Class 11: Complex Numbers – Very Short Answer Questions

$\displaystyle \textbf{Question 1. }\text{Write the values of the square root of }i.$
$\displaystyle \text{Answer:}$
$\displaystyle i=\cos\frac{\pi}{2}+i\sin\frac{\pi}{2}$
$\displaystyle \therefore \sqrt{i}=\pm\left(\cos\frac{\pi}{4}+i\sin\frac{\pi}{4}\right)$
$\displaystyle =\pm\frac{1+i}{\sqrt2}$
$\\$

$\displaystyle \textbf{Question 2. }\text{Write the values of the square root of }-i.$
$\displaystyle \text{Answer:}$
$\displaystyle -i=\cos\frac{3\pi}{2}+i\sin\frac{3\pi}{2}$
$\displaystyle \therefore \sqrt{-i}=\pm\left(\cos\frac{3\pi}{4}+i\sin\frac{3\pi}{4}\right)$
$\displaystyle =\pm\frac{1-i}{\sqrt2}$
$\\$

$\displaystyle \textbf{Question 3. }\text{If }x+iy=\sqrt{\frac{a+ib}{c+id}},\text{ then write the value of }(x^2+y^2)^2.$
$\displaystyle \text{Answer:}$
$\displaystyle x+iy=\sqrt{\frac{a+ib}{c+id}}$
$\displaystyle \therefore (x+iy)^2=\frac{a+ib}{c+id}$
$\displaystyle \therefore |x+iy|^4=\left|\frac{a+ib}{c+id}\right|^2$
$\displaystyle \therefore (x^2+y^2)^2=\frac{a^2+b^2}{c^2+d^2}$
$\\$

$\displaystyle \textbf{Question 4. }\text{If }\pi<\theta<2\pi\text{ and }z=1+\cos\theta+i\sin\theta,\text{ then write the value of }|z|.$
$\displaystyle \text{Answer:}$
$\displaystyle |z|=\sqrt{(1+\cos\theta)^2+\sin^2\theta}$
$\displaystyle =\sqrt{2+2\cos\theta}$
$\displaystyle =\sqrt{4\cos^2\frac{\theta}{2}}$
$\displaystyle =2\left|\cos\frac{\theta}{2}\right|$
$\displaystyle \text{Since }\pi<\theta<2\pi,\text{ we have }\frac{\pi}{2}<\frac{\theta}{2}<\pi$
$\displaystyle \therefore |z|=-2\cos\frac{\theta}{2}$
$\\$

$\displaystyle \textbf{Question 5. }\text{If }n\text{ is any positive integer, write the value of }\frac{i^{4n+1}-i^{4n-1}}{2}.$
$\displaystyle \text{Answer:}$
$\displaystyle i^{4n+1}=i,\qquad i^{4n-1}=i^{-1}=-i$
$\displaystyle \therefore \frac{i^{4n+1}-i^{4n-1}}{2}=\frac{i-(-i)}{2}=i$
$\\$

$\displaystyle \textbf{Question 6. }\text{Write the value of }\frac{i^{592}+i^{590}+i^{588}+i^{586}+i^{584}}{i^{582}+i^{580}+i^{578}+i^{576}+i^{574}}.$
$\displaystyle \text{Answer:}$
$\displaystyle i^{592}+i^{590}+i^{588}+i^{586}+i^{584}=1-1+1-1+1=1$
$\displaystyle i^{582}+i^{580}+i^{578}+i^{576}+i^{574}=-1+1-1+1-1=-1$
$\displaystyle \therefore \text{Required value }=\frac{1}{-1}=-1$
$\\$

$\displaystyle \textbf{Question 7. }\text{Write }1-i\text{ in polar form.}$
$\displaystyle \text{Answer:}$
$\displaystyle z=1-i$
$\displaystyle |z|=\sqrt{1^2+(-1)^2}=\sqrt{2}$
$\displaystyle \theta=\tan^{-1}\left(\frac{-1}{1}\right)=-\frac{\pi}{4}$
$\displaystyle \therefore z=\sqrt{2}\left(\cos\left(-\frac{\pi}{4}\right)+i\sin\left(-\frac{\pi}{4}\right)\right)$
$\\$

$\displaystyle \textbf{Question 8. }\text{Write }-1+i\sqrt3\text{ in polar form.}$
$\displaystyle \text{Answer:}$
$\displaystyle |-1+i\sqrt3|=\sqrt{(-1)^2+(\sqrt3)^2}=2$
$\displaystyle \arg(-1+i\sqrt3)=\frac{2\pi}{3}$
$\displaystyle \therefore -1+i\sqrt3=2\left(\cos\frac{2\pi}{3}+i\sin\frac{2\pi}{3}\right)$
$\\$

