MCA Entrance Previous Year Question Bank

Let $\arg (z)$ represent the principal argument of the complex number $z$. Then, $|z|=3$ and $\arg (z-1)-\arg (z+1)=\frac{\pi }{4}$ intersect.

Let $\alpha$ be a root of the equation $1+x^2+x^4=0$. Then the value of ${\alpha }^{1011}+{\alpha }^{2022}-{\alpha }^{3033}$ is equal to:

Let $S=\{z\in C:|z-2|\le 1,z(1+i)+\bar{z}(1-i)\le 2\}$. Let $|z-4i|$ attains minimum and maximum values, respectively, at $z_1\in S$ and $z_2\in S$. If $5(|z_1{|}^2+|z_2{|}^2)=\alpha +\beta \sqrt{5}$, where $\alpha$ and $\beta$ are integers, then the value of $\alpha +\beta$ is equal to:

Let $\alpha$ and $\beta$ be the roots of the equation $x^2+(2i-1)=0$. Then, the value of $|{\alpha }^8+{\beta }^8|$ is equal to:

Let $a,b$ be two real numbers such that ab < 0. If the complex number $\frac{1+ai}{b+i}$ is of unit modulus and $a+ib$ lies on the circle $|z-1|=|2z|$, then a possible value of $\frac{1+[a]}{4b}$, where $[t]$ is the greatest integer function, is:

If the center and radius of the circle $\left|\frac{z-2}{z-3}\right|=2$ are respectively $(\alpha ,\beta )$ and $\gamma$, then $3(\alpha +\beta +\gamma )$ is equal to:

Let $S=\{x:x\in \mathbb{R}\text{ and }(\sqrt{3}+\sqrt{2}{)}^{x^2-4}+(\sqrt{3}-\sqrt{2}{)}^{x^2-4}=10\}$. Then $n(S)$ is equal to:

The complex number $z=\frac{i-1}{\cos \frac{\pi }{3}+i\sin \frac{\pi }{3}}$ is equal to:

For all $z\in \mathbb{C}$ on the curve $C_1:|z|=4$, let the locus of the point $z+\frac{1}{z}$ be the curve $C_2$. Then:

Let $z=1+i$ and $z_1=\frac{1+i\bar{z}}{\bar{z}(1-z)+\frac{1}{z}}$. Then $\frac{12}{\pi }\text{arg}(z_1)$ is equal to: