List of top Mathematics Questions on Complex numbers

If \[ \sum_{n=1}^{10}(i^n+i^{n+1}+i^{n+2}) = x+iy, \] then $2x+3y=$

If one of the values of \[ \sqrt{-1-\sqrt{3}i} \] is \(\alpha+i\beta\), where \(\alpha<0\) and \(\beta>0\), then \(\alpha=\)

\[ \frac{\left(1+\sin\frac{4\pi}{9}-i\cos\frac{4\pi}{9}\right)^6} {\left(1+\sin\frac{4\pi}{9}+i\cos\frac{4\pi}{9}\right)^6} = \ ? \]

One of the \(5^{\text{th}}\) roots of \(\omega\) is

If $\omega$ is a complex cube root of unity, evaluate $\left(\frac{\sqrt{3}-i}{-\sqrt{2}+\sqrt{2}i}\right)^{20}$.}

If \(z_1\) and \(z_2\) are two complex numbers such that \[ |z_1-a|=|z_2-a| \] for \(a\in\mathbb R\), and \[ Arg(z_1-a)+Arg(z_2-a)=\frac{\pi}{2}, \] then \[ \frac{z_1-a}{z_2-a}= \]

If a complex number \(\alpha\) is a common root of \[ x^{2026}+x^{1964}+1=0 \] and \[ x^3+2x^2+2x+1=0, \] then the sum of the complex roots of the equation \[ z^3=\alpha^3 \] is

If \( z = x+iy \) and the point \( P \) denotes \( z \) in the Argand plane, then the locus of \( P \) satisfying the condition \( \text{Im}\left(\frac{z-3i}{z+2}\right) = 1, z \neq -2 \) is:

If \( \alpha \) and \( \beta \) are the roots of the quadratic equation \( x^2 - 2x + 2 = 0 \), then \( \alpha^{2026} + \beta^{2026} = \)

Number of values of a complex number \( z \) satisfying the condition \( z = z^2 + i\text{Im}(\overline{z}) \) is:

One of the values of \[ (\sqrt{3}-i)^{5/3} \] is:

$m,n$ and $k$ are integers and $9.5\le n\le12$. If \[ \frac{(\cos\theta+i\sin\theta)^m} {(\sin\theta+i\cos\theta)^n} = k(\sin17\theta-i\cos17\theta), \] then $n-m-k=$

If \[ A=\{z=x+iy:|z-4||z-3|\}, \] \[ B=\{z=x+iy:-3\le y\le3,\;x\in N,\;y\in N\}, \] and \(C=A\cap B\), then \(n(C)=\)

If \(z\) is a complex number such that \[ \left|9z+\frac{1}{z}\right|=8, \] then the maximum value of \(|z|\) is

If \(\omega\) is a complex cube root of unity, then \[ (1+\omega)(1+\omega^2)(1+\omega^4)(1+\omega^5)(1+\omega^7)(1+\omega^8)\cdots \] (2n factors) is equal to

If the curve represented by the locus of a point in the Argand plane corresponding to the complex number \(z\) satisfying the relation \[ \operatorname{Re}\!\left(\frac{z+4}{2z-5}\right) + \operatorname{Re}\!\left(\frac{\bar z+4}{2\bar z-5}\right) =4 \] cuts the \(X\)-axis at two points \(A\) and \(B\), then the sum of the abscissae of those two points is

Which of the following is not a possible value of \[ \sqrt{i}+\sqrt{-i}? \]

If \(z = \sum _{n = 0}^{2026}i^n\), where \(i = \sqrt{-1}\), then one of the values of \(\sqrt{z}\) is...

\(\frac{(cos2θ+isin2θ)^7}{(cos4θ+isin4θ)^3} =\)

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

If \(ω\) is a complex cube root of unity, then the value of the expression \(2(1+\frac{1}{ω})(1+\frac{1}{ω^2})+3(2+\frac{1}{ω})(2+\frac{1}{ω^2})+\ldots +(n+1)(n+\frac{1}{ω})(n+\frac{1}{ω^2})\) is...

If \(x = \sqrt{-1-\sqrt{-1-\sqrt{-1-\ldots \infty }}}\), where \(ω\) is a non-real complex cube root of unity, then the value of \(x\) is...

The polar form of the complex number \(z = \frac{1}{1+i}\), (where \(i = \sqrt{-1}\)) is

If \(α\) and \(β\) are the roots of the equation \(x^2+x+1 = 0\) then \(α^{2026}+β^{2026} =\)

If \(-1+\sqrt{-3} = re^{iθ}\), then the value of \(θ\) is