Class 11 Complex Numbers Lecture 5

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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}
(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 |.

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