If z is a complex number satisfying |z|^2 - |z| - 2 < 0 , then value of |z^2 + z sin θ| , for al

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4-1.Complex numbers

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If $z$ is a complex number satisfying $|z|^2 - |z| - 2 < 0$, then the value of $|z^2 + z sin \theta|$ , for all values of $\theta$ , is

A

equal to $4$

B

equal to $6$

C

more than $6$

D

less than $6$

Solution

$|z|^{2}-|z|-2<0$

$\Rightarrow(|z|-2)(|z|+1)<0 \Rightarrow|z|<2$

Now $\left| {{z^2} + z\sin \theta } \right| \le {\left| z \right|^2} + |z\sin \theta | \le |z{|^2} + |z| < 4 + 2 = 6$

Standard 11

Mathematics

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