Exercise No 4

Exercise No 4(Online) 1. The complex numbers z1 and z2 satisfy the equation z 2 = 1 − 2 2i . (a) Express z1 and z2 in the form of a+bi, where a and b are real numbers. (b) Represent z1 and z2 in the argand diagram. (c) For each z1 and z2, find the modulus and argument (in radian). 2. Using the method of completing the square, or otherwise, solve the equation z 2 + 4 z = 4 − 6i . Hence determine z and arg z. 3. Using the algebraic laws of sets, show that ( A ∩ B )′ − ( A′ ∩ B ) = B′ . 4. Given x = log a b, y = log b c and z = log c a , show that xzy=1. 5. Find the least integral value of n such that log10 (2n + 1) − log10 2n < log10 1.0025 . 6. One of the roots of the equation 21x 3 − 50 x 2 − 37 x − 6 = 0 is a positive integer. Find this root and hence, solve the equation completely. 7. Find the set of values of x such that − 4 < x 3 − 2 x 2 + 2 x − 4 < 0 . 8. The function f is defined as f ( x) = 2 x 2 + 4 x + 5, x ∈ ℜ . (a) Find the set of values of x such that f(x) <3x2. (b) Find the set of values of k so that the equation f(x) =kx has no real roots. x3 9. Express in partial fractions. ( x + 1)( x + 2) 10. The 2nd, 4th and 8th terms of an AP are 2x+3, 5x+1, and x2-23 respectively. Find the value of x and hence the sum of the first 12 terms of the series.