NTA JEE Main 2nd September 2020 Shift 2

Which of the following is a tautology?

Let $$A = \left\{X = (x, y, z)^T : PX = 0 \text{ and } x^2 + y^2 + z^2 = 1\right\}$$ where $$P = \begin{bmatrix} 1 & 2 & 1 \\ -2 & 3 & -4 \\ 1 & 9 & -1 \end{bmatrix}$$ then the set $$A$$:

Let $$a, b, c \in R$$ be all non-zero and satisfies $$a^3 + b^3 + c^3 = 2$$. If the matrix $$A = \begin{bmatrix} a & b & c \\ b & c & a \\ c & a & b \end{bmatrix}$$ satisfies $$A^TA = I$$, then a value of $$abc$$ can be:

Let $$f : R \to R$$ be a function which satisfies $$f(x + y) = f(x) + f(y)$$, $$\forall x, y \in R$$. If $$f(1) = 2$$ and $$g(n) = \sum_{k=1}^{(n-1)} f(k)$$, $$n \in N$$ then the value of $$n$$, for which $$g(n) = 20$$, is:

The equation of the normal to the curve $$y = (1+x)^{2y} + \cos^2(\sin^{-1}x)$$, at $$x = 0$$ is:

Let $$f : (-1, \infty) \to R$$ be defined by $$f(0) = 1$$ and $$f(x) = \frac{1}{x}\log_e(1 + x)$$, $$x \ne 0$$. Then the function $$f$$:

Consider a region $$R = \{(x, y) \in R^2 : x^2 \le y \le 2x\}$$. If a line $$y = \alpha$$ divides the area of region $$R$$ into two equal parts, then which of the following is true?

If a curve $$y = f(x)$$, passing through the point $$(1, 2)$$, is the solution of the differential equation $$2x^2dy = (2xy + y^2)dx$$, then $$f\left(\frac{1}{2}\right)$$ is equal to:

A plane passing through the point $$(3, 1, 1)$$ contains two lines whose direction ratios are 1, -2, 2 and 2, 3, -1 respectively. If this plane also passes through the point $$(\alpha, -3, 5)$$, then $$\alpha$$ is equal to:

Let $$E^C$$ denote the complement of an event $$E$$. Let $$E_1$$, $$E_2$$ and $$E_3$$ be any pairwise independent events with $$P(E_1) > 0$$ and $$P(E_1 \cap E_2 \cap E_3) = 0$$ then $$P\left((E_2^C \cap E_3^C)/E_1\right)$$ is equal to: