The least value of $|z|$ where $z$ is complex number which satisfies the inequality $\exp \left(\frac{(|z|+3)(|z|-1)}{||z|+1|} \log _{e} 2\right) \geq \log _{\sqrt{2}}|5 \sqrt{7}+9 i|, i=\sqrt{-1}$, is equal to :

JEE Main · 2021 · 16 Mar 2021 · Shift 2 · Complex Numbers

The least value of

∣ z ∣

where

z

is complex number which satisfies the inequality

exp (\frac{( ∣ z ∣ + 3 ) ( ∣ z ∣ - 1 )}{∣∣ z ∣ + 1∣} lo g_{e} 2) \geq lo g_{2} ∣5 7 + 9 i ∣, i = - 1

, is equal to :

Answer: (B)

On the right,

57 + 9 i = 175 + 81 = 256 = 16

, so

lo g_{2} 16 = \frac{lo g _{e} 16}{lo g _{e} 2} = \frac{4 lo g _{e} 2}{\frac{1}{2} lo g _{e} 2} = 8

On the left, with

r = ∣ z ∣ \geq 0

the exponential is

2^{t}

where

t = \frac{( r + 3 ) ( r - 1 )}{r + 1}

, so the inequality reads

2^{t} \geq 2^{3}

, that is

t \geq 3

. Multiplying by the positive quantity

r + 1

(r + 3) (r - 1) \geq 3 (r + 1) \Rightarrow r^{2} - r - 6 \geq 0 \Rightarrow (r - 3) (r + 2) \geq 0

With

r \geq 0

only

r \geq 3

survives, so the least value of

∣ z ∣

3

JEE Main · 2021 · 16 Mar 2021 · Shift 2 · Single correct option