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Question

The modulus of the complex number \(\left( {\frac{{3 + 4i}}{{1 - 2i}}} \right)\) is 

The correct answer is

\(\sqrt 5\)

The modulus of a quotient is the quotient of moduli: |3+4i|/|1−2i|=5/√5=√5.

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Important Questions from Complex Variables

  1. Let $z = x + iy$ be a complex variable and $\bar{z}$ be its complex conjugate. The equation$z^2 + \bar{z}^2 = 2$ represents a
  2. Consider the complex function $f(z) = \cos z + e^{z^2}$. The coefficient of $z^5$ in the Taylor series expansion of $f(z)$ about the origin is ________ (rounded off to 1 decimal place).
  3. Let $f(z) = j \frac{1-z}{1+z}$, where z denotes a complex number and j denotes $\sqrt{-1}$. The inverse function $f^{-1}(z)$ maps the real axis to the ______.
  4. The contour integral $$\oint_C e^{1/z} dz$$ with C as the counter-clockwise unit circle in the z-plane is equal to

  5. Let $$ I = \int_0^1 \frac{1}{1+t} dt + \frac{\pi i}{2} \int_0^1 \frac{1}{1+e^2} dt - i \int_0^1 \frac{1}{1+it} dt, $$ where $ t $ is a real variable and $ i = \sqrt{-1} $. The value of $ I $ is ______________

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