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Complex Numbers Part 1

Section: Mathematics

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Question : 29 of 100

Marks: +1, -0

Let a complex number

z,|z| \neq 1

\log_{\frac{1}{\sqrt{2}}}\left(\frac{|z|+11}{(|z|-1)^2}\right) \leq 2

. Then, the largest value of

|z|

[16 Mar 2021 Shift 1]

8
7
6
5

Solution:

Given,

\log_{\frac{1}{\sqrt{2}}}\left(\frac{|z|+11}{(|z|-1)^2}\right) \leq 2, |z| \neq 1

As,

\log_a x

is a decreasing function when

0 < a < 1

. So,

\log_a(x_1) \leq k

,

x_1 \geq a^k

Similarly,

\frac{|z|+11}{(|z|-1)^2} \geq \left(\frac{1}{\sqrt{2}}\right)^2

\Rightarrow \frac{|z|+11}{(|z|-1)^2} \geq \frac{1}{2}

\Rightarrow 2|z|+22 \geq (|z|-1)^2

\Rightarrow |z|^2 + 1 - 2|z| - 2|z| - 22 \leq 0

\Rightarrow |z|^2 - 4|z| - 21 \leq 0

\Rightarrow (|z|+3)(|z|-7) \leq 0

0 \leq |z| \leq 7

\text{So}, |z|_{\max} = 7

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