6. The point 0 is the centre of the base of a right circular cone C1 of given volume V cu. units and given base-radius r units. A second right circular cone C2 is drawn inside C1 with its vertex at 0 and with the edge of its base lying on the curved surface of C1. Taking the base-radius of C2 as tr units, where t can vary from 0 to 1, find an expression for the volume of C2 in terms of t and V only. Hence show that the maximum volume of C2 is cu. units.
[The volume of a right circular cone of height h units and base radius r units is cu. units.]
7. (a) Find, from first principles, the differential coefficient with respect to x of .
(b) The distance s moved in a straight line by a particle in time t is given by , where a, b and c are constants. If V is the velocity of the particle at time t, show that
8. Sketch the curve , where a is a positive number.
Find the area bounded by this curve, the line x = 4a, and the coordinate axes.
Show also that the volume of the solid of revolution generated when this area is rotated about the y-axis is cu. units.