NTA JEE Main 1st September 2021 Shift 2

The range of the function $$f(x) = \log_{\sqrt{5}}\left(3 + \cos\frac{3\pi}{4} + x + \cos\frac{\pi}{4} + x + \cos\frac{\pi}{4} - x - \cos\frac{3\pi}{4} - x\right)$$ is:

The function $$f(x) = x^3 - 6x^2 + ax + b$$ is such that $$f(2) = f(4) = 0$$. Consider two statements:
$$S_1$$: there exists $$x_1, x_2 \in (2, 4)$$, $$x_1 \lt x_2$$, such that $$f'(x_1) = -1$$ and $$f'(x_2) = 0$$.
$$S_2$$: there exists $$x_3, x_4 \in (2, 4)$$, $$x_3 \lt x_4$$, such that $$f$$ is decreasing in $$(2, x_4)$$, increasing in $$(x_4, 4)$$ and $$2f'(x_3) = \sqrt{3}f(x_4)$$. Then

Let $$f : R \rightarrow R$$ be a continuous function. Then $$\lim_{x \to \pi/4} \frac{\frac{\pi}{4}\int_2^{\sec^2 x} f(x) dx}{x^2 - \frac{\pi^2}{16}}$$ is equal to:

Let $$I_{n,m} = \int_0^{1/2} \frac{x^n}{x^m-1} dx$$, $$\forall n > m$$ and $$n, m \in N$$. Consider a matrix $$A = a_{ij_{3 \times 3}}$$ where $$a_{ij} = \begin{cases} I_{6+i,3} - I_{i+3,3}, & i \leq j \\ 0, & i > j \end{cases}$$. Then adj $$A^{-1}$$ is:

The function $$f(x)$$, that satisfies the condition $$f(x) = x + \int_0^{\pi/2} \sin x \cos y f(y) dy$$, is:

The area, enclosed by the curves $$y = \sin x + \cos x$$ and $$y = |\cos x - \sin x|$$ and the lines $$x = 0$$, $$x = \frac{\pi}{2}$$, is:

If $$y = y(x)$$ is the solution curve of the differential equation $$x^2 dy + (y - \frac{1}{x}) dx = 0$$; $$x > 0$$ and $$y(1) = 1$$, then $$y\left(\frac{1}{2}\right)$$ is equal to:

The distance of line $$3y - 2z - 1 = 0 = 3x - z + 4$$ from the point $$(2, -1, 6)$$ is:

Let the acute angle bisector of the two planes $$x - 2y - 2z + 1 = 0$$ and $$2x - 3y - 6z + 1 = 0$$ be the plane P. Then which of the following points lies on P?

Two squares are chosen at random on a chessboard (see figure). The probability that they have a side in common is: