AP EAMCET · Maths · Complex Number
If \(z=(1-i)^3(x+i)\) is a purely imaginary number for \(x=x_1\) and if \(\mathrm{z}\) is a purely real number for \(\mathrm{x}=\mathrm{x}_2\), then \(\mathrm{x}_1 \mathrm{x}_2=\)
- A \(-1\)
- B \(0\)
- C \(1\)
- D \(2\)
Answer & Solution
Correct Answer
(A) \(-1\)
Step-by-step Solution
Detailed explanation
\(\text {} \begin{aligned} z & =(1-i)^3(x+i) \\ \Rightarrow \quad z & =(1+i-3 i(1-i))(x+i) \\ & =(-2 i-2)(x+i) \\ \Rightarrow \quad & z=(2-2 x)-i(2+2 x) \end{aligned}\) If \(z\) is purely imaginary: \(2-2 x=0\) \(\Rightarrow x=1 \quad \therefore x_1=1\) If \(z\) is purely real:…
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