NTA JEE Main 6th April 2023 Shift 2

For the system of equations
$$x + y + z = 6$$
$$x + 2y + \alpha z = 10$$
$$x + 3y + 5z = \beta$$, which one of the following is NOT true?

Let the sets $$A$$ and $$B$$ denote the domain and range respectively of the function $$f(x) = \dfrac{1}{\sqrt{[x] - x}}$$, where $$[x]$$ denotes the smallest integer greater than or equal to $$x$$. Then among the statements
(S1): $$A \cap B = (1, \infty) - \mathbb{N}$$ and
(S2): $$A \cup B = (1, \infty)$$

Let $$f(x)$$ be a function satisfying $$f(x) + f(\pi - x) = \pi^2$$, $$\forall x \in \mathbb{R}$$. Then $$\int_0^\pi f(x) \sin x \, dx$$ is equal to

The area bounded by the curves $$y = |x-1| + |x-2|$$ and $$y = 3$$ is equal to

If the solution curve $$f(x, y) = 0$$ of the differential equation $$(1 + \log_e x)\dfrac{dx}{dy} - x\log_e x = e^y$$, $$x > 0$$, passes through the points (1, 0) and $$(a, 2)$$, then $$a^a$$ is equal to

Let the vectors $$\vec{a}, \vec{b}, \vec{c}$$ represent three coterminous edges of a parallelopiped of volume $$V$$. Then the volume of the parallelopiped, whose coterminous edges are represented by $$\vec{a}, \vec{b}+\vec{c}$$ and $$\vec{a}+2\vec{b}+3\vec{c}$$ is equal to

The sum of all values of $$\alpha$$, for which the points whose position vectors are $$\hat{i} - 2\hat{j} + 3\hat{k}$$, $$2\hat{i} - 3\hat{j} + 4\hat{k}$$, $$(\alpha+1)\hat{i} + 2\hat{k}$$ and $$9\hat{i} + (\alpha-8)\hat{j} + 6\hat{k}$$ are coplanar, is equal to

Let the line $$L$$ pass through the point (0, 1, 2), intersect the line $$\dfrac{x-1}{2} = \dfrac{y-2}{3} = \dfrac{z-3}{4}$$ and be parallel to the plane $$2x + y - 3z = 4$$. Then the distance of the point $$P(1, -9, 2)$$ from the line $$L$$ is

A plane $$P$$ contains the line of intersection of the plane $$\vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 6$$ and $$\vec{r} \cdot (2\hat{i} + 3\hat{j} + 4\hat{k}) = -5$$. If $$P$$ passes through the point (0, 2, -2), then the square of distance of the point (12, 12, 18) from the plane P is

Three dice are rolled. If the probability of getting different numbers on the three dice is $$\dfrac{p}{q}$$, where $$p$$ and $$q$$ are co-prime, then $$q - p$$ is equal to