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The given complex number \(\frac{{1 + 4{\rm{i}}}}{{1 - 2{\rm{i}}}}\) lies in which quadrant?
1. First quadrant
2. Second quadrant
3. Third quadrant
4. Fourth quadrant
5. None of these

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Correct Answer - Option 2 : Second quadrant

Concept:

​Let z = x + iy be a complex number.

Where x is real part of the complex number and y is the imaginary part.

Calculation:

Given: complex number \(\frac{{1 + 4{\rm{i}}}}{{1 - 2{\rm{i}}}}\)

\(\begin{array}{l} \frac{{1 + 4{\rm{i}}}}{{1 - 2{\rm{i}}}} = \frac{{1 + 4{\rm{i}}}}{{1 - 2{\rm{i}}}} \times \frac{{1 + 2{\rm{i}}}}{{1 + 2{\rm{i}}}}\\ = \frac{{1 + 2{\rm{i}} + 4{\rm{i}} - 8}}{{1 + 4}} = - \frac{7}{5} + \frac{{6{\rm{i}}}}{5} \end{array}\)

We can see that the real part is negative and imaginary part is positive. 

Hence the given complex number is in second quadrant.

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