Let a complex number $\mathrm{z},|\mathrm{z}| \neq 1$, satisfy $\log _{\frac{1}{\sqrt{2}}}\left(\frac{|\mathrm{z}|+11}{(|\mathrm{z}|-1)^{2}}\right) \leq 2$. Then, the largest value of $|\mathrm{z}|$ is equal to.........

JEE Main · 2021 · 16 Mar 2021 · Shift 1 · Complex Numbers

Let a complex number

z, ∣ z ∣ \neq = 1

, satisfy

lo g_{\frac{1}{2}} (\frac{∣ z ∣ + 11}{( ∣ z ∣ - 1 ) ^{2}}) \leq 2

. Then, the largest value of

∣ z ∣

is equal to.........

Answer: (B)

The base

\frac{1}{2}

is less than

1

, so removing the logarithm reverses the inequality:

\frac{∣ z ∣ + 11}{( ∣ z ∣ - 1 ) ^{2}} \geq (\frac{1}{2})^{2} = \frac{1}{2}

With

t = ∣ z ∣

and

(t - 1)^{2} > 0

this is

2 (t + 11) \geq (t - 1)^{2}

, that is

t^{2} - 4 t - 21 \leq 0 ⟹ (t - 7) (t + 3) \leq 0

- 3 \leq t \leq 7

. Together with

t \geq 0

and

t \neq = 1

, the admissible moduli form

[0, 1) \cup (1, 7]

, and the largest value of

∣ z ∣

7

JEE Main · 2021 · 16 Mar 2021 · Shift 1 · Single correct option