NTA JEE Main 26th August 2021 Shift 2

If $$(\sqrt{3} + i)^{100} = 2^{99}(p + iq)$$, then $$p$$ and $$q$$ are roots of the equation:

A 10 inches long pencil $$AB$$ with mid point $$C$$ and a small eraser $$P$$ are placed on the horizontal top of a table such that $$PC = \sqrt{5}$$ inches and $$\angle PCB = \tan^{-1}(2)$$. The acute angle through which the pencil must be rotated about $$C$$ so that the perpendicular distance between eraser and pencil becomes exactly 1 inch is:

The value of $$$2\sin\left(\frac{\pi}{8}\right)\sin\left(\frac{2\pi}{8}\right)\sin\left(\frac{3\pi}{8}\right)\sin\left(\frac{5\pi}{8}\right)\sin\left(\frac{6\pi}{8}\right)\sin\left(\frac{7\pi}{8}\right)$$$ is:

A circle $$C$$ touches the line $$x = 2y$$ at the point $$(2, 1)$$ and intersects the circle $$C_1 : x^2 + y^2 + 2y - 5 = 0$$ at two points $$P$$ and $$Q$$ such that $$PQ$$ is a diameter of $$C_1$$. Then the diameter of $$C$$ is:

The point $$P\left(-2\sqrt{6}, \sqrt{3}\right)$$ lies on the hyperbola $$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$$ having eccentricity $$\frac{\sqrt{5}}{2}$$. If the tangent and normal at $$P$$ to the hyperbola intersect its conjugate axis at the points $$Q$$ and $$R$$ respectively, then $$QR$$ is equal to:

The locus of the mid points of the chords of the hyperbola $$x^2 - y^2 = 4$$, which touch the parabola $$y^2 = 8x$$, is:

$$\lim_{x \to 2}\left(\sum_{n=1}^{9} \frac{x}{n(n+1)x^2 + 2(2n+1)x + 4}\right)$$ is equal to:

Consider the two statements:
$$(S_1) : (p \rightarrow q) \vee (\sim q \rightarrow p)$$ is a tautology.
$$(S_2) : (p \wedge \sim q) \wedge (\sim p \vee q)$$ is a fallacy.
Then:

Let $$A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 0 \end{bmatrix}$$. Then $$A^{2025} - A^{2020}$$ is equal to:

Two fair dice are thrown. The numbers on them are taken as $$\lambda$$ and $$\mu$$, and a system of linear equations
$$x + y + z = 5$$
$$x + 2y + 3z = \mu$$
$$x + 3y + \lambda z = 1$$
is constructed. If $$p$$ is the probability that the system has a unique solution and $$q$$ is the probability that the system has no solution, then: