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 + 2y + 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.SR=AS.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

8. State and prove Lagrange's formula for the remainder in Taylor's series.

Expand x cot x as far as the term in a3.

9. State and prove a rule for finding maxima and minima values of a function of one variable.

S is the focus of an ellipse of excentricity e, G is a fixed point on the major axis, and P is a point on the curve. Shew that when GP is a minimum SG = eSP.

11. Shew how to find the partial fractions corresponding to a repeated simple factor in the decomposition of a rational fraction.

The Board of Examiners.

1. Find the polar equation of the normal at any point of the conic

If the normals at the points a, ß, y meet on the curve, prove that

ax2+by+cz2 + 2fyz + 2gzx + 2hxy=0. Given three points on a conic, if one asymptote pass through a fixed point the other will envelop

a conic.

3. Find the polar reciprocal of one circle with respect to another.

A conic touches three given lines and its director circle passes through a fixed point; shew that the conic touches another fixed line.

4. Find the condition of intersection of two straight lines whose equations are given in the symmetrical form.

A straight line moves parallel to a fixed plane and intersects two fixed straight lines which are not in one plane; shew that the locus of a point which divides the intercepted segment in a fixed ratio is a straight line.

5. Find the equation of the cone whose vertex is the point f, g, h and base the section of the surface ax2 + by2 + cz2 = 1

made by the plane

lx + my + nz = 1,

and shew that the equation of the plane in which this cone again meets the surface is

(lx + my + nz

1)(af2+bg2+ ch2 - 1) =2(If + mg + nh − 1)(afx + bgy + chz −1). Hence find the enveloping cone whose vertex is f, g, h.

6. Investigate the relations between the coordinates of the extremities of conjugate diameters of an ellipsoid.

From a fixed point perpendiculars are let fall on any three conjugate diameters of an ellipsoid; prove that the plane through the feet of the perpendiculars passes through a fixed point.