*4. A car weighing 1 ton travels, with engine switched off, down a hill of 1 in 20 (i.e. sin-1 1/20) at a uniform speed of 5 m.p.h. When the car reaches level ground, the engine is switched on and works at 8 H.P. Calculate the initial acceleration in ft. per sec.2, correct to one decimal place.

5. The axles of a truck are at a distance a apart and the centre of gravity of the truck is at a distance b from the ground and equidistant from the axles. Show that, with only the lower pair of wheels locked, the truck will slip on an inclined plane if

the inclination is greater than a, where tan α =

μα

u

2 (a-μb) and μ is

the coefficient of friction between the wheels and the incline. Calculate the corresponding greatest incline on which the truck will rest with only the upper pair of wheels locked.

6. Show that the centre of gravity of a uniform semicircular plate of radius r is at a distance 4r/(3) from the bounding diameter.

A uniform plate weighing w per unit area, is in the shape of an equilateral triangle OAB of side 2r, with a semicircle described on AB and outside the triangle. The plate is suspended freely from O and a horizontal force X, in the plane of the plate, acts at B. Calculate X when the plate is in equilibrium with OA vertical,

7. A small ring of mass m can slide on a smooth circular wire, with centre O and of radius a. The wire is fixed in a vertical plane and the ring is held at A where OA makes an angle of 60° with the upward vertical through O. The ring is now given a velocity u in a downward direction tangential to the wire. Calculate u, if the reaction between the wire and the ring in the lowest position is eight times the weight of the ring.

In this case, calculate

(i) the reaction between the ring and the wire when the ring is at the point diametrically opposite to A;

(ii) the impulse on the ring which would bring it to rest at the lowest point.

8. The diagram shows a freely jointed framework of seven light rods in a vertical plane. AB = BC; AB is vertical and there is a load of 4 cwt. at C. The joint A is hinged at a fixed point and X is a horizontal force to maintain equilibrium. Find, graphically or otherwise, the stresses in the rods BA, BD and BE, distinguishing between tension and compression.

9. On a plane of inclination α, AOB is a line of greatest slope. A particle is projected from O with velocity V so that motion takes place in a vertical plane through AOB. Calculate the greatest range up the plane.

If OA is the greatest range down the plane and OB is the greatest range up the plane, show that

1 1 1

+
OA OB h'

where h is the greatest height which would be reached by a particle projected vertically upward with velocity V.

10. A particle of mass m is attached at B to two light elastic strings AB and BC each of natural length a and modulus of elasticity λ. The ends A and C are fixed so that AC is vertical and of length 3a, with A above C. If the particle is released

from rest at the middle-point of AC, show that it oscillates in simple harmonic motion if >>2mg.

In this case, find the amplitude and the period of the motion, and show that the greatest tension in AB is mg +1λ.

11. A small ring of mass m, which can move on a smooth vertical wire, is attached to one end of a light inextensible string. The string passes over a small smooth pulley fixed at A and supports a particle of mass 2m hanging freely. The ring is released from rest at B, where AB is the perpendicular from A to the wire and AB=a. Show that, when the ring has descended a distance x to P,

where AP is y.

x2+2ỷ2 = 2g (x − 2y+2a),

Calculate the velocity of the ring after it has fallen a distance a, and show that it is instantaneously at rest after falling a distance 4a/3.

12. Show that the radius of gyration of a uniform rod of length l about an axis through the centre and perpendicular to the rod is 1/(2/3).

A uniform rod AB of mass 2m and of length 4a can turn freely in a vertical plane about a fixed horizontal axis perpendicular to the plane and through a point 0 of the rod where OA=3a. A particle of mass m is attached at A and an equal particle at B. If the body is released when the rod is horizontal, show that the rod will pass through the vertical position with an angular speed of [6g/(11a)]*.

Find the period when the body makes small oscillations as a compound pendulum.

Candidates may attempt as many questions as they please, but marks will be assessed on the eight questions best answered.

87639 O 62- -16

Begin each answer on a fresh sheet of paper and arrange your answers in numerical order.

Mathematical tables and squared paper are provided.

1. The real numbers a, b, c, f, g, h satisfy the three conditions: b>0; bc-f2>0; abc+2fgh-af2 — bg2 — ch2 > 0.

Prove that

ax2+by2+cz2 + 2fyz+2gzx+2hxy >0

for all real values of x, y, z except x=y=z=0, and hence prove that

a>0, c>0, ac-g2>0 and ab-h2>0.

2. (i) Eliminate x from the equations

ax2+ bx+c=0,) a'x2+b'x+c'=0.)

(ii) Given that x+B+y=2, a2 +ß2+y2=14 and

(n − 1) ("C1) + (n − 2) ("C2) + ... + 2 ("Cn-2)+"Cn-1

(ii) By considering (1—et)n+1, or otherwise, prove that

4. (i) Show that for any complex numbers 21, 22 and any rea.

number a

| Az1 − 22 |2 + | 21 + az。 |2 = (1+a2) (|z1|2+|22|2).

(ii) If x2+ y2=1, z=x+iy and u+iv=z2+z+1, show that

(u2 - 2u+v2)2=u2+v2.

5. If u u(x) satisfies the differential equation

find the differential equation of the second order satisfied by

dmu

y= √xm

where m is a positive integer.

dmu

If z = (1-x2) m show that z satisfies the equation

dxm'

6. A family of curves is given by y=x+ae, where a is a parameter. Find the differential equations of the two families of curves which intersect the members of the first family at an angle of 45°.

Of the two differential equations so obtained, solve that one which applies to the family the members of which cut the line y=x in a direction parallel to the x-axis.

7. A circle C and an ellipse S lie in the same plane and are concentric. A point P moves in the plane so that the tangents from P to C separate harmonically those from P to S. Show that the locus of P is an ellipse with the same principal axes as S.

8. The points A, B have, respectively, the coordinates (1, 0), (0, 1). A conic S, is drawn through A, B and the points (0, 2), (1, 1), (3, 0); a conic S2 is drawn through A, B and the points (−1, 0), (2, 1), (−1, −1).

(i) Find the equations of S1 and S2.

(ii) Find the equation of that conic through the common points of S1 and S, which passes through the origin.

(iii) Show that a rectangular hyperbola can be drawn through the common points of S1 and S2, and find its equation.