KC Sinha Mathematics Solution Class 11 Chapter 11 Complex Numbers Exercise 11.1

 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 $x^{4}-4 x^{3}+4 x^{2}+8 x+39$