Complex Numbers and Quadratic Equations

Class 11 NCERT

NCERT

1   Express the given complex number in the form $a+ib:(5i)(-\dfrac{3}{5}i)$

Solution :

$ ( 5 i ) ( - \dfrac {3} {5} i ) \\ = - 5 * \dfrac {3} {5} * i * i \\ = - 3 i^2 \\ = - 3 (-1) \ \ \ \ [ i^2 = - 1 ] \\ = 3 \\ = 3 + i(0)$

2   Express the given complex number in the form $a+ib:i^9+i^{19}$

Solution :

$i^9+i^{19}\\ =i^{4*2+1}+i^{4*4+3}\\ =(i^4)^2.i+(i^4)^4.i^3\\ =1*i+1*(-i) \ \ \ \ \ \ \ [i^4=1,i^3=-i]\\ =i+(-i)\\ =0$

3   Express the given complex number in the form $a+ib:i^{-39}$

Solution :

$i^{-39}=i^{-4*9-3}\\ =(i^4)^{-9}.i^{-3}\\ =(1)^{-9}.i^{-3} \ \ \ [i^4=1]\\ =\dfrac{1}{i^3}=\dfrac{1}{-i} \ \ \ \ \ \ \ \ [i^3=-i]\\ =\dfrac{-1}{i}*\dfrac{i}{i}\\ =\dfrac{-i}{i^2}\\ =\dfrac{-i}{-1}=i \ \ \ \ \ \ [i^2=-1]$

4   Express the given complex number in the form a $A+ib:3(7+i7)+i(7+i7)$

Solution :

$ 3 ( 7 + i 7 ) + i ( 7 + i 7 ) \\ = 21 + 21 i + 7 i + 7 i^2 \\ = 21 + 28 i + 7 * ( -1 ) \ \ \ \ \ \ [ \therefore i^2 = - 1 ] \\ = 14 + 28 i $

5   Express the given complex number in the form $a+ib:(1-i)-(-1+i6).$

Solution :

$ ( 1 - i ) - ( - 1 + i 6 ) \\ = 1 - i + 1 - 6 i \\ = 2 - 7 i $