NTA JEE Main 1st September 2021 Shift 2

The number of pairs $$a, b$$ of real numbers, such that whenever $$\alpha$$ is a root of the equation $$x^2 + ax + b = 0$$, $$\alpha^2 - 2$$ is also a root of this equation, is:

Let $$P_1, P_2, \ldots, P_{15}$$ be 15 points on a circle. The number of distinct triangles formed by points $$P_i, P_j, P_k$$ such that $$i + j + k \neq 15$$, is:

Let $$S_n = 1 \cdot (n-1) + 2 \cdot (n-2) + 3 \cdot (n-3) + \ldots + (n-1) \cdot 1$$, $$n \geq 4$$.
The sum $$\sum_{n=4}^{\infty} \frac{2 S_n}{n!} - \frac{1}{(n-2)!}$$ is equal to:

Let $$a_1, a_2, \ldots, a_{21}$$ be an A.P. such that $$\sum_{n=1}^{20} \frac{1}{a_n a_{n+1}} = \frac{4}{9}$$. If the sum of this A.P. is 189, then $$a_6 a_{16}$$ is equal to:

If $$n$$ is the number of solutions of the equation $$2\cos x \cdot 4\sin\frac{\pi}{4} + x\sin\frac{\pi}{4} - x - 1 = 1$$, $$x \in [0, \pi]$$ and $$S$$ is the sum of all these solutions, then the ordered pair $$(n, S)$$ is:

Consider the parabola with vertex $$\left(\frac{1}{2}, \frac{3}{4}\right)$$ and the directrix $$y = \frac{1}{2}$$. Let P be the point where the parabola meets the line $$x = -\frac{1}{2}$$. If the normal to the parabola at P intersects the parabola again at the point Q, then $$(PQ)^2$$ is equal to:

Let $$\theta$$ be the acute angle between the tangents to the ellipse $$\frac{x^2}{9} + \frac{y^2}{1} = 1$$ and the circle $$x^2 + y^2 = 3$$ at their point of intersection in the first quadrant. Then $$\tan\theta$$ is equal to:

Which of the following is equivalent to the Boolean expression $$p \wedge \sim q$$?

Consider the system of linear equations
$$-x + y + 2z = 0$$
$$3x - ay + 5z = 1$$
$$2x - 2y - az = 7$$
Let $$S_1$$ be the set of all $$a \in R$$ for which the system is inconsistent and $$S_2$$ be the set of all $$a \in R$$ for which the system has infinitely many solutions. If $$nS_1$$ and $$nS_2$$ denote the number of elements in $$S_1$$ and $$S_2$$ respectively, then

$$\cos^{-1}(\cos(-5)) + \sin^{-1}(\sin(6)) - \tan^{-1}(\tan(12))$$ is equal to:
(The inverse trigonometric functions take the principal values)