Question

Assertion :If $z=i+2i^2+3i^3+.............+32i^{32}$, then $z,\overline{z},z$ & $\overline{z}$ forms the vertices of square on argand plane. Reason: $z,\overline{z},z,\overline{z}$ are situated at the same distance from the origin on argand plane.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

$z=i+2i^2+3i^3+...32i^{32}$
And
$iz=i^2+2i^3+3i^4+...31i^{32}+32i^{33}$
Now
$z(1-i)=i+i^2+i^3+..i^{32}-32i^{33}$
$z(1-i)=\frac{i(i^{32}-1)}{i-1}-32i^{32}.i$
$z(1-i)=-32.i$ since $i^{32}=1$
Or
$z=\frac{-32i}{1-i}$
$=-16.\left(i\right)(1+i)$
$=-16(i-1)$
$=16(1-i)$
$=16\sqrt{2}\left[\frac{1-i}{\sqrt{2}}\right]$
$=16\sqrt{2}.e^{-i\frac{\pi }{4}}$
Hence
$z=16\sqrt{2}.e^{-i\frac{\pi }{4}}$
$\overline{z}=16\sqrt{2}.e^{i\frac{\pi }{4}}$.
Now both $z$ and $\overline{z}$ are equidistant from the origin, and origin, $z,\overline{z}$ forms a right angled isosceles triangle.