NTA JEE Main 27th August 2021 Shift 2

If $$y(x) = \cot^{-1}\left(\frac{\sqrt{1+\sin x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}}\right)$$, $$x \in \left(\frac{\pi}{2}, \pi\right)$$, then $$\frac{dy}{dx}$$ at $$x = \frac{5\pi}{6}$$ is:

A box open from top is made from a rectangular sheet of dimension $$a \times b$$ by cutting squares each of side $$x$$ from each of the four corners and folding up the flaps. If the volume of the box is maximum, then $$x$$ is equal to:

Let $$M$$ and $$m$$ respectively be the maximum and minimum values of the function $$f(x) = \tan^{-1}(\sin x + \cos x)$$ in $$\left[0, \frac{\pi}{2}\right]$$. Then the value of $$\tan(M - m)$$ is equal to:

The value of the integral $$\int_0^1 \frac{\sqrt{x} dx}{(1+x)(1+3x)(3+x)}$$ is:

The area of the region bounded by the parabola $$(y-2)^2 = (x-1)$$, the tangent to it at the point whose ordinate is 3 and the x-axis, is:

If the solution curve of the differential equation $$(2x - 10y^3)dy + ydx = 0$$, passes through the points $$(0, 1)$$ and $$(2, \beta)$$, then $$\beta$$ is a root of the equation?

A differential equation representing the family of parabolas with axis parallel to y-axis and whose length of latus rectum is the distance of the point $$(2, -3)$$ from the line $$3x + 4y = 5$$, is given by:

The equation of the plane passing through the line of intersection of the planes $$\vec{r} \cdot (2\hat{i} + 3\hat{j} - \hat{k}) + 4 = 0$$ and $$\vec{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 1$$ and parallel to the x-axis, is

The angle between the straight lines, whose direction cosines $$l, m, n$$ are given by the equations $$2l + 2m - n = 0$$ and $$mn + nl + lm = 0$$, is:

Each of the persons $$A$$ and $$B$$ independently tosses three fair coins. The probability that both of them get the same number of heads is: