Exercise 1.4 (Solutions)

The solutions of the Exercise 1.4 of book “Model Textbook of Mathematics for Class XI” published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan are given on this page. This exercise consists of the question related to polar form of the complex numbers.

Question 1. Write the following complex number in polar form.
(i) $2+i 2 \sqrt{3}$
(ii) $3-i \sqrt{3}$
(iii) $-2-i 2$
(iv) $\dfrac{i-1}{\cos \dfrac{\pi}{3}+i \sin \dfrac{\pi}{3}}$
Solution: Question 1

Question 2. Write the following complex number in rectangular form.
(i) $\left(\cos \dfrac{\pi}{6}+i \sin \dfrac{\pi}{6}\right)\left(\cos \dfrac{\pi}{3}+i \sin \dfrac{\pi}{3}\right)$
(ii) $\dfrac{\cos \dfrac{\pi}{6} - i \sin \dfrac{\pi}{6}}{2\left(\cos \dfrac{\pi}{3}+i \sin \dfrac{\pi}{3}\right)}$
Solution: Question 2

Question 3. If $\left(x_{1}+i y_{1}\right)\left(x_{2}+i y_{2}\right)\left(x_{3}+i y_{3}\right) \ldots\left(x_{n}+i y_{n}\right)=a+i b$, show that:
(i) $\left(x_{1}^{2}+y_{1}^{2}\right)\left(x_{2}^{2}+y_{2}^{2}\right)\left(x_{3}^{2}+y_{3}^{2}\right) \ldots\left(x_{n}^{2}+y_{n}^{2}\right)=a^{2}+b^{2}$
(ii) $\sum_{r=1}^{n} \tan ^{-1}\left(\frac{y_{r}}{x_{r}}\right)=\tan ^{-1}\left(\frac{b}{a}\right)+2 k \pi, k \in \mathbb{Z}$
Solution: Question 3

Question 4. If $\dfrac{1+z}{1-z}=\cos 2 \theta+i \sin 2 \theta$, show that $z=i \tan \theta$
Solution: Question 4

Question 5. If $\cos \alpha+\cos \beta+\cos \gamma=\sin \alpha+\sin \beta+\sin \gamma=0$, show that:
(i) $\cos 3 \alpha+\cos 3 \beta+\cos 3 \gamma=3 \cos (\alpha+\beta+\gamma)$.
(ii) $\sin 3 \alpha+\sin 3 \beta+\sin 3 \gamma=3 \sin (\alpha+\beta+\gamma)$.
Solution: Question 5

Question 6. Write a following given complex number in the algebraic form:
(i) $\sqrt{2}\left(\cos 315^{\circ}+i \sin 315^{\circ}\right)$
(ii) $5\left(\cos 210^{\circ}+i \sin 210^{\circ}\right)$
(iii) $2\left(\cos \dfrac{3 \pi}{2}+i \sin \dfrac{3 \pi}{2}\right)$
(iv) $4\left(\cos \dfrac{5 \pi}{6}+i \sin \dfrac{5 \pi}{6}\right)$
(v) $2\left(\cos \dfrac{\pi}{6}+i \sin \dfrac{\pi}{6}\right)$
(vi) $\cos 135^{\circ}+i \sin 135^{\circ}$
(vii) $10\left(\cos 50^{\circ}+i \sin 50^{\circ}\right)$
(viii) $\sqrt{2}\left(\cos \dfrac{3 \pi}{4}+i \sin \dfrac{3 \pi}{4}\right)$
(ix) $4\left(\cos \dfrac{2 \pi}{3}+i \sin \dfrac{2 \pi}{3}\right)$
Solution: Question 6(i-ix)

Question 6. Write a following given complex number in the algebraic form:
(x) $7 \sqrt{2}\left(\cos \dfrac{5 \pi}{4}+i \sin \dfrac{5 \pi}{4}\right)$
(xi) $10 \sqrt{2}\left(\cos \dfrac{7 \pi}{4}+i \sin \dfrac{7 \pi}{4}\right)$
(xii) $2\left(\cos\dfrac{5\pi}{2}+i \sin \dfrac{5\pi}{2}\right)$
(xiii) $\dfrac{1}{\sqrt{2}}\left(\cos \dfrac{\pi}{4}+i \sin \dfrac{\pi}{4}\right)$
(xiv) $7\left(\cos 180^{\circ}+i \sin 180^{\circ}\right)$
(xv) $2 e^{i \dfrac{\pi}{4}}$
(xvi) $3 e^{i \dfrac{\pi}{2}}$
(xvii) $5 e^{i{\dfrac{\pi}{3}}}$
Solution: Question 6(x-xvii)

Question 7. Convert the following equation in Cartesian form:
(i) $\arg (z-1)=-\dfrac{\pi}{4}$
(ii) $z \bar{z}=4\left|e^{i \theta}\right|$
(iii) $-\dfrac{\pi}{3} \leq \arg (z-4) \leq \dfrac{\pi}{3}$
(iv) $0 \leq \arg \left(\dfrac{z-4}{1+i}\right) \leq \dfrac{\pi}{6}$
(v) $\arg \left(\dfrac{1-iz}{1-z}\right)=\dfrac{\pi}{4} ; z \neq i$
(vi) $\dfrac{1}{2} \arg (z-i)=\dfrac{\pi}{3}-\dfrac{1}{2} \arg (z+i)$
Solution: Question 7

Question 8. Calculate the following position of a particle from mean position when amplitude is $0.004 \mathrm{~mm}$ and angle is:
(i) $\dfrac{\pi}{4}$
(ii) $\dfrac{\pi}{3}$
(iii) $\dfrac{\pi}{6}$
Solution: Question 8

Question 9. When particle is at a position of $x=2+3 i$ from its mean position and $x_{\max }=1+4 i$ is the position at maximum distance from mean position as it can be seen under microscope at this point.
(i) Calculate the angle at time $\mathrm{t}=0$ and find the position of the particle
(ii) If $x=2+3 i$ and $x_{\max }=1+4 i$. Calculate the frequency when $\mathrm{t}=2$
Solution: Question 9

Question 10. Find the impedance $Z$ for the following values:
(i) $E=(-50+100 i)$ volts, $I=(-6-2 i)$ amps
(ii) $E=(100+10 i)$ volts, $I=(-8+3 i) \mathrm{amps}$
Solution: Question 10