Theory of Equations and Symmetries — DU B.Sc. (Hons.) Mathematics Previous Year Question Papers

1. (a) Without actual division find the remainder when 4x3 - 17x? + 9 is divided by (b) Form an equation with real coefficients two of whose roots are 1 and 3 + 2i. (c) Solve the equation 2x3 - x2 - 22x - 24, two of its roots being in the ratio 3:4 2. (a) Find all the integral roots of the equation x3 + 8x2 + 13x +6 = 0 (b) Divide 2x4 - x3 - 6x2 + 4x - 8 by x2 - 4 using Synthetic Division method. (c) Find modulus and argument of (cos30 + i sin30)8 (cose + i sing)s 3. (a) Remove the fractional coefficients from the equation: (b) Find the equation whose roots are diminished by 5, the roots of: (c) Remove the second term from the equation: 4. (a) Find the solution of equation x3 - 3x? + 12x + 16 = 0 by Cardan's Method. (b) Find the equation whose roots are reciprocals of the roots of: (c) Find the solution of equation x4 - 10x3 + 35x2 - 50x+24 = 0 by Descartes' 5 (a) If ay, az, az are the roots of the equation x3 + ax? + bx + c = 0 then find the equation whose roots are a - aza3, 0z -1 (b) Find an equation whose roots are the reciprocals of the roots of the equation (c) If ay, a2, ag, as are the roots of the equation ** +ax+bx?+cx+d = 0, then find the value of Da: 6 (a) If a, B and y are the roots of the equation x3 - px? + qx - r = 0. Find (b) Find Sk for x5 - 3 = 0, where Sk = 2a,* and k = 2,3,4,5. (c) If a, B, y are the roots of x3 + px? + qx + r = 0, find the value of 2(a +B - Y)3.