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.