Theory of Equations and Symmetries — DU B.A. (Prog) Mathematics as Major Previous Year Question Papers

(PROGRAMME) WITH MAJOR - DSC 1. (a) Form a equation of the lowest degree with real coefficient which has (1 + 3i) and (2 - i) as two of its roots. (b) Find all the integral roots of x* - x3- 19x2 + 49x - 30 = 0. (c) Without actual division, find the remainder when 2. (a) If x, = cos then prove that (b) Find all the values of 2+ SN (c) Use De-Moivre's theorem to solve the equation z* + 1 = 0 for z e C. 3. (a) Solve the cubic equation x3 - 9x + 28 = 0 by Cardan's method. (b) Find a necessary condition for the roots of equation a„x° + 3a, x? + 3a,x + a, = 0 to be in the geometric progression. (c) Solve the equation x4 - 15x2 + 20x - 6 = 0 by the method of Descartes. 4. (a) Solve the equation x3 - 18x - 35 = 0 using Cardan's method. (b) Transform the equation 25x4 + 5x3 - 7x2 + 1 = 0 into one with integral coefficients and unity for the coefficient of the first term. (c) Find the solution of equation x4 + 3x2 + 2x + 12 = 0 by Descartes method. 5. (a) If a, B, y are the roots of equation x3 + px+ qx + r = 0 then prove that (b) Solve the equation x3-5x2 - 16x + 80 = 0 for the sum of two of its roots being zero. (c) If a, B, y be the roots of equation x + qx + r= 0, then find the value of the symmetric function 6. (a) Find the equation whose roots are the roots of the equation 14 - 5x1 + 7x2 - 17x + 11 = 0, each diminished by 4. (b) Find the value of the symmetric function Za3 for the equation x° + 5x2 - 6x + 3 = 0, whose roots are a, B, Y. (c) Use Newton's formula to prove that for a cubic equation x° + a,x? + a,x + a, = 0, S4 = a,* - 4a,}a, + 4a,a, + 2a,}; where s* = E(a,)k, ay, a, ay the roots of the given cubic equation.