Let $$\mathrm{C}$$ be a circle with radius $$\sqrt{10}$$ units and centre at the origin. Let the line $$x+y=2$$ intersects the circle $$\mathrm{C}$$ at the points $$\mathrm{P}$$ and $$\mathrm{Q}$$. Let $$\mathrm{MN}$$ be a chord of $$\mathrm{C}$$ of length 2 unit and slope $$-1$$. Then, a distance (in units) between the chord PQ and the chord $$\mathrm{MN}$$ is

The area (in sq. units) of the region $$S=\{z \in \mathbb{C}:|z-1| \leq 2 ;(z+\bar{z})+i(z-\bar{z}) \leq 2, \operatorname{lm}(z) \geq 0\}$$ is

If the value of the integral $$\int_{-1}^1 \frac{\cos \alpha x}{1+3^x} d x$$ is $$\frac{2}{\pi}$$.Then, a value of $$\alpha$$ is

Let three real numbers $$a, b, c$$ be in arithmetic progression and $$a+1, b, c+3$$ be in geometric progression. If $$a>10$$ and the arithmetic mean of $$a, b$$ and $$c$$ is 8, then the cube of the geometric mean of $$a, b$$ and $$c$$ is