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).* \( Re \left ( \frac{z_1z_2}{ \overline{z_1}} \right ) \).

*\(\Im \left ( \frac{1}{z_1 \overline{z_1}} \right )\)*

**(ii).****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*.