6. If two tangents be drawn to a circle, prove that any third tangent is divided harmonically by the two tangents, the chord of contact, and the circle.

7. If A1, B1, C1 be the angles of the triangle formed by the centres of the escribed circles of the triangle ABC, prove that 2A1, 2B1, and 2C1 are the supplements, respectively, of A, B, and C.

If the triangle A2B2C2 be similarly formed from A1B1C1, A3B3C3 from A2B2C2, and so on, find An, Bn, and Cn, and prove that, as n increases indefinitely, the triangle AnBnCn ultimately becomes equilateral.

8. Find the size of a cube which will completely stop up a tube of uniform bore, the section of which is a regular hexagon whose magnitude is given.

9. Resolve into partial fractions the expressions

and expand each in powers of x, obtaining in each case the coefficient of xr.

10. Through the angular point C of a triangle ABC a straight line CPQ is drawn, on which are let fall the perpendiculars AP, BQ; prove that

12. Prove De Moivre's Theorem for a positive integral index, and for

a negative integral index.

If

prove that

13.

sin (a +ẞ- 1) = x+y√-1,
x2 cosec2 a - y2 sec2 a = 1.

Prove that, in an equation with real coefficients, imaginary roots

occur in pairs.

Having given that 2 + √ 3 is one of the roots of the equation,

find all its roots.

x2-4x2+8x+35=0,

14. State the relations between the coefficients and the roots of a rational algebraic equation.

If a, β, y be the roots of the equation,

x3- px2+r=o,

prove that the equation, of which the roots are

is

β+γ για α+β

a β' γ'

rx3 +3rx2+(3r - p3) x+13=0.

I.

2.

VII. PURE MATHEMATICS (3).

[Great importance will be attached to accuracy in results.]
Full marks may be obtained for four-fifths of this paper.
Find from first principles the differential coefficient of

(1) tan x + sec x, (2) tan-1.

a

3. Prove Leibnitz's theorem for obtaining the nth differential coefficient

of the product of two functions.

Find the nth differential coefficient of ex sin x ; and shew that

4.

State fully the conditions necessary for the truth of Taylor's theorem.

6.

Shew how to determine the maxima and minima values of a function of a single variable, of which the nth derived function is the first not to vanish for the critical value of the variable.

Find the area and position of the maximum triangle having a given angle which can be inscribed in a given circle, and prove that the area cannot have a minimum value.

7.

Shew how to determine the number and position of the asymptotes of a plane curve.

Find the asymptotes of the curve

y3 - 7yx2 - 6x3 + 4ax2=0.

8. What is a point of inflexion of a plane curve? Find the condition that a plane curve should have points of inflexion.

Express the condition also in polar co-ordinates.

9. Shew that the envelop of perpendiculars drawn to tangents of the parabola, y2 = 4ax, at the points where they cut the axis is 27ay2 + 4x3 = 0.

C

12. Shew that the area of the curve y=ae included between two ordinates varies as the difference of the ordinates; and find the length of the curve between two points.

VIII. STATICS.

Full marks may be obtained by doing four-fifths of this paper.

I. If three commensurable forces in one plane are in equilibrium acting on a point, prove, without assuming the "parallelogram of forces," that either force is in the direction of the diagonal of the parallelogram whose sides represent the other two forces.

Assuming the "parallelogram of forces," shew that a single force acting on a point may be resolved in an unlimited number of ways into two forces acting on the same point.

Equal forces act on the centre of a regular pentagon along the lines drawn from the centre to the angles of the pentagon; prove that the resultant of any two of these adjacent forces is equal and opposite to the resultant of the other three.

2.

When three forces, which do not meet, act on a body in one plane, determine the relation that must exist among them, both as to magnitude and position, in order to preserve equilibrium, the extreme forces being like.

A horizontal rod without weight, 6 feet long, rests on two supports at its extremities; a weight of 6 cwt. is suspended from the rod at a distance of 2 feet from one end: find the reaction at each point of support. If one support could only bear a pressure of one cwt., what is the greatest distance from the other support at which the weight could be suspended?

3. When are couples said to be "like" or "unlike"? Shew that two unlike couples in the same plane will balance each other if their moments are equal.

The sides of a regular polygon taken in order represent the forces acting in the plane of the polygon; shew that the sum of their moments will be the same round any point within the figure. Find the couple, having one side of the polygon for an arm, that will keep the system in equilibrium.

4.

Find the centre of gravity of a solid triangular pyramid whose faces are equilateral triangles.

Shew that the position of the centre of gravity for the four faces considered as plane areas will be the same as it is for the solid pyramid.

5. On a smooth inclined plane a weight is just sustained by a force making a given angle with the plane; find the relation of the power to the weight. Determine also the pressure on the plane.

If the weight, the force, and the pressure be respectively as the numbers 4, 3, and 2, find the direction of the force and the inclination of the plane.

6. State the conditions of equilibrium of any number of forces acting on a body in one plane. Explain also when either of these conditions may, in solving problems, be dispensed with.

A uniform beam rests with a smooth end against the junction of the horizontal ground and a vertical wall; it is supported by a string fastened to the other end of the beam and to a staple in the vertical wall. Find the tension of the string, and shew that it will be half the weight of the beam if the length of the string be equal to the height of the staple above the ground.

7. When a weight is placed on a rough surface, explain what is meant by the coefficient of friction and by the limiting angle of resistance. Express one in terms of the other.

Shew that the least value of a force (P) to move a weight (W) along a rough horizontal plane is (W sin ) where (6) is the limiting angle of resistance.

Two equal heavy particles on two equally rough inclined planes of the same height, and placed back to back, are connected by a string passing over the top of the planes; shew that when the particles are on the point of moving, the limiting angle of resistance will be half the difference of the inclination of the planes.

8. State the principle of virtual velocities. Shew that it holds good in that system of pulleys where each pulley hangs by a separate string (usually called the first system). If the principle be stated Px P's displacement = W× W's displacement, will it be true in this case though the displacement should not be small?

In the above system, the distance of the highest pulley from the fixed end which passes round it is 16 feet, and the whole height through which the weight can be raised is one foot; find the number of pulleys, the size of the pulley being neglected.

9. If a right cone be placed with its base on an inclined plane, friction being sufficient to prevent sliding, examine the conditions that the cone may just remain at rest on the plane.

I

If be the coefficient of friction, find the angle of the cone when it is √3

on the point both of sliding and falling over.

10. How is "work" measured? How is the efficiency of a working agent estimated? With reference to what units is 33,000 taken as the measure of horse-power?

When weights are raised through different heights, prove that the whole work expended is equal to the work that would be expended in lifting a weight equal to the sum of the weights through the same height as that through which the centre of gravity of the weights has been raised.

Find the horse-power of an engine that would empty a cylindrical shaft full of water in 32 hours, if the diameter of the shaft be 8 feet and its depth 600 feet; the weight of a cubic foot of water being 62.5 pounds.

IX. DYNAMICS.

Full marks may be obtained by doing four-fifths of this paper.
(Note. The acceleration of gravity may be taken at 32 feet.)

I. How is velocity estimated (1) when uniform, (2) when variable? Give an instance where, in ordinary language, an estimate of variable velocity is made.