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.

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

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.

l/a+m/ß+n/y=0,

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

7. Interpret x+y=ey 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, M2, M3, μ4 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.

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

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,+a2u2+... is convergent.

1

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

2

5. Find the nth differences of x(m), xn, xn+1, and r2 sin ax.

6. If f(a), f(a1),

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

2

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

and show how to find the coefficients.

provided x is one of the numbers a, b

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

8. Find the first five of Bernoulli's numbers, and use them to obtain the approximate value of

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FINAL HONOUR MATHEMATICS.

Theory of Functions of a Complex Variable.

(Any seven questions).

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

ηκ

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

(c) Define sin 2, 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=3(x+1/2). Sketch the curves in the w-plane corresponding to the radiating lines and concentric circles in the z-plane.