Theory of Equations and Symmetries — DU B.Sc. (Prog) Physical Sciences courses pertaining to Mathematical Sciences Previous Year Question Papers

Semester Ill Programme Theory of Equations and symmetries BACIHLOR OF ARTS (PROGRAMRE) WITH MATREMATICS AS MAJOR Instruction for Candidates: 3. Each part carries 7.5 marks. 1. (a) Find the quotient and remainder when x4 -3x3 + 3x2 -3x + 2 is divided by x2 -3x + 2, using the synthetic division. (b) Find the nature of the roots of the equation #4 + 3x -1 = 0, using Descartes rule of signs. (c) Find the equation of the lowest degree with real coefficients having 1 + 2£ and 3 -I as two its roots. 2. (a) State and prove De-Moivre's Theorem for an integral index. (b) Prove that cos(6G) + 6 cos(4G) + 15 cos(2G) + 10 = 32 (cos G)6. (c) Solve the equation z4 + 1 = 0, using De-moivre's Theorem for a rational index. 3. (a) Solve the equation x3 -6x -9 = 0, using Cardan's method. a) Find the necessary condition for the roots of the equation x3 -px2 + qx -r = 0 to be in the geometric progression. (c) Solve the equation x4 -6x2 -16x -15 = 0 by the method of Descartes. 4. (a) Solve the equation 2x3 + 3x2 + 3x + 1 = 0 using Cardan's method. (b) Solve the equation 2x3 + #2 -7x -6 = 0, given that the difference of two of its roots is 3. (c) Find the solution of the equation x4 -10x2 -20x -16 = 0 by Descartes Method. 5. (a) If cr, 4, rare the roots of the equation 2x3 + 7x2 + x + 8 = 0, then flnd the value of c¥2 + 42 + y2. (b) Ifc¥,4, y, 6 be the roots of equation EG12p. x4 + px3 + q#2 + rx + s = 0, then find the value of 6. (a) Find the equation whose roots are squares of the roots of the equation x3 -x2 -8x + 6 = 0. (b) If cr, P, y be the roots of equation r3 + 4x2 + 2x + 3 = 0, then flnd the value of I:(P + y)2. (c) Use Newton's formulae to find the value ofs4, for the cubic equation x3 + C|ix2 + C12X + C13 = 0; Where Sk = I(cri)k,cri, c¥2, c¥3be roots of the given cubic equation.