Rd sharma solution of class 11 chapter 13 complex number

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)

Leave a Reply

Your email address will not be published. Required fields are marked *