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) $-8$
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