NTA JEE Main 31st January 2023 Shift 1

Let $$y = fx$$ represent a parabola with focus $$(-\dfrac{1}{2}, 0)$$ and directrix $$y = -\dfrac{1}{2}$$. Then
$$S = \{x \in \mathbb{R}: \tan^{-1}(\sqrt{fx}) + \sin^{-1}(\sqrt{fx+1}) = \dfrac{\pi}{2}\}$$:

If the domain of the function $$f(x) = \dfrac{x}{1 + x^2}$$, where $$x$$ is greatest integer $$\le x$$, is $$[2, 6)$$, then its range is

Let $$y = f x = \sin^3\dfrac{\pi}{3}\cos\dfrac{\pi}{3\sqrt{2}} - 4x^3 + 5x^2 + 1^{\frac{3}{2}}$$. Then, at $$x = 1$$,

A wire of length 20 m is to be cut into two pieces. A piece of length $$\ell_1$$ is bent to make a square of area $$A_1$$ and the other piece of length $$\ell_2$$ is made into a circle of area $$A_2$$. If $$2A_1 + 3A_2$$ is minimum then $$\pi\ell_1 : \ell_2$$ is equal to:

Let $$\alpha \in (0, 1)$$ and $$\beta = \log_e(1 - \alpha)$$. Let $$P_n(x) = x + \dfrac{x^2}{2} + \dfrac{x^3}{3} + \ldots + \dfrac{x^n}{n}$$, $$x \in (0, 1)$$. Then the integral $$\int_0^{\alpha} \dfrac{t^{50}}{1-t} dt$$ is equal to

The value of $$\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \dfrac{2 + 3\sin x}{\sin x(1 + \cos x)} dx$$ is equal to

Let a differentiable function $$f$$ satisfy $$f(x) + \int_3^x \dfrac{f(t)}{t} dt = \sqrt{x+1}$$, $$x \ge 3$$. Then $$12f(8)$$ is equal to:

Let $$\vec{a} = 2\hat{i} + \hat{j} + \hat{k}$$, and $$\vec{b}$$ and $$\vec{c}$$ be two nonzero vectors such that $$\vec{a} + \vec{b} + \vec{c} = \vec{a} + \vec{b} - \vec{c}$$ and $$\vec{b} \cdot \vec{c} = 0$$. Consider the following two statements:
A: $$\vec{a} + \lambda\vec{c} \ge \vec{a}$$ for all $$\lambda \in \mathbb{R}$$.
B: $$\vec{a}$$ and $$\vec{c}$$ are always parallel.

Let the shortest distance between the lines L: $$\dfrac{x-5}{-2} = \dfrac{y-\lambda}{0} = \dfrac{z+\lambda}{1}$$, $$\lambda \ge 0$$ and L$$_1$$: $$x+1 = y-1 = 4-z$$ be $$2\sqrt{6}$$. If $$(\alpha, \beta, \gamma)$$ lies on L, then which of the following is NOT possible?

A bag contains 6 balls. Two balls are drawn from it at random and both are found to be black. The probability that the bag contains at least 5 black balls is