NTA JEE Main 3rd September 2020 Shift 2

Let $$R_1$$ and $$R_2$$ be two relations defined as follows:
$$R_1 = \{(a, b) \in R^2 : a^2 + b^2 \in Q\}$$ and $$R_2 = \{(a, b) \in R^2 : a^2 + b^2 \notin Q\}$$, where Q is the set of all rational numbers, then

Let A be a $$3 \times 3$$ matrix such that adj $$A = \begin{bmatrix} 2 & -1 & 1 \\ -1 & 0 & 2 \\ 1 & -2 & -1 \end{bmatrix}$$ and $$B = $$ adj(adjA). If $$|A| = \lambda$$ and $$\left|(B^{-1})^T\right| = \mu$$, then the ordered pair $$(|\lambda|, \mu)$$ is equal to

Suppose $$f(x)$$ is a polynomial of degree four having critical points at -1, 0, 1. If $$T = \{x \in R | f(x) = f(0)\}$$, then the sum of squares of all the elements of $$T$$ is:

If the surface area of a cube is increasing at a rate of 3.6 cm$$^2$$/sec, retaining its shape; then the rate of change of its volume (in cm$$^3$$/sec), when the length of a side of the cube is 10 cm, is:

If $$\int \sin^{-1}\left(\frac{\sqrt{x}}{1+x}\right)dx = A(x)\tan^{-1}(\sqrt{x}) + B(x) + C$$, where C is a constant of integration, then the ordered pair $$(A(x), B(x))$$ can be:

If the value of the integral $$\int_0^{\frac{1}{2}}\frac{x^2}{(1-x^2)^{\frac{3}{2}}}dx$$ is $$\frac{k}{6}$$, then $$k$$ is equal to:

If $$x^3 dy + xy \cdot dx = x^2 dy + 2y dx$$; $$y(2) = e$$ and $$x > 1$$, then $$y(4)$$ is equal to:

Let $$a, b, c \in R$$ be such that $$a^2 + b^2 + c^2 = 1$$. If $$a\cos\theta = b\cos\left(\theta + \frac{2\pi}{3}\right) = c\cos\left(\theta + \frac{4\pi}{3}\right)$$, where $$\theta = \frac{\pi}{9}$$, then the angle between the vectors $$a\hat{i} + b\hat{j} + c\hat{k}$$ and $$b\hat{i} + c\hat{j} + a\hat{k}$$ is:

The plane which bisects the line joining the points (4, -2, 3) and (2, 4, -1) at right angles also passes through the point:

The probability that a randomly chosen 5-digit number is made from exactly two digits is: