Exercise 13.3

QUESTION 1

(i) $-5+12 i$

Sol :

$\sqrt{z}=\pm\left[\sqrt{\dfrac{|z|+\operatorname{Re}(z)}{2}}+i \sqrt{\dfrac{|z|-\operatorname{Re}(z)}{2}}\right]$ $\text { if } \operatorname{Im}(z)>0$

$\sqrt{z}=\pm\left[\sqrt{\dfrac{|z|+\operatorname{Re}(z)}{2}}-i \sqrt{\dfrac{|z|-\operatorname{Re}(z)}{2}}\right]$ $, \text { if } \operatorname{Im}(z)<0$

$z=-5+12 i$ , ${ Re(z) }=-5$ and $|z|=\sqrt{25+144}=13$

$\text{ Here } , \text{ Im(z)}>0$

$\therefore \sqrt{z}=\pm\left[\sqrt{\dfrac{|z|+\operatorname{Re}(z)}{2}}+i \sqrt{\dfrac{|z|-\operatorname{Re}(z)}{2}}\right]$

$=\pm\left[\sqrt{\dfrac{13-5}{2}}+i \sqrt{\dfrac{13+5}{2}}\right]$

$=\pm[\sqrt{4}+i \sqrt{9}]$

$=\pm(2+3 i)$

(ii) $-7-24 i$

Sol :

$z=-7-24 i$ , $\operatorname{Re}(z)=1$ , ṇṇ $|z|=\sqrt{1+1}=\sqrt{2}$

$\text{ Here } , \text{ Im(z)}<0$

$\sqrt{z}=\pm\left[\sqrt{\dfrac{|z|+\operatorname{Re}(z)}{2}}-i \sqrt{\dfrac{|z|-\operatorname{Re}(z)}{2}}\right]$

$=\pm\left[\sqrt{\dfrac{25-7}{2}}-i \sqrt{\dfrac{25+7}{2}}\right]$

$=\pm(3-4 i)$

(iii) $1-{i}$

Sol :

$z=1-i$ , $\operatorname{Re}(z)=1$ , $|z|=\sqrt{1+1}=\sqrt{2}ṇ$

$\text{ Here } , \text{ Im(z)}<0$

$\sqrt{z}=\pm\left[\sqrt{\dfrac{|z|+\operatorname{Re}(z)}{2}}-i \sqrt{\dfrac{|z|-\operatorname{Re}(z)}{2}}\right]$

$=\pm\left[\sqrt{\dfrac{\sqrt{2}+1}{2}}-i \sqrt{\dfrac{\sqrt{2}-1}{2}}\right]$

(iv) $-8-6i$

Sol :

$z=-8-6 i$ , $\operatorname{Re}(z)=-8$ , $|z|=\sqrt{64+36}=10$

$\text{ Here } , \text{ Im(z)}<0$

$\sqrt{z}=\pm\left[\sqrt{\dfrac{|z|+\operatorname{Re}(z)}{2}}-i \sqrt{\dfrac{|z|-\operatorname{Re}(z)}{2}}\right]$

$=\pm\left[\sqrt{\dfrac{10-8}{2}}-i \sqrt{\dfrac{10+8}{2}}\right]$

$=\pm(1-3 i)$

(v) $8-15i$

Sol :

$z=8-15 i$ , $\operatorname{Re}(z)=8ṇ$ , $|z|=\sqrt{64+225}=17$

$\text{ Here } , \text{ Im(z)}<0$

$\sqrt{z}=\pm\left[\sqrt{\dfrac{|z|+\operatorname{Re}(z)}{2}}-i \sqrt{\dfrac{|z|-\operatorname{Re}(z)}{2}}\right]$

$=\pm\left[\sqrt{\dfrac{17+8}{2}}-i \sqrt{\dfrac{17+8}{2}}\right]$

$=\pm \dfrac{1}{\sqrt{2}}(5-3 i)$

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

Sol :

$-11-60 \sqrt{-1}=-11-60 i$ , $\operatorname{Re}(z)=-11$ , $|z|=\sqrt{121+3600}=61$

$\text{ Here } , \text{ Im(z)}<0$

$\sqrt{z}=\pm\left[\sqrt{\dfrac{|z|+\operatorname{Re}(z)}{2}}-i \sqrt{\dfrac{|z|-\operatorname{Re}(z)}{2}}\right]$

$=\pm\left[\sqrt{\dfrac{61-11}{2}}-i \sqrt{\dfrac{61+11}{2}}\right]$

$=\pm(5-6 i)$

(vii) $1+4\sqrt{-3}$

Sol :

$z=1+4 \sqrt{3} \sqrt{-1}=1+4 \sqrt{3} i$ , $\operatorname{Re}(z)=1$ , $|z|=\sqrt{1+16 \times 3}=7$

$\text{ Here } , \text{ Im(z)}>0$

$\sqrt{z}=\pm\left[\sqrt{\dfrac{|z|+\operatorname{Re}(z)}{2}}+i \sqrt{\dfrac{|z|-\operatorname{Re}(z)}{2}}\right]$

$=\pm\left[\sqrt{\dfrac{7+1}{2}}+i \sqrt{\dfrac{7-1}{2}}\right]$

$=\pm(2+\sqrt{3} i)$

(viii) $4i$

Sol :

$z=0+4 i$ , $\operatorname{Re}(z)=0$ , $|z|=4$

$\text{ Here } , \text{ Im(z)}>0$

$\sqrt{z}=\pm\left[\sqrt{\dfrac{|z|+\operatorname{Re}(z)}{2}}+i \sqrt{\dfrac{|z|-\operatorname{Re}(z)}{2}}\right]$

$=\pm\left[\sqrt{\dfrac{4+0}{2}}+i \sqrt{\dfrac{4-0}{2}}\right]$

$=\pm(\sqrt{2}+i \sqrt{2})$

$=\pm \sqrt{2}(1+i)$

(ix) $-i$

Sol :

$z=-i$ , $\operatorname{Re}(z)=0$ , $|z|=1$

$\text{ Here } , \text{ Im(z)}<0$

$\sqrt{z}=\pm\left[\sqrt{\dfrac{|z|+\operatorname{Re}(z)}{2}}-i \sqrt{\dfrac{|z|-\operatorname{Re}(z)}{2}}\right]$

$=\pm\left[\sqrt{\dfrac{1}{2}}-i \sqrt{\dfrac{1}{2}}\right]$

$=\pm \dfrac{1}{\sqrt{2}}(1-i)$

Rd sharma solution of class 11 chapter 13 complex number

Exercise 13.3

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