$\displaystyle \textbf{Question 9. }\text{Write the argument of }-i.$
$\displaystyle \text{Answer:}$
$\displaystyle -i=\cos\frac{3\pi}{2}+i\sin\frac{3\pi}{2}$
$\displaystyle \therefore \arg(-i)=\frac{3\pi}{2}$
$\\$

$\displaystyle \textbf{Question 10. }\text{Write the least positive integral value of }n\text{ for which }\left(\frac{1+i}{1-i}\right)^n\text{ is real.}$
$\displaystyle \text{Answer:}$
$\displaystyle \frac{1+i}{1-i}=\frac{(1+i)^2}{(1-i)(1+i)}=\frac{1+2i+i^2}{2}=i$
$\displaystyle \therefore \left(\frac{1+i}{1-i}\right)^n=i^n$
$\displaystyle i^n\text{ is real for }n=2,4,6,\ldots$
$\displaystyle \therefore \text{Least positive integral value of }n=2$
$\\$

$\displaystyle \textbf{Question 11. }\text{Find the principal argument of }(1+i\sqrt3)^2.$
$\displaystyle \text{Answer:}$
$\displaystyle 1+i\sqrt3=2\left(\cos\frac{\pi}{3}+i\sin\frac{\pi}{3}\right)$
$\displaystyle (1+i\sqrt3)^2=4\left(\cos\frac{2\pi}{3}+i\sin\frac{2\pi}{3}\right)$
$\displaystyle \therefore \text{Principal argument }=\frac{2\pi}{3}$
$\\$

$\displaystyle \textbf{Question 12. }\text{Find }z,\text{ if }|z|=4\text{ and }\arg(z)=\frac{5\pi}{6}.$
$\displaystyle \text{Answer:}$
$\displaystyle z=4\left(\cos\frac{5\pi}{6}+i\sin\frac{5\pi}{6}\right)$
$\displaystyle =4\left(-\frac{\sqrt3}{2}+\frac{i}{2}\right)$
$\displaystyle =-2\sqrt3+2i$
$\\$

$\displaystyle \textbf{Question 13. }\text{If }|z-5i|=|z+5i|,\text{ then find the locus of }z.$
$\displaystyle \text{Answer:}$
$\displaystyle \text{Let }z=x+iy$
$\displaystyle |z-5i|=|z+5i|$
$\displaystyle |x+i(y-5)|=|x+i(y+5)|$
$\displaystyle x^2+(y-5)^2=x^2+(y+5)^2$
$\displaystyle y=0$
$\displaystyle \therefore \text{Locus of }z\text{ is the real axis.}$
$\\$

$\displaystyle \textbf{Question 14. }\text{If }\frac{(a^2+1)^2}{2a-i}=x+iy,\text{ find the value of }x^2+y^2.$
$\displaystyle \text{Answer:}$
$\displaystyle x^2+y^2=\left|\frac{(a^2+1)^2}{2a-i}\right|^2$
$\displaystyle =\frac{(a^2+1)^4}{|2a-i|^2}$
$\displaystyle =\frac{(a^2+1)^4}{4a^2+1}$
$\\$

$\displaystyle \textbf{Question 15. }\text{Write the value of }\sqrt{-25}\times\sqrt{-9}.$
$\displaystyle \text{Answer:}$
$\displaystyle \sqrt{-25}\times\sqrt{-9}=5i\cdot3i=15i^2=-15$
$\\$

$\displaystyle \textbf{Question 16. }\text{Write the sum of the series }i+i^2+i^3+\ldots\text{ upto }1000\text{ terms.}$
$\displaystyle \text{Answer:}$
$\displaystyle i+i^2+i^3+i^4=i-1-i+1=0$
$\displaystyle \text{Since }1000\text{ terms form }250\text{ complete groups of }4\text{ terms,}$
$\displaystyle \therefore i+i^2+i^3+\cdots+i^{1000}=0$
$\\$

$\displaystyle \textbf{Question 17. }\text{Write the value of }\arg(z)+\arg(\overline{z}).$
$\displaystyle \text{Answer:}$
$\displaystyle \text{If }\arg(z)=\theta,\text{ then }\arg(\overline{z})=-\theta$
$\displaystyle \therefore \arg(z)+\arg(\overline{z})=0$
$\\$

