For me, life is continuously being hungry. The meaning of life is not simply to exist, to survive, but to move ahead, to go up, to achieve, to conquer. – Arnold Schwarzenegger

# Lecture - 5 Chapter 5 Complex Numbers

MISCELLANEOUS EXERCISE |

**Question 1.** Evaluate: \(\left [ i^{18}+ \left ( \frac{1}{i} \right )^{25} \right ]^3\).

**Question 2.** For any two complex numbers \(z_1 \text{ and } z_2\), prove that \( Re(z_1 z_2) = Re z_1 Re z_2 – Im z_1 Im z_2\).

**Question 3.** Reduce \(\left ( \frac{1}{1-4i} – \frac{2}{1+i} \right ) \left ( \frac{3-4i}{5+i} \right )\) to the standard form.

**Question 4.** If \(x-iy = \sqrt{\frac{a-ib}{c-id}}\) prove that \( (x^2+y^2)^2=\frac{a^2+b^2}{c^2+d^2}\).

**Question 5.** Convert the following in the polar form: * (i).* \(\frac{1+7i}{(2-i)^2}\)

*\( \frac{1+3i}{1-2i}\)*

**(ii).**Solve each of the equation in Exercises 6 to 9:

**Question 6.** \(3x^2-4x+\frac{20}{3}=0\)

**Question 7.** \(x^2-2x+\frac{3}{2}=0\)

**Question 8.** \(27x^2-10x+1=0\)

**Question 9.** \(21x^2-28x+10=0\)

**Question 10.** If \(z_1=2-i, z_2=1+i\), find \( \left | \frac{z_1+z_2+1}{z_1-z_2+1} \right |\).