NTA JEE Main 24th June 2022 Shift 2

If $$y = \tan^{-1}\left(\sec x^3 - \tan x^3\right), \frac{\pi}{2} < x^3 < \frac{3\pi}{2}$$, then

The number of distinct real roots of the equation $$x^7 - 7x - 2 = 0$$ is

Let $$\lambda^*$$ be the largest value of $$\lambda$$ for which the function $$f_\lambda(x) = 4\lambda x^3 - 36\lambda x^2 + 36x + 48$$ is increasing for all $$x \in \mathbb{R}$$. Then $$f_{\lambda^*}(1) + f_{\lambda^*}(-1)$$ is equal to:

The value of the integral $$\int_{-\pi/2}^{\pi/2} \frac{dx}{(1+e^x)(\sin^6 x + \cos^6 x)}$$ is equal to

$$\lim_{n \to \infty} \left(\frac{n^2}{(n^2+1)(n+1)} + \frac{n^2}{(n^2+4)(n+2)} + \frac{n^2}{(n^2+9)(n+3)} + \cdots + \frac{n^2}{(n^2+n^2)(n+n)}\right)$$ is equal to

The slope of normal at any point $$(x, y), x > 0, y > 0$$ on the curve $$y = y(x)$$ is given by $$\frac{x^2}{xy - x^2y^2 - 1}$$. If the curve passes through the point $$(1, 1)$$, then $$e \cdot y(e)$$ is equal to

Let $$\mathbf{a}$$ and $$\mathbf{b}$$ be two unit vectors such that $$|\mathbf{a} + \mathbf{b}| + 2|\mathbf{a} \times \mathbf{b}| = 2$$. If $$\theta \in (0, \pi)$$ is the angle between $$\hat{a}$$ and $$\hat{b}$$, then among the statements:
$$(S1) : 2|\hat{a} \times \hat{b}| = |\hat{a} - \hat{b}|$$
$$(S2)$$ : The projection of $$\hat{a}$$ on $$(\hat{a} + \hat{b})$$ is $$\frac{1}{2}$$

If the shortest distance between the lines $$\frac{x-1}{2} = \frac{y-2}{3} = \frac{z-3}{\lambda}$$ and $$\frac{x-2}{1} = \frac{y-4}{4} = \frac{z-5}{\frac{1}{\sqrt{3}}}$$, then the sum of all possible values of $$\lambda$$ is:

Let the points on the plane $$P$$ be equidistant from the points $$(-4, 2, 1)$$ and $$(2, -2, 3)$$. Then the acute angle between the plane $$P$$ and the plane $$2x + y + 3z = 1$$ is

A random variable $$X$$ has the following probability distribution: