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 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 converges when I 2 |≈ = 1 but z is not I. (1·3·5) 2 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+ 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) 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) 7. (a) Find the most general function of x whose second difference is Ax being = 1. I x(x+2)(x+3)' (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 + where FINAL HONOUR MATHEMATICS. Theory of Functions of a Complex Variable. 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. |