Ashish Kumar - let's learn, implement then understand Maths and Physics
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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: $$\left [ i^{18}+ \left ( \frac{1}{i} \right )^{25} \right ]^3$$.

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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.

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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). $$\frac{1+3i}{1-2i}$$

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

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