List of practice Questions

Which of the following is the correct representation of the given expression?
\[ \frac{(1+i)^8}{(1-i)^6} \]

Let \(X\) be a discrete random variable, \(\mu\) be its mean, and \(P(X=x)\) be the probability function. The variance of \(X\) is

If \(z_1=8+4i,\; z_2=6+4i\) and \[ \operatorname{Arg}\left(\frac{z-z_1}{z-z_2}\right)=\frac{\pi}{4}, \] then \(z\) satisfies

If $z$ is a complex number such that $|z| + z = 3 + i$ where $i = \sqrt{-1}$, then $|z| =$

In the complex plane, if the points A and B represent $(1 + i)$ and $(-1 + i)$ then the angle between OA and OB is

Let \(z\) be a complex number such that \(Re(z)=3\) and \(Im(z)\neq 0\). If \[ \frac{3z-n}{z+n}=\frac{5-i}{2} \] for a real number \(n\), then \(n-Im(z)= \)

Among the roots of \[ 3\sqrt{3}\,z^3-i=0, \] the sum of the squares of the two roots having non-zero real part is

Given that \[ \sum_{k=1}^{n} k(k-1)=\frac{n(n-1)(n+1)}{3} \] and $\omega$ and $\omega^2$ are complex cube roots of unity. If \[ \sum_{k=1}^{2026} \left( k+\frac{1}{\omega} \right) \left( k+\frac{1}{\omega^2} \right) = \frac{2026}{3}(N+3), \] then $N=$

If \[ \sqrt[3]{i}=cis~\alpha,\qquad \alpha \text{ belongs to second quadrant} \] and \[ \sqrt[3]{-i}=cis~\beta,\qquad \beta \text{ belongs to third quadrant} \] then \[ cis~\alpha+cis~\beta= \]

If \[ z=i^i \] then \[ z^i= \]

If \[ \sqrt{-4x+2i\sqrt{x^{4}+2x^{2}+9}} = \pm(a+ib) \] then \[ a^2+b^2-6= \]

Product of all the five values of \[ (1-i)^{4/5} \] is

If \[ \cos\alpha+\cos\beta+\cos\gamma=0 \] and \[ \sin\alpha+\sin\beta+\sin\gamma=0, \] then which of the following is true?

The modulus of the product of all the values of \( (2+3i)^{3/5} \) is:

If \( \text{Arg}(\frac{z-1}{z+5}) = \pi/3 \) and \( |z-1|=|z+5| \), then \( |z|^2 = \)

Among the solutions of the equation \( x^6 = 64 \), the sum of all those solutions whose real part is negative is:

If \( z_{1}=x+iy \), \( z_{2}=a+ib \) and \( x^{2}+y^{2}=a^{2}+b^{2} \), then \( z_{2} = \)

The locus of \( |z-2i|+|z+4i|=10 \) is

If \( z=(1+\sqrt{3}i)^{4/3} \), then the product of all the values of \( z \) is

If \[ \frac{3}{2+\cos\theta+i\sin\theta}=x+iy, \] then \[ (x-1)(x-3)= \]

If $z_1,z_2$ are two roots of the equation \[ z^2+az+b=0 \] and on the Argand plane the points represented by $z_1,z_2$ and the origin form an equilateral triangle, then $a^2=$

If \[ (1+\sin\alpha+i\cos\alpha)^8 = A(\cos\theta+i\sin\theta), \] then $A$ and $\theta$ are respectively

If $\omega$ is a complex cube root of unity, then $(1 - \omega + \omega^2)^5 + (1 + \omega - \omega^2)^5 =$

If the conjugate of a complex number $z$ is $\frac{1}{z - i}$, then $z$ can be:

If $z = \frac{\sqrt{3} + i}{2}$, then $z^{101} + z^{103} =$