$\displaystyle \textbf{Question 18. }\text{If }|z+4|\leq3,\text{ then find the greatest and least values of } \\ |z+1|.$
$\displaystyle \text{Answer:}$
$\displaystyle |z+4|\leq3$
$\displaystyle \text{represents a circle with centre }(-4,0)\text{ and radius }3$
$\displaystyle |z+1|\text{ is the distance from }(-1,0)$
$\displaystyle \text{Distance between centres }=3$
$\displaystyle \therefore \text{Greatest value of }|z+1|=3+3=6$
$\displaystyle \therefore \text{Least value of }|z+1|=3-3=0$
$\\$

$\displaystyle \textbf{Question 19. }\text{For any two complex numbers }z_1\text{ and }z_2\text{ and any two real numbers }a,b,$
$\displaystyle \text{find the value of }|az_1-bz_2|^2+|az_2+bz_1|^2.$
$\displaystyle \text{Answer:}$
$\displaystyle |az_1-bz_2|^2=(az_1-bz_2)(a\overline{z_1}-b\overline{z_2})$
$\displaystyle =a^2|z_1|^2+b^2|z_2|^2-ab(z_1\overline{z_2}+\overline{z_1}z_2)$
$\displaystyle |az_2+bz_1|^2=(az_2+bz_1)(a\overline{z_2}+b\overline{z_1})$
$\displaystyle =a^2|z_2|^2+b^2|z_1|^2+ab(z_1\overline{z_2}+\overline{z_1}z_2)$
$\displaystyle \therefore |az_1-bz_2|^2+|az_2+bz_1|^2$
$\displaystyle =(a^2+b^2)(|z_1|^2+|z_2|^2)$
$\\$

$\displaystyle \textbf{Question 20. }\text{Write the conjugate of }\frac{2-i}{(1-2i)^2}.$
$\displaystyle \text{Answer:}$
$\displaystyle (1-2i)^2=1-4i+4i^2$
$\displaystyle =1-4i-4=-3-4i$
$\displaystyle \therefore \frac{2-i}{(1-2i)^2}=\frac{2-i}{-3-4i}$
$\displaystyle =\frac{(2-i)(-3+4i)}{(-3-4i)(-3+4i)}$
$\displaystyle =\frac{-6+8i+3i+4}{9+16}$
$\displaystyle =\frac{-2+11i}{25}$
$\displaystyle \therefore \text{Conjugate}=\frac{-2-11i}{25}$
$\\$

$\displaystyle \textbf{Question 21. }\text{If }n\in N,\text{ then find the value of }i^n+i^{n+1}+i^{n+2}+i^{n+3}.$
$\displaystyle \text{Answer:}$
$\displaystyle i^n+i^{n+1}+i^{n+2}+i^{n+3}$
$\displaystyle =i^n(1+i+i^2+i^3)$
$\displaystyle =i^n(1+i-1-i)=0$
$\\$

$\displaystyle \textbf{Question 22. }\text{Find the real value of }a\text{ for which }3i^3-2ai^2+(1-a)i+5\text{ is real.}$
$\displaystyle \text{Answer:}$
$\displaystyle 3i^3-2ai^2+(1-a)i+5$
$\displaystyle =3(-i)-2a(-1)+(1-a)i+5$
$\displaystyle =(2a+5)+(-3+1-a)i$
$\displaystyle =(2a+5)+(-2-a)i$
$\displaystyle \text{For this to be real, }-2-a=0$
$\displaystyle \therefore a=-2$
$\\$

$\displaystyle \textbf{Question 23. }\text{If }|z|=2\text{ and }\arg(z)=\frac{\pi}{4},\text{ find }z.$
$\displaystyle \text{Answer:}$
$\displaystyle z=2\left(\cos\frac{\pi}{4}+i\sin\frac{\pi}{4}\right)$
$\displaystyle =2\left(\frac{1}{\sqrt2}+\frac{i}{\sqrt2}\right)$
$\displaystyle =\sqrt2+i\sqrt2$
$\\$

$\displaystyle \textbf{Question 24. }\text{Write the argument of }(1+\sqrt3)(1+i)(\cos\theta+i\sin\theta).$
$\displaystyle \text{Answer:}$
$\displaystyle \arg(1+\sqrt3)=0$
$\displaystyle \arg(1+i)=\frac{\pi}{4}$
$\displaystyle \arg(\cos\theta+i\sin\theta)=\theta$
$\displaystyle \therefore \arg\{(1+\sqrt3)(1+i)(\cos\theta+i\sin\theta)\}$
$\displaystyle =0+\frac{\pi}{4}+\theta$
$\displaystyle =\theta+\frac{\pi}{4}$
$\\$