NTA JEE Main 28th July 2022 Shift 2

The function $$f(x) = xe^{x(1-x)}$$, $$x \in \mathbb{R}$$, is

The sum of the absolute maximum and absolute minimum values of the function $$f(x) = \tan^{-1}(\sin x - \cos x)$$ in the interval $$[0, \pi]$$ is

Let $$I_n(x) = \int_0^x \frac{1}{(t^2+5)^n} dt$$, $$n = 1, 2, 3, \ldots$$. Then

The area enclosed by the curves $$y = \log_e(x+e^2)$$, $$x = \log_e\left(\frac{2}{y}\right)$$, above the line $$x = \log_e 2$$ and $$y = 1$$ is

Let $$y = y(x)$$ be the solution curve of the differential equation $$\frac{dy}{dx} + \frac{1}{x^2-1}y = \left(\frac{x-1}{x+1}\right)^{1/2}$$, $$x > 1$$ passing through the point $$\left(2, \sqrt{\frac{1}{3}}\right)$$. Then $$\sqrt{7}y(8)$$ is equal to

The differential equation of the family of circles passing through the points (0, 2) and (0, -2) is

Let S be the set of all $$a \in \mathbb{R}$$ for which the angle between the vectors $$\vec{u} = a(\log_e b)\hat{i} - 6\hat{j} + 3\hat{k}$$ and $$\vec{v} = (\log_e b)\hat{i} + 2\hat{j} + 2a(\log_e b)\hat{k}$$, $$(b > 1)$$ is acute. Then S is equal to

Let the lines $$\frac{x-1}{\lambda} = \frac{y-2}{1} = \frac{z-3}{2}$$ and $$\frac{x+26}{-2} = \frac{y+18}{3} = \frac{z+28}{\lambda}$$ be coplanar and P be the plane containing these two lines. Then which of the following points does NOT lie on P?

A plane P is parallel to two lines whose direction ratios are $$(-2, 1, -3)$$, and $$(-1, 2, -2)$$ and it contains the point $$(2, 2, -2)$$. Let P intersect the co-ordinate axes at the points A, B, C making the intercepts $$\alpha, \beta, \gamma$$. If V is the volume of the tetrahedron OABC, where O is the origin and $$p = \alpha + \beta + \gamma$$, then the ordered pair $$(V, p)$$ is equal to

Let A and B be two events such that $$P(B|A) = \frac{2}{5}$$, $$P(A|B) = \frac{1}{7}$$ and $$P(A \cap B) = \frac{1}{9}$$. Consider $$(S_1): P(A' \cup B) = \frac{5}{6}$$, $$(S_2): P(A' \cap B') = \frac{1}{18}$$. Then