Set No 3_new

Set No 3 1. If (2 x + 3 yi) 2 = 2 x + 3 yi where x and y are real, find the possible values of x and y. 2 2. Solve the equation log 3 x + log3 x − log9 27 = 0 . Leave your answer in the surd form. 3. Show that 1 is the only real root of x 3 + 3x 2 + x − 5 = 0 . 4. Show that x +1 is a factor of the polynomial x 3 − 12 x 2 + 23x + 36 . (a) Find all the real roots of the equation x 6 − 12 x 4 + 23x 2 + 36 = 0 . (b) Determine the set of values of x such that x 3 − 12 x 2 + 23 x + 36 < −8 x − 8 . 5. Prove that x 2 + y 2 ≥ 2 xy . If a, b and c are real numbers, prove that a 2 + b 2 + c 2 − ab − bc − ca ≥ 0 . 6. Show that the roots of the quadratic equation mx 2 + nx + k = 0 are given by − n ± n 2 − 4mk . 2m Deduce that if p+qi, p and q are real numbers is a root of the quadratic equation, then the other root is p-qi. (a) Show that 1+i is a zero of the polynomial f ( x) = 2 x 3 − 3 x 2 + 2 x + 2 = 0 and find the other zero. (b) Find the polynomial g(x) such that f(x)-xg(x) = 2- x. By expressing g(x) in the form of a ( x − b) 2 + c , where a, b and c are real numbers, find the 1 maximum value of . g ( x) x=

2

x −1  x −1 7. Find the set of values of x so that the series 1 + +  +  converges. x  x  Find the sum to infinity of this series when x = 3.