NTA JEE Main 26th August 2021 Shift 2

If $$\sum_{r=1}^{50} \tan^{-1} \frac{1}{2r^2} = p$$, then the value of $$\tan p$$ is:

The domain of the function $$\operatorname{cosec}^{-1}\left(\frac{1+x}{x}\right)$$ is:

Let $$[t]$$ denote the greatest integer less than or equal to $$t$$. Let $$f(x) = x - [x]$$, $$g(x) = 1 - x + [x]$$, and $$h(x) = \min\{f(x), g(x)\}$$, $$x \in [-2, 2]$$. Then $$h$$ is:

The local maximum value of the function, $$f(x) = \left(\frac{2}{x}\right)^{x^2}$$, $$x \gt 0$$, is:

The value of $$\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \left(\frac{1 + \sin^2 x}{1 + \pi^{\sin x}}\right) dx$$ is:

If the value of the integral $$\int_0^5 \frac{x + [x]}{e^{x-[x]}} dx = \alpha e^{-1} + \beta$$, where $$\alpha, \beta \in R$$, $$5\alpha + 6\beta = 0$$, and $$[x]$$ denotes the greatest integer less than or equal to $$x$$; then the value of $$(\alpha + \beta)^2$$ is equal to:

Let $$y(x)$$ be the solution of the differential equation $$2x^2 dy + (e^y - 2x)dx = 0$$, $$x > 0$$. If $$y(e) = 1$$, then $$y(1)$$ is equal to:

A hall has a square floor of dimension 10 m $$\times$$ 10 m (see the figure) and vertical walls. If the angle GPH between the diagonals AG and BH is $$\cos^{-1}\frac{1}{5}$$, then the height of the hall (in meters) is:

Let $$P$$ be the plane passing through the point $$(1, 2, 3)$$ and the line of intersection of the planes $$\vec{r} \cdot (\hat{i} + \hat{j} + 4\hat{k}) = 16$$ and $$\vec{r} \cdot (-\hat{i} + \hat{j} + \hat{k}) = 6$$. Then which of the following points does NOT lie on $$P$$?

A fair die is tossed until six is obtained on it. Let $$X$$ be the number of required tosses, then the conditional probability $$P(X \geq 5 \mid X \gt 2)$$ is: