Let $$B$$ and $$C$$ be the two points on the line $$y+x=0$$ such that $$B$$ and $$C$$ are symmetric with respect to the origin. Suppose $$A$$ is a point on $$y-2 x=2$$ such that $$\triangle A B C$$ is an equilateral triangle. Then, the area of the $$\triangle A B C$$ is

A light ray emits from the origin making an angle 30$$^\circ$$ with the positive $$x$$-axis. After getting reflected by the line $$x+y=1$$, if this ray intersects $$x$$-axis at Q, then the abscissa of Q is

Let $$m_{1}, m_{2}$$ be the slopes of two adjacent sides of a square of side a such that $$a^{2}+11 a+3\left(m_{1}^{2}+m_{2}^{2}\right)=220$$. If one vertex of the square is $$(10(\cos \alpha-\sin \alpha), 10(\sin \alpha+\cos \alpha))$$, where $$\alpha \in\left(0, \frac{\pi}{2}\right)$$ and the equation of one diagonal is $$(\cos \alpha-\sin \alpha) x+(\sin \alpha+\cos \alpha) y=10$$, then $$72\left(\sin ^{4} \alpha+\cos ^{4} \alpha\right)+a^{2}-3 a+13$$ is equal to :

Let $$\mathrm{A}(\alpha,-2), \mathrm{B}(\alpha, 6)$$ and $$\mathrm{C}\left(\frac{\alpha}{4},-2\right)$$ be vertices of a $$\triangle \mathrm{ABC}$$. If $$\left(5, \frac{\alpha}{4}\right)$$ is the circumcentre of $$\triangle \mathrm{ABC}$$, then which of the following is NOT correct about $$\triangle \mathrm{ABC}$$ ?