Exercise 11.1
Page no-11.13
Type 1
Question 1
Write the following as complex numbers :
(i) $\sqrt{-27}$
(ii) $\sqrt{-16}$
(iii) $4-\sqrt{-5}$
(iv) $-1-\sqrt{-5}$
(v) $1+\sqrt{-1}$
Question 2
Write the real and imaginary parts of the following complex numbers :
(i) $2-i \sqrt{2}$
(ii) $-\frac{1}{5}+\frac{i}{5}$
(iii) $\frac{\sqrt{5}}{7} i$
(iv) $\sqrt{37}+\sqrt{-19}$
(v) $\frac{\sqrt{37}}{3}+\frac{3}{\sqrt{70}} i$
Question 3
Find the additive inverse of the following :
(i) $-5+7 i$
(ii) $4-3 i$
(iii) 10
Question 4
Find the sum of the following numbers $\frac{2}{3}+\frac{5}{3} i,-\frac{2}{3} i$ and $-\frac{5}{4}-i$.
Question 5
Find the difference of the following complex numbers (i) $-3+2 i$ and $13-i$
(ii) $1-i$ and $-1+6 i$
Page no-11.14
Question 6
Find the product and quotient of the complex numbers $1+i$ and $3+i$.
Question 7
Find multiplicative inverse of the following :
(i) $2+\sqrt{3} i$
(ii) $-3+4 i$
(iii) $-i$
(iv) $4-i 3$
(v) $(\sqrt{5}+i 3)$
(vi) $2-3 i$
Question 8
(i) Prove that $\operatorname{Re}\left(z_{1} z_{2}\right)=\operatorname{Re} z_{1} \operatorname{Re} z_{2}-\operatorname{Im} z_{1} \operatorname{Im} z_{2}$
(ii) Let $z_{1}=2-i, z_{2}=-2+i$, find (a) $\operatorname{Re}\left(\frac{z_{1} z_{2}}{\overline{z_{1}}}\right)$
(b) $\operatorname{Im}\left(\frac{1}{z_{1} \overline{z_{1}}}\right)$
Type 2
Question 9
Express the following in the form $a+i b$ :
(i) $(3+2 i)(3-2 i)$
(ii) $(i-2)^{2}$
(iii) $\frac{2-i}{4+3 i}$
(iv) $\frac{1+2 i+3 i^{2}}{1-2 i+3 i^{2}}$
(v) $\left(\frac{1+i}{1-i}\right)^{2}$
(vi) $\left(\frac{1+2 i}{2+i}\right)^{2}$
(vii) $\frac{6+3 i}{2-i}$
(viii) $\frac{5+\sqrt{2} i}{1-\sqrt{2} i}$
Question 10
Simplify the following :
(i) $2 i^{2}+6 i^{3}+3 i^{16}-6 i^{19}+4 i^{25}$
(ii) $1+i^{10}+i^{110}+i^{1000}$
(iii) $i^{n}+i^{n+1}+i^{n+2}+i^{n+3}$
(iv) $\left\{i^{17}-\left(\frac{1}{i}\right)^{34}\right\}^{2}$
(v) $(-i)^{4 n+3}$, where $n$ is a positive integer.
(vi) $\left(\frac{1+i}{1-i}\right)^{4 n+1}$, where $n$ is a positive integer.
(vii) $(2 i)^{3}$
(viii) (8i) $\left(-\frac{1}{8} i\right)$
(ix) $(5 i)\left(-\frac{3}{5} i\right)$
(x) $(-5 i)\left(\frac{1}{8} i\right)$
(xi) $(-i)(2 i)\left(-\frac{1}{8} i\right)^{3}$
(xii) $i^{-35}$
(xiii) $i^{-39}$
(xiv) $i^{9}+i^{19}$
$(\mathrm{xv})\left[i^{18}+\left(\frac{1}{i}\right)^{25}\right]^{3}$
(xvi) $i^{6}+i^{8}$
(xvii) $i+i^{2}+i^{3}+i^{4}$
(xviii) $i^{12}+i^{13}+i^{14}+i^{15}$
(xix) $i^{4}+i^{8}+i^{12}+i^{16}$
Page no-11.15
Question 11
Write the following in the form $a+i b: \frac{1}{(2+i)^{2}}-\frac{1}{(2-i)^{2}}$.
Question 12
Express the following in the form $a+i b$ :
(i) $\left(\frac{1}{5}+i \frac{2}{5}\right)-\left(4+i \frac{5}{2}\right)$
(ii) $(7-i 2)-(4+i)+(-3+i 5)$
(iii) $\left[\left(\frac{1}{3}+i \frac{7}{3}\right)+\left(4+i \frac{1}{3}\right)\right]-\left(-\frac{4}{3}+i\right)$ (iv) $i^{3}+(6+i 3)-(20+i 5)+(14+i 3)$
(v) $(7+i 5)(7-i 5)$
(vi) $3 i^{3}\left(15 i^{6}\right)$
(vii) $\sqrt{3}+(\sqrt{3}-i 2)-(3-i 2)$
(viii) $(1+i)^{4}$
(ix) $\left(\frac{1}{2}+i 2\right)^{3}$
$(x)\left(-2-i \frac{1}{3}\right)^{3}$
(xi) $3(7+7 i)+i(7+7 i)$
(xii) $(3+5 i)(2+6 i)$
(xiii) $\left(\frac{1}{3}+3 i\right)^{3}$
(xiv) $(5-3 i)^{3}$
(xv) $(1-i)^{4}$
Question 13
Find the following as a single complex number $x+i y$ :
(i) $(\sqrt{6}+i 5)\left(\sqrt{6}-i \frac{1}{5}\right)$
(ii) $(5+i 9)(-3+i 4)$
(iii) $(-2-i 5)(3-i 6)$
(iv) $\left[\left(\sqrt{5}+\frac{i}{2}\right)(\sqrt{5}-i 2)\right]+(6+i 5)$
(v) $\frac{[(\sqrt{2}+i \sqrt{3})+(\sqrt{2}-i \sqrt{3})]}{[(\sqrt{3}+i \sqrt{2})+(\sqrt{3}-i \sqrt{2})]}$
(vi) $\frac{(3+i \sqrt{5})(3-i \sqrt{5})}{(\sqrt{3}+i \sqrt{2})-(\sqrt{3}-i \sqrt{2})}$
(vii) $\left(\frac{1}{1-4 i}-\frac{2}{1+i}\right)\left(\frac{3-4 i}{5+i}\right)$
Question 14
If $\left(\frac{1+i}{1-i}\right)^{m}=1$, then find the least positive integral value of $m$.
Question 15
Find $x$ and $y$ if :
(i) $(x+i y)+(7-5 i)=9+4 i$
(ii) $(x+i y)(2-3 i)=4+i$
(iii) $\left(\frac{3}{\sqrt{5}} x-5\right)+i 2 \sqrt{5} y=\sqrt{2}$
(iv) $4 x+i(3 x-y)=3-i 6$
(v) $(3 y-2)+i(7-2 x)=0$.
Type 3
Question 16
If $a=\frac{1+i}{\sqrt{2}}$, find the value of $a^{6}+a^{4}+a^{2}+1$
Question 17
If $x=\sqrt{-2}-1$, find the value of $x^{4}+4 x^{3}+6 x^{2}+4 x+9$
Question 18
If $x=3+4 i$, find the value of $x^{4}-12 x^{3}+70 x^{2}-204 x+225$
Question 19
If $x=3+2 i$, find the value of $x^{4}-4 x^{3}+4 x^{2}+8 x+39$
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