Master cadre maths 2024 important MCQ on complex numbers

 Find the square root of the complex number \(3 + 4i\):

[A] 1 + 2i

[B] 2 + i

[C] 2 - i

[D] 1 - 2i

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Solution:

Let the square root of \(3 + 4i\) be \(a + bi\), where \(a\) and \(b\) are real numbers.

Squaring both sides:
\((a + bi)^2 = 3 + 4i\)

Expanding and equating the real and imaginary parts:

Real Part: \(a^2 - b^2 = 3\)

Imaginary Part: \(2ab = 4\) => \(ab = 2\)

From the second equation, if \(a = 1\), then \(b = 2\), or if \(a = 2\), then \(b = 1\).

Plugging these into the first equation, the pair \(a = 2\) and \(b = 1\) satisfies the equation \(a^2 - b^2 = 3\).

So, the square root of \(3 + 4i\) is:

\(2 + i\)

**Correct Answer: [B] 2 + i