2. Shew that any rational integral symmetrical function of the roots of an equation can be expressed as a rational integral function of the coefficients of the equation.

Prove that

(yz — xu)(zx — yu)(xy — zu) =(yzu + zxu + xyu + xyz)2

xyzu (x + y + z + u)2.

3. If u be the nth term of an infinite series in which every term is positive, shew that if the limit when n is infinite of un+1/un be less than unity the series is convergent, and if this limit be greater than unity the series is divergent.

Determine whether the series whose nth terms arena" and "n are convergent or divergent.

4. Prove the binomial theorem for any exponent. Find to five places of decimals the cube root of 1003.

5. State and prove Fermat's theorem.

If m and n be primes, prove that

6. If the probabilities of n mutually exclusive events be P1, P2, ..., pn, the chance that one of these n events happens on any particular occasion on which all of them are in question is

P1 + P2 + .... + Pr.

If n cards, numbered consecutively, be thrown into a bag and drawn out successively, the chance that one card at least is drawn in the order that its number indicates is

7. Shew that a determinant is not altered in value by adding to all the elements of any column the same multiples of the corresponding elements of number of other columns.

any

Resolve into five factors the determinant

bc, a2, a2 - (bc)2

ca, b2, b2 - (ca)2

ab, c2, c2 - (a - b)2

8. State and prove De Moivre's theorem, and shew how to find all the values of the expression

Find all the values of x which satisfy the equation

(cos a + sin a) (cos ẞ + x sin ẞ) (cos y + x sin y) cos (a + 6 + y) + x sin (a + B + y).

9. Prove that

....

2 cos no (2 cos 0)" · n(2 cos 0)”—2 + n(nr 1)(n-r-2)....(n − 2r+1)

+(−1)"

Ľ

10. Find the sum of the sines of a series of angles which are in arithmetical progression.

Sum to n terms the series

sin3a sin32a + sin33a — sin34a +

and deduce the sum of the series

13233343 + .... +(−1)"−1ñ3.

12. Prove that in any spherical triangle

20

02

sin (a - b) C tan (AB)=sin(a + b) cot 2 ·

If be the angle between the bisector of the angle C and the perpendicular from C on AB,

then

sin (a - b)

tan=sin(a + b) tan (A + B).

PURE MATHEMATICS.-PART I.

SECOND PAper.

The Board of Examiners.

1. Find the condition that the general equation of the second degree may represent a pair of straight lines.

Shew that the equation

(a2 + b2 − 1)(x2 + y2 − 1) = (ax + by − 1)2 represents two straight lines.

2. Find the general equation of a circle which passes through two given points.

Find the equation of the circle having for a diameter the portion of the straight line

Ax + By + C = 0

which is intercepted by the circle

x2 + y2 + 2gx + 2fy + c = 0.

3. Shew that three normals can be drawn from a given point to a parabola.

If OP, OQ, OR be the normals drawn from the point O to a parabola, prove that

SP.SQ.SRAS.SO2,

where A is the vertex and S the focus.

4. Define the excentric angle of any point on an ellipse, and find the equation of the line joining two points whose excentric angles are given.

Three sides of a quadrilateral inscribed in an ellipse are parallel to given lines; shew that the fourth side is parallel to a fixed line.

5. Find the equation of a hyperbola referred to its asymptotes as axes.

A variable line has its extremities on two fixed intersecting lines and passes through a fixed point; find the locus of the point in which it is divided in a given ratio.

and shew how to find the nth differential coefficient of