JEE Mains · Maths · STD 11 - 4.1 complex nubers
Let \(z \in C\) be such that \(\left| z \right| < 1\). If \(w = \frac{{5 + 3z}}{{5\,\left( {1 - z} \right)}}\), then
- A \(5\,lm\,\left( w \right) < 1\)
- B \(4\,lm\,\left( w \right) > 5\)
- C \(5\,{\mathop{\rm Re}\nolimits} \,\left( w \right) > 1\)
- D \(5\,{\mathop{\rm Re}\nolimits} \,\left( w \right) > 4\)
Answer & Solution
Correct Answer
(C) \(5\,{\mathop{\rm Re}\nolimits} \,\left( w \right) > 1\)
Step-by-step Solution
Detailed explanation
\(W=\frac{5+3 z}{5(1-z)}\) \(\Rightarrow 5 w-5 w z=5+3 z\) \(\Rightarrow z=\frac{5 w-5}{3+5 w}\) given \(|z|<1\) \(\Rightarrow\left|\frac{5 w-5}{3+5 w}\right|<1\) \(\Rightarrow|5 w-5|<|3+5 w|\) \(\Rightarrow|w-1|<\left|\frac{3}{5}+w\right|\)
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