Let a complex number , satisfy . Then, the largest value of is equal to

Question

Let a complex number $z,$ $| z |$ $\neq 1$ , satisfy $\log_{\sqrt{2}} (\frac{| z | + 11}{{(| z | - 1)}^{2}}) \geq 0$ . Then, the largest value of $| z |$ is equal to

Smart Achievers · Accepted Answer

(1)

$\log_{\sqrt{2}} (\frac{| z | + 11}{| z |^{2} - 2 | z | + 1}) \geq 0 \Rightarrow \frac{| z | + 11}{| z |^{2} - 2 | z | + 1} \geq 1$

$\Rightarrow | z |^{2} - 2 | z | + 1 \leq | z | + 11$

$\Rightarrow (| z | - 5) (| z | + 2) \leq 0 \Rightarrow | z | \leq 5$ $[∵ | z | + 2 > 0]$

Solution

Q.
Let a complex number $z,$ $| z |$ $\neq 1$ , satisfy $\log_{\sqrt{2}} (\frac{| z | + 11}{{(| z | - 1)}^{2}}) \geq 0$ . Then, the largest value of $| z |$ is equal to

1 5

2 2

3 0

4 1

Ans.
(1)

$\log_{\sqrt{2}} (\frac{| z | + 11}{| z |^{2} - 2 | z | + 1}) \geq 0 \Rightarrow \frac{| z | + 11}{| z |^{2} - 2 | z | + 1} \geq 1$

$\Rightarrow | z |^{2} - 2 | z | + 1 \leq | z | + 11$

$\Rightarrow (| z | - 5) (| z | + 2) \leq 0 \Rightarrow | z | \leq 5$ $[∵ | z | + 2 > 0]$

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Solution

Q. Let a complex number z, |z|≠1, satisfy log2(|z|+11(|z|-1)2)≥0. Then, the largest value of |z| is equal to

1 5

2 2

3 0

4 1

Ans. (1) log2(|z|+11|z|2-2|z|+1)≥0⇒|z|+11|z|2-2|z|+1≥1 ⇒|z|2-2|z|+1≤|z|+11 ⇒(|z|-5)(|z|+2)≤0⇒|z|≤5 [∵|z|+2>0]

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Q.
Let a complex number $z,$ $| z |$ $\neq 1$ , satisfy $\log_{\sqrt{2}} (\frac{| z | + 11}{{(| z | - 1)}^{2}}) \geq 0$ . Then, the largest value of $| z |$ is equal to

Ans.
(1)

$\log_{\sqrt{2}} (\frac{| z | + 11}{| z |^{2} - 2 | z | + 1}) \geq 0 \Rightarrow \frac{| z | + 11}{| z |^{2} - 2 | z | + 1} \geq 1$

$\Rightarrow | z |^{2} - 2 | z | + 1 \leq | z | + 11$

$\Rightarrow (| z | - 5) (| z | + 2) \leq 0 \Rightarrow | z | \leq 5$ $[∵ | z | + 2 > 0]$