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.