PURE MATHEMATICS. (3.)

1. If a be a root of the equation f(x) = 0, prove that f(x) is divisible

by x

Find the ratio of b to a in order that the equations ax2 + bx +α = O and x3

2x2 + 2x − 1 = 0 may have either

one or two roots in common.

2. If p be the numerically greatest of the negative coefficients in the equation

Prove that p+ 1 is a superior limit of the positive roots.

3. If f(x) = 0 have equal roots, prove that the equations f(x) = 0 and f'(x) = 0 must have a root in common;

4. Define a differential coefficient, and from your definition find the differential coefficient of a quotient of one given function of x divided by another.

Find also, from first principles, the differential coefficients of logx and tan x.

7. Prove that f(a + h) − f(a)

dx

=

1

hf (a+Oh),

being some fraction less than unity, and f(x) and f'(x) being finite between the limits a and a + h of x.

Assuming Maclaurin's theorem, prove that is very nearly equal to when h is very small.

8. If e = e + xy, find the first six terms of the expansion of y powers of x.

in

9. If ABC be a triangle inscribed in a given circle, and the side BC be of constant length, prove that any small variations db, and de in the lengths b and c of the remaining sides are connected

11. Find the conditions that f(x) may be a maximum or minimum when x = a.

If P and Q be two fixed points situated either, both without, or both within, the circumference of a circle whose centre is O, and if R be a point upon that circumference, prove that the sum of the distances PR and QR is the least possible when PR and QR make equal angles with the radius OR.

12. Find expressions—

(1) For the perpendicular from the pole upon the tangent, (2) For the radius of curvature,

at any point of a curve referred to polar co-ordinates.

If S be the pole, Y the foot of the perpendicular (p) from S on the tangent PY at P, and 4 the angle between SY and any

dp аф

fixed line, prove that = PY.

13. Find the asympotes to the curve

STATICS.

1. State under what circumstances two forces have a single resultant.

Two forces, whose magnitudes are P lbs. and P/2 lbs., act at a point in directions inclined to each other at an angle of 135°; find the direction and magnitude of their resultant.

2. Define a couple, and the moment of a couple.

Prove that the effect of a couple is not altered if its arm be turned about one end on the plane of the couple through any angle.

Find the resultant of any number of couples acting in one plane.

3. Enunciate the triangle of forces.

P is a particle acted upon by forces emanating from two fixed centres A and B, and proportioned to the distances from those points respectively; if the force from A be repulsive and the force from B attractive, prove that the resultant force on P is constant in magnitude and direction.

4. The three perpendiculars from the angular points of a triangle ABC upon the opposite sides meet in the point T, and O is the centre of the circle circumscribing the triangle; prove that if three forces acting at a point be represented in magnitude and direction by the lines AT, BT, and CT, their resultant is represented in magnitude and direction by twice the line OT. 5. It is required to hang a picture on a vertical wall so that it shall lean forwards at a given angle to the wall, and the picture is to be supported by a single cord attached to a point of the wall at a given height above its lowest edge; determine by a geometrical construction the point on the back of the picture to which the cord must be attached, and find the length of cord which will be wanted.

6. Define a lever, and give instances of different kinds of levers.

Describe the action of the muscles of the arm when it is held out horizontally from the elbow and a weight is held in the hand.

7. Prove that a system of heavy particles has one and only one centre of gravity. A straight rod, one foot in length, and weighing one ounce, has an ounce of lead fastened to it at one end, and another ounce of lead fastened to it at a distance from that end equal to one-third of its length; find its centre of gravity. 8. Two equal rods AB, BC are jointed at B, and have their middle points connected by an inelastic string of such a length, that when it is straightened, the angle ABC is a right angle; if the system be suspended by a string attached to the point A, prove

1 that the inclination of AB to the vertical will be tan-1 and 3'

find the tension of the string and the action at the joint.

9. Define the horse-power of an engine. An engine is required to raise a weight of a ton from a depth of 150 fathoms in five minutes; what is its horse-power?

10. Find the ratio of the Power to the Weight when there is equilibrium on the Wheel and Axle.

If the radius of the wheel be six times that of the axle, and if by means of a power of 5 lbs. a weight is lifted through 50 feet, find the amount of energy expended.

11. Find the ratio of the Power to the Weight on a smooth Screw, the axis of which is vertical.

If the power be applied at the end of an arm one foot in length, and if the screw make ten complete turns in one inch of its length, find the power which will support a weight of a

ton.

12. Describe a method of determining practically the coefficient of friction between two given substances.

A heavy body rests on a rough inclined plane; if tan ẞ be the coefficient of friction and a the inclination of the plane, prove that the greatest and least force acting upwards parallel to the plane, which will support the weight are in the ratio of

sin a+B to sin a B.

13. Explain the use of a balancing pole to a man standing on a tight

rope.

DYNAMICS.

NOTE. In all the following questions the measure of the force of gravity, when required, may be taken as 32 feet.

1. When a body's motion is variable, how is the velocity at any point of its course estimated? What will be the law of the motion of a particle acted on by a single uniform force in the direction of its motion? A particle under the action of a force in its direction has a velocity of 200 feet at the end of the third second, and of 260 feet at the end of the fourth second, what will be its velocity at the end of the fifth second if the force be uniform? Account briefly for the variation in the measure of the force of gravity at different stations on the earth.

2. If a particle be projected in the direction of a uniform force (ƒ), with a velocity V, investigate the following expressions for the space (8) described, and the velocity (v) acquired, after (t) seconds.

(1) 8 = Vt +

of s.

ft2 2

(2) v + V = ft, hence find (v) in terms

What will (f) be in case of a body sliding down a smooth inclined plane under the action of gravity?

3. Divide a given inclined plane into three parts, so that a particle at the top of the plane sliding from rest may describe the three parts in equal times. In what number of seconds will each part be described if the height of the plane is 64 feet, and its length 288 feet?

4. Enunciate the second law of motion, and apply it to determine the position of a body, acted on by gravity, projected with a given velocity at a given angle of projection after it has been in motion for t seconds.

If (a) be the angle and V the velocity of projection, find the 2V sin a position after a time represented by

5. Find the equation to the path of a projectile in vacuo, when referred to horizontal and vertical co-ordinates.

From the equation obtained, determine the range on an inclined plane that passes through the point of projection, and show that the vertical line, drawn from the point in the curve at which the projectile is moving parallel to the inclined plane, will pass through the middle point of the range on that plane.