SATHEE: Complex Numbers 2 Question 2

Complex Numbers 2 Question 2

2.

If $a > 0$ and $z = \frac{(1 + i)^{2}}{a - i}$ , has magnitude $\sqrt{\frac{2}{5}}$ , then $\bar{z}$ is equal to

(2019 Main, 10 April I)

(a) $\frac{1}{5} - \frac{3}{5} i$

(b) $- \frac{1}{5} - \frac{3}{5} i$

(d) $- \frac{3}{5} - \frac{1}{5} i$

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Answer:

Correct Answer: 2. (b)

Solution:

The given complex number $z = \frac{(1 + i)^{2}}{a - i}$

$\begin{aligned} = \frac{(1 - 1 + 2 i) (a + i)}{a^{2} + 1} [∵ i^{2} = - 1] \\ = \frac{2 i (a + i)}{a^{2} + 1} = \frac{- 2 + 2 a i}{a^{2} + 1} \\ ∵ | z | = \sqrt{2 / 5} \\ \Rightarrow \sqrt{\frac{4 + 4 a^{2}}{{(a^{2} + 1)}^{2}}} = \sqrt{\frac{2}{5}} \Rightarrow \frac{2}{\sqrt{1 + a^{2}}} = \sqrt{\frac{2}{5}} \\ \Rightarrow \frac{4}{1 + a^{2}} = \frac{2}{5} \Rightarrow a^{2} + 1 = 10 \\ \Rightarrow a^{2} = 9 \Rightarrow a = 3 \\ ∴ z = \frac{- 2 + 6 i}{10} \end{aligned}$

So, $\bar{z} = (\frac{- 2 + 6 i}{10}) = (\frac{- 1}{5} + \frac{3}{5} i)$

$\Rightarrow \bar{z} = \frac{- 1}{5} - \frac{3}{5} i$

Complex Numbers 2 Question 2

2.

Answer:

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Complex Numbers 2 Question 2

2.

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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