KC Sinha Mathematics Solution Class 11 Chapter 11 Complex Numbers Exercise 11.3

 Exercise 11.3

Page no 11.50

Type 1

Question 1

Find the modulus of the following :

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

(ii) $\frac{2+i}{4 i+(1+i)^{2}}$

(iii) $\frac{1+i}{1-i}-\frac{1-i}{1+i}$

Question 2

Find the argument of the following :

(i) $-\sqrt{3}-i$

(ii) $\frac{1+i}{1-\sqrt{3} i}$

Page no 11.51

Question 3

Find the modulus and argument of the following complex numbers :

(i) $z=-1-i \sqrt{3}$

(ii) $z=-\sqrt{3}+i$

(iii) $z=\frac{(1+i)^{13}}{(1-i)^{7}}$

(iv) $z=\frac{1}{1+i}$

(v) $z=\frac{1+i}{1-i}$

(vi) $z=\frac{1+2 i}{1-3 i}$

Question 4

Find the conjugate of $\frac{(3-2 i)(2+3 i)}{(1+2 i)(2-i)}$

Question 5

Find real numbers $x$ and $y$ if $(x-i y)(3+5 i)$ is the conjugate of $-6-24 i$.

Question 6

If $|2 z-1|=|z-2|$, prove that $|z|=1$, where $z$ is a complex.

Question 7

Let $z=x+i y$ and $w=\frac{1-i z}{z-i}$, show that if $|w|=1$, then $z$ is real.

Question 8

If $\left|\frac{z-5 i}{z+5 i}\right|=1$, prove that $z$ is real.

Question 9

If $|z|<4$, prove that $|i z+3-4 i|<9$

$[$ Hint : $|i z+(3-4 i)| \leq|i z|+|3-4 i|]$

Question 10

(i) If $|z-1|<3$, prove that $|i z+3-5 i|<8$.

(ii) Prove that $\left|1-\bar{z}_{1} z_{2}\right|^{2}-\left|z_{1}-z_{2}\right|^{2}=\left(1-\left|z_{1}\right|^{2}\right)\left(1-\left|z_{2}\right|^{2}\right)$.

Question 11

If $z_{1}$ and $z_{2}$ are any two complex numbers, show that

$\left|z_{1}+z_{2}\right|^{2}+\left|z_{1}z_{2}\right|^{2}=2\left|z_{1}\right|^{2}+2\left|z_{2}\right|^{2} \text {. }$

Question 12

Solve the equation $|z|+z=2+i$.

Type 2

Question 13

Change the following complex numbers in Cartesian form :

(i) $2\left(\cos 0^{\circ}+i \sin 0^{\circ}\right)$

(ii) $5\left(\cos 270^{\circ}+i \sin 270^{\circ}\right)$

(iii) $4\left(\cos 300^{\circ}+i \sin 300^{\circ}\right)$.

Question 14

Write the number $z=(i-\sqrt{3})^{13}$ in algebraic form.

Question 15

Put the following numbers in the polar form :

(i) $1+i$

(ii) $-1-\sqrt{3} i$

(iii) $1-i$

$($ iv $)-3$

$(\mathrm{v})-1+i$

(vi) $-1-i$

(vii) $\sqrt{3}+t$

(viii) $-4+i 4 \sqrt{3}$

(ix) $i$

$(x)\left(\frac{2+i}{3-i}\right)^{2}$

(xi) $1+i \sqrt{3}$

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

(xiii) $\frac{-16}{1+i \sqrt{3}}$

Question 16

Give the following products in polar form :

(i) $\left[2\left(\cos 0^{\circ}+i \sin 0^{\circ}\right)\right]\left[4\left(\cos 90^{\circ}+i \sin 90^{\circ}\right)\right]$

(ii) $\left[3\left(\cos 225^{\circ}+i \sin 225^{\circ}\right)\right]\left[6\left(\cos 45^{\circ}+i \sin 45^{\circ}\right)\right]$

(iii) $\left[2\left(\cos 210^{\circ}+i \sin 210^{\circ}\right)\right]\left[4\left(\cos 120^{\circ}+i \sin 120^{\circ}\right)\right]$

Question 17

Give the following quotients in polar form :

(i) $\frac{7\left(\cos 135^{\circ}+i \sin 135^{\circ}\right)}{14\left(\cos 90^{\circ}+i \sin 90^{\circ}\right)}$

(ii) $\frac{9\left(\cos 90^{\circ}+i \sin 90^{\circ}\right)}{3\left(\cos 45^{\circ}+i \sin 45^{\circ}\right)}$

Page no 11.52

Question 18

Write the complex number $z=\frac{i-1}{\cos \frac{\pi}{3}+i \sin \frac{\pi}{3}}$ in polar form.

Type 3

Question 19

If $(a+i b)(c+i d)=x+i y$, show that

(i) $(a-i b)(c-i d)=x-i y$

(ii) $\left(a^{2}+b^{2}\right)\left(c^{2}+d^{2}\right)=x^{2}+y^{2}$

Question 20

If $x$ is real and $\frac{1-i x}{1+i x}=m+i n$, show that $m^{2}+n^{2}=1$

Question 21

If $x+i y=\frac{a+i b}{a-i b}$, prove that $x^{2}+y^{2}=1$

Question 22

If $(a+i b)(c+i d)(e+i f)(g+i h)=A+i B$, then show that

$\left(a^{2}+b^{2}\right)\left(c^{2}+d^{2}\right)\left(e^{2}+f^{2}\right)\left(g^{2}+h^{2}\right)=A^{}+B^{2}$

Question 23

If $a+i b=\frac{(x+i)^{2}}{2 x^{2}+1}$, prove that $a^{2}+b^{2}=\frac{\left(x^{2}+1\right)^{2}}{\left(2 x^{2}+1\right)^{2}}$

Question 24

If $\frac{1}{m+i n}-\frac{x-i y}{x+i y}=0$, where $x, y, m, n$ are real and $x+i y \neq 0$ and $m+i n \neq 0$, prove that $m^{2}+n^{2}=1$

Question 25

If $\left(1+i \frac{x}{a}\right)\left(1+i \frac{x}{b}\right)\left(1+i \frac{x}{c}\right) \ldots=A+i B$, then prove that

$\left(1+\frac{x^{2}}{a^{2}}\right)\left(1+\frac{x^{2}}{b^{2}}\right)\left(1+\frac{x^{2}}{c^{2}}\right) \ldots=A^{2}+B^{2}$

Question 26

If $\frac{a-i b}{a+i b}=\frac{1+i}{1-i}$, then show that $a+b=0$

Type 4

Question 27

Express as the sum of two squares :

(i) $\left(1+a^{2}\right)\left(1+b^{2}\right)$

(ii) $\left(1+x^{2}\right)\left(1+y^{2}\right)\left(1+z^{2}\right)$.

Question 28

Show that :

$\left(x^{2}+y^{2}\right)^{4}=\left(x^{4}-6 x^{2} y^{2}+y^{4}\right)^{2}+\left(4 x^{3} y-4 xy^{3}\right)^{2}$