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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.

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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.


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.

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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

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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.


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


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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


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

and show how to find the coefficients.

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provided x is one of the numbers a, b

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

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

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Theory of Functions of a Complex Variable.

(Any seven questions).

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


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

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(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.

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