Complex Numbers - Algebra - Mathematics Previous Year Questions

The amplitude of \({(1 + i)^5}\) is

The imaginary part of \(i^i\) is

The conjugate of the complex number \({{{{(1 + i)}^2}} \over {1 - i}}\) is

If \(1,\omega ,{\omega ^2}\) are the cube roots of unity, then \((1 + \omega )(1 + {\omega ^2})(1 + {\omega ^4})(1 + {\omega ^8})\) is equal to

What is the argument of the complex number \({{(1 + i)(2 + i)} \over {3 - i}}\), where \(i = \sqrt { - 1}\) ?

Evaluate \({\left[ {{i^{18}} + {{\left( {{1 \over i}} \right)}^{25}}} \right]^3}\).

If \({(\sqrt 3 + i)^{100}} = {2^{99}}(a + ib)\), then \({a^2} + {b^2}\) is equal to

\({(i + \sqrt 3 )^{100}} + {(i - \sqrt 3 )^{100}} + {2^{100}}\) is equal to

The argument of \({{1 - i\sqrt 3 } \over {1 + i\sqrt 3 }}\) is

Evaluate \({\left[ {{i^{22}} + {{\left( {{1 \over i}} \right)}^{25}}} \right]^3}\)

If the conjugate of \((x+i y)(1-2 i)\) be \(1+i\), then

$$\text { The value of } \frac{i^{1004}+i^{1006}+i^{1008}+i^{1010}+i^{1012}}{i^{510}+i^{508}+i^{506}+i^{504}+i^{502}} \text { is }$$

$$\text { If }(1-4 i)^3=a+i b \text { then the value of } \mathrm{a} \text { and } \mathrm{b} \text { is }$$

$$\text { The modulus of the following complex number } \frac{1+i}{1-i}-\frac{1-i}{1+i} \text { is }$$

If \(\left(\frac{3}{2}+i \frac{\sqrt{3}}{2}\right)^{50}=3^{25}(x+i y)\), where \(x\) and \(y\) are real, then the ordered pair \((2 x, 2 y)\) is

If \(z=\sqrt{3}+i\), then the argument of \(z^2 e^{z-i}\) is equal to

If \(i=\sqrt{-1}\) and \(n\) is a positive integer, then \(i^n+i^{n+1}+i^{n+2}+i^{n+3}\) is equal to