The least value of |z|, where z is complex number which satisfies the inequality exp(((| z |+3)(| z |-1)/|| z |+1|) log e 2) ≥ log √2|5 √7+9 i | i =√-1, is equal to :

Question

The least value of |z|, where z is complex number which satisfies the inequality exp(((| z |+3)(| z |-1)/|| z |+1|) log e 2) ≥ log √2|5 √7+9 i | i =√-1, is equal to : (A) 3 (B) √5 (C) 2 (D) 8

Tardigrade Admin · Accepted Answer

exp (((|z|+3)(|z|-1)/|| z|+1|) ln 2) ≥ log √2|5 √7+9 i| ⇒ 2((|z|+3)(|z|-1)/(|z|+1)) ≥ log √2(16) ⇒ 2((|z|+3)(|z|-1)/(|z|+1)) ≥ 23 ⇒ ((|z|+3)(|z|-1)/(|z|+1)) ≥ 3 ⇒ (|z|+3)(|z|-1) ≥ 3(|z|+1) |z|2+2|z|-3 ≥ 3|z|+3 ⇒ |z|2+|z|-6 ≥ 0 ⇒ (|z|-3)(|z|+2) ≥ 0 ⇒ |z|-3 ≥ 0 ⇒ |z| ≥ 3 ⇒ |z| min =3

Q. 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 :

3

$5$

2

8

Q. The least value of ∣z∣, where z is complex number which satisfies the inequality exp(∣∣z∣+1∣(∣z∣+3)(∣z∣−1)​loge​2)≥log2​​∣57​+9i∣ i=−1​, is equal to :

3

5​

2

8

Q. 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 :

$5$