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4-1.Complex numbers
hard
Let $z_1$ and $z_2$ be any two non-zero complex numbers such that $3\left| {{z_1}} \right| = 4\left| {{z_2}} \right|$. If $z = \frac{{3{z_1}}}{{2{z_2}}} + \frac{{2{z_2}}}{{3{z_1}}}$ then
A
Re$(z) = 0$
B
$\left| z \right| = \sqrt {\frac{5}{2}} $
C
$\left| z \right| = \frac{1}{2}\sqrt {\frac{{17}}{2}} $
D
Im$(z) \neq 0$
(JEE MAIN-2019)
Solution
$\left|\frac{3 z_{1}}{2 z_{2}}\right|=2$
Let $\frac{3 z_{1}}{2 z_{2}}=2 \cos \theta+2(\sin \theta) i$
$\Rightarrow \frac{2 z_{2}}{3 z_{1}}=\frac{1}{2} \cos \theta-\frac{1}{2}(\sin \theta)$
Given, $z=\frac{2 z_{1}}{3 z_{2}}+\frac{3 z_{2}}{2 z_{1}}$ $=\frac{5}{2} \cos \theta+\frac{3}{2}(\sin \theta) i$
Which is neither purely real nor purely imaginary and $|z|$ depends on $\theta$.
Standard 11
Mathematics
