2. For any two complex numbers \(z_1\) and \(z_2\), prove that \(Re(z_1 z_2)=Re(z_1)Re (z_2)\)\(-Im(z_1)Im(z_2)\).
Class 11
Complex Numbers and Quadratic Equations
Miscellaneous Exercise
1. Evalaute: \([i^{18}+(\frac{1}{i})^{25}]^3\).
3. Reduce \((\frac{1}{1-4i}-\frac{2}{1+i})(\frac{3-4i}{5+i})\) to the standard form.
4. If \(x-iy=\sqrt{\frac{a-ib}{c-id}}\), prove that \((x^2+y^2)^2=\frac{a^2+b^2}{c^2+d^2}\).
5. If \(z_1=2-i, z_2=1+i\), find \(|\frac{z_1+z_2+1}{z_1-z_2+1}|\).
6. If \(a+ib=\frac{(x+i)^2}{2x^2+1}\), prove that \(a^2+b^2=\frac{(x^2+1)^2}{(2x^2+1)^2}\).
7. Let \(z_1=2-i, z_2=-2+i\). Find
(i) \(Re(\frac{z_1 z_2}{\bar{z_1}})\)
(ii) \(Im(\frac{1}{z_1 \bar{z_1}})\)
8. Find the real numbers x and y if (x – iy) (3 + 5i) is the conjugate of –6 – 24i.
9. Find the modulus of \(\frac{1+i}{1-i}-\frac{1-i}{1+i}\).
10. If \((x+iy)^3=u+iv\), then show that \(\frac{u}{x}+\frac{v}{y}=4(x^2-y^2)\).
11. If α and β different complex numbers with |β|=1, then find \(|\frac{β-α}{1-\bar{α}β}|\)
12. Find the number of non-zero integral solutions of the equation \(|1-i|^x=2^x\).
14. If \((\frac{1+i}{1-i})^m=1\), then find the least positive integral value of m.