# Lecture-6

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 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.