COMEDK · Maths · 3. Complex Number
The conjugate of the multiplicative inverse of the complex number \(z = \dfrac{1 + 7i}{3 + i}\) is:
- A \(1 + 2i\)
- B \(\dfrac{1}{5} + \dfrac{2}{5}i\)
- C \(\dfrac{2}{5} + \dfrac{1}{5}i\)
- D \(\dfrac{1}{5} - \dfrac{2}{5}i\)
Answer & Solution
Correct Answer
(B) \(\dfrac{1}{5} + \dfrac{2}{5}i\)
Step-by-step Solution
Detailed explanation
\(z = \dfrac{1 + 7i}{3 + i}\) Multiplying the numerator and denominator by the conjugate of the denominator: \(z = \dfrac{(1 + 7i)(3 - i)}{(3 + i)(3 - i)}\) \(z = \dfrac{3 - i + 21i - 7i^2}{3^2 - i^2}\) \(z = \dfrac{3 + 20i + 7}{9 + 1}\) \(z = \dfrac{10 + 20i}{10} = 1 + 2i\) The…
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