NTA JEE Main 17th March 2021 Shift 2

Let $$S_1, S_2$$ and $$S_3$$ be three sets defined as
$$S_1 = \{z \in \mathbb{C} : |z-1| \leq \sqrt{2}\}$$,
$$S_2 = \{z \in \mathbb{C} : Re((1-i)z) \geq 1\}$$ and
$$S_3 = \{z \in \mathbb{C} : Im(z) \leq 1\}$$.
Then, the set $$S_1 \cap S_2 \cap S_3$$:

If the sides $$AB$$, $$BC$$ and $$CA$$ of a triangle $$ABC$$ have 3, 5 and 6 interior points respectively, then the total number of triangles that can be constructed using these points as vertices, is equal to:

The value of $$\sum_{r=0}^{6} {^{6} C_{r}} \cdot {^{6} C_{6-r}}$$ is equal to:

The number of solutions of the equation $$x + 2\tan x = \frac{\pi}{2}$$ in the interval $$[0, 2\pi]$$ is:

Two tangents are drawn from a point $$P$$ to the circle $$x^2 + y^2 - 2x - 4y + 4 = 0$$, such that the angle between these tangents is $$\tan^{-1}\left(\frac{12}{5}\right)$$, where $$\tan^{-1}\left(\frac{12}{5}\right) \in (0, \pi)$$. If the centre of the circle is denoted by $$C$$ and these tangents touch the circle at points $$A$$ and $$B$$, then the ratio of the areas of $$\triangle PAB$$ and $$\triangle CAB$$ is:

Let the tangent to the circle $$x^2 + y^2 = 25$$ at the point $$R(3, 4)$$ meet $$x$$-axis and $$y$$-axis at point $$P$$ and $$Q$$, respectively. If $$r$$ is the radius of the circle passing through the origin $$O$$ and having centre at the incentre of the triangle $$OPQ$$, then $$r^2$$ is equal to:

Let $$L$$ be a tangent line to the parabola $$y^2 = 4x - 20$$ at $$(6, 2)$$. If $$L$$ is also a tangent to the ellipse $$\frac{x^2}{2} + \frac{y^2}{b} = 1$$, then the value of $$b$$ is equal to:

The value of $$\lim_{n \to \infty} \frac{[r] + [2r] + \ldots + [nr]}{n^2}$$, where $$r$$ is non-zero real number and $$[r]$$ denotes the greatest integer less than or equal to $$r$$, is equal to:

The value of the limit $$\lim_{\theta \to 0} \frac{\tan(\pi\cos^2\theta)}{\sin(2\pi\sin^2\theta)}$$ is equal to:

If the Boolean expression $$(p \wedge q) \circledast (p \otimes q)$$ is a tautology, then $$\circledast$$ and $$\otimes$$ are respectively given by: