Algebra Answers

4. Which of the following statements are true and which are false ? Justify your answers with a short proof or counter-example. 5x2=10 (a) If A and B are two subsets of a universal set U, then Ac \B = A\ Bc. (b) The roots of a quadratic equation are always real numbers. (c) lx+yl = lx1 + ly1 for all x,yE R. (d) The contrapositive of 'If two triangles have the same area, then they are congruent' is 'If two triangles are congruent, then they have the same area'. (e) If A is a square matrix with I A l = 0, then two of its rows or two of its columns must be the same. 5. (a) If x, y and z are positive real numbers, show that (x2y + y2z + z2x) (xy2+ yz2+ zx2) > 9x2y2z2. (b) Use Cardano's method to obtain the roots of x3— 3x + 2 = 0.

Solve the following linear systems by the process of elimination : 3 x + 2y + 5z = 9 x — y + $z = 2 3x — 6y z = 26

solve the following question: ||x^2-2x-8|+x^4-x^3+4x+8| =6