JEE PYQ: Complex Numbers Question 27
Question 27 - 2020 (09 Jan Shift 2)
If $z$ be a complex number satisfying $|\text{Re}(z)| + |\text{Im}(z)| = 4$, then $|z|$ cannot be:
(1) $\sqrt{\frac{17}{2}}$
(2) $\sqrt{10}$
(3) $\sqrt{7}$
(4) $\sqrt{8}$
Show Answer
Answer: (3)
Solution
$|x| + |y| = 4$. $|z|^2 = x^2 + y^2$. With constraint $|x|+|y| = 4$: min of $x^2+y^2$ is 8 (at $|x|=|y|=2$) and max is 16 (at $|x|=4, y=0$). So $|z| \in [2\sqrt{2}, 4]$. $\sqrt{7} < 2\sqrt{2}$, so $|z|$ cannot be $\sqrt{7}$.
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