Let $$S=\left\{z\in \mathbb{C}:\bar{z}=i\left(z^2+\mathrm{Re}(\bar{z})\right)\right\}$$. Then \sum_\limits{z \in \mathrm{S}}|z|^{2} is equal to :

Let $$\mathrm{C}$$ be the circle in the complex plane with centre $${\mathrm{z}}_0=\frac{1}{2}(1+3i)$$ and radius $$r=1$$. Let $${\mathrm{z}}_1=1+\mathrm{i}$$ and the complex number $$z_2$$ be outside the circle $$C$$ such that $$\left|z_1-z_0\right|\left|z_2-z_0\right|=1$$. If $$z_0,z_1$$ and $$z_2$$ are collinear, then the smaller value of $${\left|z_2\right|}^2$$ is equal to :

For $$a\in \mathbb{C}$$, let $$\mathrm{A}=\{z\in \mathbb{C}:\mathrm{Re}(a+\bar{z})>\mathrm{Im}(\bar{a}+z)\}$$ and $$\mathrm{B}=\{z\in \mathbb{C}:\mathrm{Re}(a+\bar{z})(S1):If$$\operatorname{Re}(a), \operatorname{Im}(a) > 0$$,thenthesetAcontainsalltherealnumbers(S2):If$$\operatorname{Re}(a), \operatorname{Im}(a)

Let $$w_1$$ be the point obtained by the rotation of $$z_1=5+4i$$ about the origin through a right angle in the anticlockwise direction, and $$w_2$$ be the point obtained by the rotation of $$z_2=3+5i$$ about the origin through a right angle in the clockwise direction. Then the principal argument of $$w_1-w_2$$ is equal to :

Let S = \left\{ {z = x + iy:{{2z - 3i} \over {4z + 2i}}\,\mathrm{is\,a\,real\,number}} \right\}. Then which of the following is NOT correct?

Let the complex number $$z=x+iy$$ be such that {{2z - 3i} \over {2z + i}} is purely imaginary. If $$x+y^2=0$$, then $$y^4+y^2-y$$ is equal to :

Let $$A=\{\theta \in (0,2\pi ):\frac{1+2i\sin \theta }{1-i\sin \theta }$$ is purely imaginary $$\}$$. Then the sum of the elements in $$\mathrm{A}$$ is :

If for $$z=\alpha +i\beta ,|z+2|=z+4(1+i)$$, then $$\alpha +\beta$$ and $$\alpha \beta$$ are the roots of the equation :

Let $$a\ne b$$ be two non-zero real numbers. Then the number of elements in the set $$X=\{z\in \mathbb{C}:\mathrm{Re}\left(a{z}^2+bz\right)=a$$ and $$\mathrm{Re}\left(b{z}^2+az\right)=b\}$$ is equal to :

The set of all values of $a^2$ for which the line $x+y=0$ bisects two distinct chords drawn from a point $\mathrm{P}\left(\frac{1+a}{2},\frac{1-a}{2}\right)$ on the circle $2x^2+2y^2-(1+a)x-(1-a)y=0$, is equal to :

Let a circle $$C_1$$ be obtained on rolling the circle $$x^2+y^2-4x-6y+11=0$$ upwards 4 units on the tangent $$\mathrm{T}$$ to it at the point $$(3,2)$$. Let $$C_2$$ be the image of $$C_1$$ in $$\mathrm{T}$$. Let $$A$$ and $$B$$ be the centers of circles $$C_1$$ and $$C_2$$ respectively, and $$M$$ and $$N$$ be respectively the feet of perpendiculars drawn from $$A$$ and $$B$$ on the $$x$$-axis. Then the area of the trapezium AMNB is :

Let $$y=x+2,4y=3x+6$$ and $$3y=4x+1$$ be three tangent lines to the circle $$(x-h{)}^2+(y-k{)}^2=r^2$$. Then $$h+k$$ is equal to :

Let the tangents at the points $$A(4,-11)$$ and $$B(8,-5)$$ on the circle $$x^2+y^2-3x+10y-15=0$$, intersect at the point $$C$$. Then the radius of the circle, whose centre is $$C$$ and the line joining $$A$$ and $$B$$ is its tangent, is equal to :

The points of intersection of the line $$ax+by=0,(a\ne b)$$ and the circle $$x^2+y^2-2x=0$$ are $$A(\alpha ,0)$$ and $$B(1,\beta )$$. The image of the circle with AB as a diameter in the line $$x+y+2=0$$ is :

The locus of the mid points of the chords of the circle $$C_1:(x-4{)}^2+(y-5{)}^2=4$$ which subtend an angle $${\theta }_i$$ at the centre of the circle $$C_1$$, is a circle of radius $$r_i$$. If {\theta _1} = {\pi \over 3},{\theta _3} = {{2\pi } \over 3} and $$r_1^2=r_2^2+r_3^2$$, then $${\theta }_2$$ is equal to :

The number of common tangents, to the circles $x^2+y^2-18x-15y+131=0$ and $x^2+y^2-6x-6y-7=0$, is :

Let the centre of a circle C be $$(\alpha ,\beta )$$ and its radius $$r

Let A be the point $$(1,2)$$ and B be any point on the curve $$x^2+y^2=16$$. If the centre of the locus of the point P, which divides the line segment $$\mathrm{AB}$$ in the ratio $$3:2$$ is the point C$$(\alpha ,\beta )$$, then the length of the line segment $$\mathrm{AC}$$ is :

A line segment AB of length $$\lambda$$ moves such that the points A and B remain on the periphery of a circle of radius $$\lambda$$. Then the locus of the point, that divides the line segment AB in the ratio 2 : 3, is a circle of radius :

Let O be the origin and OP and OQ be the tangents to the circle $$x^2+y^2-6x+4y+8=0$$ at the points P and Q on it. If the circumcircle of the triangle OPQ passes through the point \left( {\alpha ,{1 \over 2}} \right), then a value of $$\alpha$$ is :