Ashish Kumar - let's learn, implement then understand Maths and Physics

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

 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). $$\frac{1+7i}{(2-i)^2}$$
(ii). $$\frac{1+3i}{1-2i}$$

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 $$\left | \frac{z_1+z_2+1}{z_1-z_2+1} \right |$$.