MISCELLANEOUS EXERCISE

Question 11. If a+ib = \frac{(x+i)^2}{2x^2+1}, prove that a^2+b^2 = \frac{(x^2+1)^2}{(2x^2+1)^2}.

Question 12. Let z_1 = 2-i, z_2=-2+i. Find
(i). Re \left ( \frac{z_1z_2}{ \overline{z_1}} \right ) .
(ii). \Im \left ( \frac{1}{z_1 \overline{z_1}} \right )

Question 13. Find the modulus and argument of the complex number \frac{1+2i}{1-3i}.

Question 14. Find the real numbers x and y if (x-iy)(3+5i) is the conjugate of -6-24i.

Question 15. Find the modulus of \frac{1+i}{1-i} – \frac{1-i}{1+i}.

Question 16. If (x+iy)^3=u+iv, then show that \frac{u}{x}+ \frac{v}{y}=4(x^2-y^2).

Question 17. If \alpha \text{ and } \beta, are different complex numbers with |\beta| = 1, then find \left | \frac{\beta – \alpha}{1 – \overline{\alpha} \beta} \right |

Question 18. Find the number of non-zero integral solutions of the equation |1-i|^x = 2^x.

Question 19. If (a+ib)(c+id)(e+if)(g+ih) = A+iB , then show that (a^2+b^2)(c^2+d^2)(e^2+f^2)(g^2+h^2) = A^2 + B^2.

Question 20. If \left ( \frac{1+i}{1-i} \right )^m = 1, then find the least positive integral value of m.