PURE MATHEMATICS.-PART I.

SECOND PAPER.

The Board of Examiners.

1. Find the angle between the lines ax2 + 2hxy + by2 = 0,

the axes being oblique.

Shew that the bisectors of the angles between the two lines are

2. Prove that the intersections of any line with the three pairs of opposite sides of a quadrangle form a range in involution.

Hence or otherwise find the mate of a given point in the involution determined by two given pairs of points.

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

If OP, OQ, OR are the three normals, shew that the diameter through O passes through the orthocentre of the triangle formed by the tangents at P, Q, R.

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

All chords of a rectangular hyperbola which subtend a right angle at a fixed point on the curve are parallel to a fixed line.

5. Find the polar equation of any chord of a conic, the focus being the pole.

All chords of a conic which subtend a fixed angle at a focus touch a fixed conic having a focus and directrix in common with the given conic.

6. State and prove any formula for the remainder in Taylor's series.

If be the same for all values of h in the equation

+ 0h)

f ( +b) = f (x) + hf' ( then f" (x) must be the same for all values of x.

dxdy dydx

If z is a function of u, v, and u, v are functions of x, y, find

d2z dxdy

8. Shew how to find the maxima and minima values of a function of n variables which are connected

by n 1 given equations.

If f (x, y) is a maximum when xa, y = b, x, y being connected by the equation ø (x, y)=0,

shew that

Τα + λφα = 0, το + λφο = 0

and that the value of μ given by

where F(x, y) is a rational algebraic function

of x, y
and

11. Prove that

y2 = ax2 + bx + c.

PURE MATHEMATICS.-PART II.

FIRST PAPER.

The Board of Examiners.

1. Find the general equation of a conic having contact of the third order at a given point with a given conic.

Hence shew that the equation of the axes of the conic

is

S=(a, b, c, f, g, h) (x, y, z)2 = 0

n2b+m2c-2mnf, l'c+n2a-2nlg, m2a+l2b—2lmh

1

,

1

1

x, y, z being any kind of point coordinates, and 1, m, n constants such that

n2b+m2c-2mnf l2c+n2a-2nlg=m2a+12b-2lmh

are the conditions that S 0 represents a circle. 2. From any point T the two tangents TP, TP' are drawn to a conic, and the two tangents TQ, TQ' to any confocal conic; shew that PQ, PQ' are equally inclined to the tangent at P.

If two conics have double contact the foci of each conic must lie on a conic confocal with the other.

3. If a line drawn through any point of an ellipse so as to make a fixed angle 0 with the tangent at that point be called a @ normal, prove that through

any point in the plane of an ellipse four
normals can be drawn, that the feet of these
normals lie on a hyperbola, which for different
values of passes through two fixed points, and
that if α, B, Y, ♪ be the eccentric angles of the
feet then

a + B + y + d = (2n + 1) π
sin (3+ y) + sin (y + a) + sin (a+ß)=

2abcote

a2 - b2'

4. Prove that through a given point in the plane of a

Iconic there can be drawn six circles which

osculate the conic, and that the centres of these six circles lie on a conic.

5. A system of conics through four fixed points is cut by any straight line in pairs of points in involution.

The envelope of a line cutting three fixed conics in pairs of points in involution is a curve of the third class.

6. If two pairs of opposite sides of a tetrahedron are at right angles to one another, shew that the remaining two edges are at right angles, and that

a2 + a'2 = b2 + b′ 2 = c2 + c′ 2

where a, b, c, a', b', c' are the six edges of the tetrahedron, and shew that the middle points of the six edges all lie on a sphere.

7. Shew how to find the equation of the cone whose vertex is the point x'y' z' and base the intersection of two given surfaces

u = 0, v = 0.