Complex Numbers

If $\alpha$ is a root of $x^4=1$ with negative principal argument, then the principle argument of $\Delta(A)$, where $\Delta(A)=\left|\begin{array}{ccc}1 & 1 & 1 \\ \alpha^n & \alpha^{n+1} & \alpha^{n+3} \\ \frac{1}{\alpha^{n+1}} & \frac{1}...

If $a$ is the root of equation $z^n+2 z^{n-1}+3 z^{n-2}+12-18 z=0$ which lies inside $|z|=1$, then

The complex number $z$ satisfying $z+|z|$ $=1+7 i$, then the value of $|z|^2$ equals

Let \(z_k=\cos \left(\frac{2 k \pi}{10}\right)+i \sin \left(\frac{2 k \pi}{10}\right) ; k=1,2, \ldots \ldots \ldots\) 9, then \(\frac{1}{10}\left\{\left|1-z_1\right|\left|1-z_2\right| \ldots .\left|1-z_a\right|\right\}\) equals to

If \(x+\frac{1}{x}=1\) and \(p=x^{4000}+\frac{1}{x^{4000}}\) and \(q\) is the digit at unit place in the number \(2^{2 n}+1\), then the value of \((p+q)\) is equal to

If \(\alpha, \beta\) and \(\gamma\) are the cube roots of \(P,(P<0)\), then for any \(x, y\) and \(z\) which does not make denominator zero, the expression \(\frac{x \alpha+y \beta+z \gamma}{x \beta+y \gamma+z \alpha}\) equals to

If \(\alpha\) is a non -real fifth root of unity, then the value of \(3^{\left|1+\alpha+\alpha^2+\alpha^{-2}-\alpha^{-1 \mid}\right|}\), is

If \(z_1, z_2\) and \(z_3\) are the vertices \(A, B\) and \(C\) respectively of an isosceles right angled triangle with right angled at \(C\), then \(\left(z_1-z_3^{\prime}\right)\left(z_2-z_3\right)\) equals to

The condition in order that \(Z_1, Z_2, Z_3\) are vertices of an isosceles triangle right angled at \(z_2\), is

The non-zero solutions of the equation \(z^2+|z|=0\), where \(z\) is a complex number, are

If \((1+i)(2 i+1)(1+3 i) \ldots(1+n i)=x+i y\), then \(2 \cdot 5 \cdot 10 \ldots\left(1+n^2\right)\) is equal to