Consider the following statements : 1. z₁z₂z₃ is purely imaginary. 2. z₁z₂+z₂z₃+z₃z₁ is purely real Which of the above statements is/are correct [NDA I 2016]

Correct Answer is : Both 1 and 2

Correct Answer is : Both 1 and 2

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UPSC NDA I 2016 Mathematics Solved Paper

For. the next two (2) items that follow :

Let

z_1,z_2

and

z_3

be non zero complex numbers satisfying

z^2=\overset{-}{iz},

where

i=\sqrt{-1}

[NDA I 2016]

Question : 49 of 120

Marks: +1, -0

z_1z_2z_3

z_1z_2+z_2z_3+z_3z_1

Solution:

z_{1}z_{2}z_{3}=(-i)\left(\; \;\frac{\sqrt{3}+i}{2}\right)\left(\; \;\frac{-\sqrt{3}+i}{2}\right)

=\; \;\frac{-i}{4}(i^{2}-(\sqrt{3})^{2})

=\; \;\frac{-i}{4}(-3-1)=i

Hence

z_1z_2z_3

is purely imaginary

z_{1}z_{2}=-i\left(\; \;\frac{\sqrt{3}+i}{2}\right)=\; \;\frac{-\sqrt{3}i+1}{2}

z_{2}z_{3}=\; \;\frac{(\sqrt{3}+i)(-\sqrt{3}+i)}{4}

=\left(\; \;\frac{-3-\sqrt{3}i+\sqrt{3}i+i^{2}}{4}\right)=-1

z_{3}z_{1}=\; \;\frac{(-\sqrt{3}+i)(-i)}{2}=\; \;\frac{\sqrt{3}i+1}{2}

z_{1}z_{2}+z_{2}z_{3}+z_{3}z_{1}=\left(\; \;\frac{-\sqrt{3}i+1}{2}\right)+(-1)+\left(\; \;\frac{\sqrt{3}i+1}{2}\right)

=\left(\; \;\frac{-\sqrt{3}i+1+\sqrt{3}i+1}{2}\right)-1

=0\in R

Hence

z_{1}z_{2}+z_{2}z_{3}+z_{3}z_{1}=0

is purely real

Hence both statements are correct.

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