FINAL HONOUR MATHEMATICS.

CONICS II.

Note. Any seven questions.

1. Use the polar equation as a basis for a short discussion of the general conic under the two heads, (a) asymptotes, (b) conjugate diameters.

2. (a) Find a conic which has contact of the (i) 1st order () 2nd order, (iii) 3rd order with the conic

́..x2+by2+2hxy+2gx+2fy=0 at the origin.

(b) Show from (a) that the osculating circle is unique for a given point.

3. Show that

(a2 - b2
b2 - a2
x/3,
a4

64

is the centre of curvature for the point (x', y') on the central conic; and thence find the equation of the evolute.

'3

4. Find a symmetric expression for the angle between two lines whose trilinear equations are given; and deduce the conditions for parallelism and perpendicularity.

5. Show that aß=kyd is a conic circumscribing the quadrilateral whose sides are a, ẞ, y, 8, and that this conic is a circle if k±1.

Distinguish between the cases k:

+1 and k—–—1.

6. Find the tangential equations of the conics

1/a+m/B+n/y=0,

and Vla+VmB+√ny=0; and explain the similarity in the forms of the trilinear and tangential equations.

7. Interpret x2+y2=e2y2 as a trilinear equation, x and y being at right angles, and e a parameter.

In what sense may all doubly confocal conics be said to be inscribed in the same quadrilateral? What are the vertices of this quadrilateral for a system of concentric circles?

8. μ1, μ2, μ3, μ are fixed points on a conic. Show that (a) they subtend at a fifth point a pencil with constant anharmonic ratio, (b) the tangents at these points intersect any fifth tangent in a range with the same anharmonic ratio as the pencil of (a).

9. Show that the polar reciprocal of a circle with respect to a circle is a conic, and state how the nature of the conic is determined.

Find any two properties of the parabola by the method of polar reciprocation.

2

Prove the theorem used in testing this series.

+

+

2. Prove that a series of positive integral powers of z converges in a circular region of the z-plane; and find an expression for the radius.

Find the radii of convergence of

n

Σ(log n)" and Ex".

22

3. If u1+u2+ ... is a complex series which converges or oscillates between finite limits, and a1, a,.. a sequence of decreasing positive quantities tending to zero, show that a1u, +a2u 2 + . . . is convergent.

Show that

23
+ +
3

converges when

I

2

|≈ = 1 but z is not I.

(1·3·5) 2
(2·4·6)

1

Illustrate by means of

+

+

...

4. Show that the necessary and sufficient condition for the absolute convergence of (1+u,)(1 + u „)..... is that u,u,+...... should converge absolutely.

) (

x)

1+
2"

5. Find the nth differences of x(m), x2, x111, and r2 sin ax.

6. If f(a.), fa1), .. f(a) are known, and f(x) is a polynomial of degree n, show that f(x) may be expressed in the form

1

Co + C1 (x − ao)+C2(x− a)(x − a1)
+....+C" (x-a)..(x-an-1);

and show how to find the coefficients. Prove that

I

I

x-a

x

a ab

provided x is one of the numbers a, b

=

+

(x-a)(x-b)
abc

7. (a) Find the most general function of x whose second difference is

Ax being = 1.

I

x(x+2)(x+3)'
u1

(b) Sum to infinity u。+ +

2

12

un is a polynomial of degree z.

+

U 2 +

8. Find the first five of Bernoulli's numbers, and

use them to obtain the approximate value of

I

+
992

where

FINAL HONOUR MATHEMATICS.

Theory of Functions of a Complex Variable.
(Any seven questions).

1. (a) Evaluate (1—i)% and represent the roots graphically.

(b) If ŋ(k=1, 2, 3, 4) are the imaginary fifth roots of unity, show graphically that

1+ 71+2+3+n1 = 0.

(c) Define sin z, e, log z where z=x+iy.

2. (a) Define a monogenic function.. Develop the conditions that w u(x, y)+iv(x, y) shall be monogenic.

(b) What condition must u satisfy in order that a v shall exist such that u+iv shall be monogenic?

Given u

log(x2+y2), find v.

3. If w=f(z) and f(z) is monogenic, show that the w-plane is a conformal representation of the z-plane.

4. What is meant by the Neumann Sphere and what is its particular advantage? Show that to every circle in the plane there corresponds a circle on the sphere and conversely.

5. Show that there is only one linear fractional transformation which will transform three distinct points in the z-plane into three distinct points in the w-plane.

6. Construct a Riemann surface for w=(≈+1/2). Sketch the curves in the w-plane corresponding to the radiating lines and concentric circles in the z-plane.