KC Sinha Mathematics Solution Class 11 Chapter 11 Complex Numbers Exercise 11.2

 Exercise 11.2

Page no 11.29

Type 1

Question 1

 Find the square root of the following : 

(i) $7-24 i$

(ii) $-15-8 i$

(iii) $i$

(iv) $-i$

(v) $1+i$

(vi) $1-i$

(vii) $-11-60 \sqrt{-1}$

(viii) $-8-6 i$

(ix) $-5+12 i$

(x) $4-6 \sqrt{-5}$

(xi) $6 \sqrt{-2}-7$

(xii) $4 a b-2\left(a^{2}-b^{2}\right) i$

(xiii) $\frac{x^{2}}{y^{2}}+\frac{y^{2}}{x^{2}}-\frac{1}{i}\left(\frac{x}{y}-\frac{y}{x}\right)-\frac{9}{4}$

(xiv) $x^{2}+\frac{1}{x^{2}}+4 i\left(x-\frac{1}{x}\right)-6$

(xv) $a^{2}-1+2 a \sqrt{-1}$

Question 2

Find $\sqrt{2+3 \sqrt{-5}}+\sqrt{2-3 \sqrt{-5}}$

Question 3

Find the value of :

(i) $\omega^{21}$

(ii) $\omega^{18}$

(iii) $\omega^{768}$

(iv) $\omega^{-105}$

(v) $\omega^{-364}$

(vi) $\omega^{-30}$

Question 4

If $\alpha=\frac{-1+\sqrt{-3}}{2}, \beta=\frac{-1-\sqrt{-3}}{2}$, prove that $\frac{\alpha}{\beta}+\frac{\beta}{\alpha}+1=0$

Question 5

If $1, \omega, \omega^{2}$ be three cube roots of 1 , show that :

(i) $\left(1+\omega-\omega^{2}\right)\left(1-\omega+\omega^{2}\right)=4$

(ii) $\left(3+\omega+3 \omega^{2}\right)^{6}=64$

(iii) $\left(1-\omega^{2}+\omega^{4}\right)\left(1+\omega^{2}-\omega^{4}\right)=4$ (iv) $\left(1-\omega+\omega^{2}\right)^{2}+\left(1+\omega-\omega^{2}\right)^{2}=-4$

(v) $\left.(1+\omega)^{3}-(1+\omega)^{2}\right)^{3}=0$

Question 6

Evaluate $\sqrt{-2+2 \sqrt{-2+2 \sqrt{-2+\ldots \text { to } \infty}}}$

Question 7

Show that $\left(\frac{\sqrt{3}+i}{2}\right)^{6}+\left(\frac{i-\sqrt{3}}{2}\right)^{6}=-2$

Question 8

If $1, \omega, \omega^{2}$ be the three cube roots of 1 , then show that :

(i) $(1+\omega)\left(1+\omega^{2}\right)\left(1+\omega^{4}\right)\left(1+\omega^{5}\right)=1$

(ii) $(1+\omega)\left(1+\omega^{2}\right)\left(1+\omega^{4}\right)\left(1+\omega^{8}\right)=1$

(iii) $\left(2+\omega+\omega^{2}\right)^{3}+\left(1+\omega-\omega^{2}\right)^{8}-\left(1-3 \omega+\omega^{2}\right)^{4}=1$

(iv) $\left(1-\omega+\omega^{2}\right)\left(1-\omega^{2}+\omega^{4}\right)\left(1-\omega^{4}+\omega^{8}\right)\left(1-\omega^{8}+\omega^{16}\right)=16$

(v) $(a+b+c)\left(a+b \omega+c \omega^{2}\right)\left(a+b \omega^{2}+c \omega\right)=a^{3}+b^{3}+c^{3}-3 a b c$

Question 9

If $\omega$ be an imaginary cube root of unity, show that

$\frac{a+b \omega+c \omega^{2}}{a \omega+b \omega^{2}+c}=\omega^{2}$

Page no 11.29

Question 10

If $\omega$ be an imaginary cube root of unity, show that $1+\omega^{n}+\omega^{2 n}=0$, for $n=2,4$

Question 11

Resolve into linear factors :

(i) $a^{2}-a b+b^{2}$

(ii) $a^{2}+a b+b^{2}$

(iii) $a^{3}+b^{3}$

(iv) $a^{3}-b^{3}$

(v) $a^{3}+b^{3}+c^{3}-3 a b c$

Question 12

If $x=a+b, y=a \omega+b \omega^{2}$ and $z=a \omega^{2}+b \omega$, where $\omega$ is an imaginary cube root of unity, prove that $x^{2}+y^{2}+z^{2}=6 a b$.

Question 13

If $\omega$ be an imaginary cube root of unity, show that :

$\frac{1}{1+2 \omega}+\frac{1}{2+\omega}-\frac{1}{1+\omega}=0$

Question 14

Find the cube roots of following :

(i) 8

(ii) $-8$