If $\alpha +i\beta$ and $\gamma +i\delta$ are the roots of $x^2-(3-2i)x-(2i-2)=0$, $i=\sqrt{-1}$, then $\alpha \gamma +\beta \delta$ is equal to:

Let $\left|\frac{\bar{z}-i}{2\bar{z}+i}\right|=\frac{1}{3},z\in C$, be the equation of a circle with center at $C$. If the area of the triangle, whose vertices are at the points $(0,0), C$ and $(\alpha ,0)$ is 11 square units, then ${\alpha }^2$ equals:

The number of complex numbers $z$, satisfying $|z|=1$ and $\left|\frac{z}{\bar{z}}+\frac{\bar{z}}{z}\right|=1$, is :

If $\alpha$ and $\beta$ are the roots of the equation $2z^2-3z-2i=0$, where $i=\sqrt{-1}$, then $16\cdot \mathrm{Re}\left(\frac{{\alpha }^{19}+{\beta }^{19}+{\alpha }^{11}+{\beta }^{11}}{{\alpha }^{15}+{\beta }^{15}}\right)\cdot \mathrm{lm}\left(\frac{{\alpha }^{19}+{\beta }^{19}+{\alpha }^{11}+{\beta }^{11}}{{\alpha }^{15}+{\beta }^{15}}\right)$ is equal to

Let $O$ be the origin, the point $A$ be $z_1=\sqrt{3}+2\sqrt{2}i$, the point $B\left(z_2\right)$ be such that $\sqrt{3}\left|z_2\right|=\left|z_1\right|$ and $\arg \left(z_2\right)=\arg \left(z_1\right)+\frac{\pi }{6}$. Then

Let $A=\left\{\theta \in [0,2\pi ]:1+10\mathrm{Re}\left(\frac{2\cos \theta +i\sin \theta }{\cos \theta -3i\sin \theta }\right)=0\right\}$. Then $\sum _{\theta \in A}{\theta }^2$ is equal to

Let $z$ be a complex number such that $|z|=1$. If $\frac{2+{\mathrm{k}}^{2}z}{\mathrm{k}+\bar{z}}=\mathrm{k}z,\mathrm{k}\in \mathbf{R}$, then the maximum distance of $\mathrm{k}+i{\mathrm{k}}^2$ from the circle $|z-(1+2 i)|=1$ is :

If the locus of z ∈ ℂ, such that Re$\left(\frac{z-1}{2z+i}\right)+\text{Re}\left(\frac{\overline{z}-1}{2\overline{z}-i}\right)=2$, is a circle of radius r and center $(a, b)$, then $\frac{15ab}{r^2}$ is equal to :

Among the statements (S1) : The set $\{z\in \mathbb{C}-\{-i\}:|z|=1$ and $\frac{z-i}{z+i}$ is purely real $\}$ contains exactly two elements, and (S2) : The set $\{z\in \mathbb{C}-\{-1\}:|z|=1$ and $\frac{z-1}{z+1}$ is purely imaginary $\}$ contains infinitely many elements.

Let $z\in C$ be such that $\frac{z^2+3i}{z-2+i}=2+3i$. Then the sum of all possible values of $z^2$ is :

Let the product of ${\omega }_1=(8+i)\sin \theta +(7+4i)\cos \theta$ and ${\omega }_2=(1+8i)\sin \theta +(4+7i)\cos \theta$ be $\alpha +i\beta$, $i=\sqrt{-1}$. Let p and q be the maximum and the minimum values of $\alpha +\beta$ respectively. Then $\mathrm{p}+\mathrm{q}$ is equal to :

Let a circle C pass through the points (4, 2) and (0, 2), and its centre lie on 3x + 2y + 2 = 0. Then the length of the chord, of the circle C, whose mid-point is (1, 2), is:

Let the line x+y=1 meet the circle $x^2+y^2=4$ at the points A and B. If the line perpendicular to AB and passing through the mid-point of the chord AB intersects the circle at C and D, then the area of the quadrilateral ABCD is equal to :

A circle C of radius 2 lies in the second quadrant and touches both the coordinate axes. Let r be the radius of a circle that has centre at the point $(2,5)$ and intersects the circle $C$ at exactly two points. If the set of all possible values of r is the interval $(\alpha ,\beta )$, then $3\beta -2\alpha$ is equal to :

Let circle $C$ be the image of $x^2+y^2-2x+4y-4=0$ in the line $2 x-3 y+5=0$ and $A$ be the point on $C$ such that $O A$ is parallel to $x$-axis and $A$ lies on the right hand side of the centre $O$ of $C$. If $B(\alpha ,\beta )$, with $$\beta$

Let the equation of the circle, which touches $x$-axis at the point $(a, 0), a>0$ and cuts off an intercept of length $b$ on $y-a x i s$ be $x^2+y^2-\alpha x+\beta y+\gamma =0$. If the circle lies below $x-a x i s$, then the ordered pair $\left(2a,b^2\right)$ is equal to

Let $C_1$ be the circle in the third quadrant of radius 3 , that touches both coordinate axes. Let $C_2$ be the circle with centre $(1,3)$ that touches ${\mathrm{C}}_1$ externally at the point $(\alpha ,\beta )$. If $(\beta -\alpha {)}^2=\frac{m}{n}$ , $\gcd (m,n)=1$, then $m+n$ is equal to

If the four distinct points $(4,6),(-1,5),(0,0)$ and $(k, 3 k)$ lie on a circle of radius $r$, then $10k+r^2$ is equal to

Let the ellipse $E_1:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, $a > b$ and $E_2:\frac{x^2}{A^2}+\frac{y^2}{B^2}=1$, $A