NTA JEE Main 26th June 2022 Shift 2

Let $$f : \mathbb{R} \to \mathbb{R}$$ be defined as $$f(x) = x - 1$$ and $$g : R \to \{1, -1\} \to \mathbb{R}$$ be defined as $$g(x) = \frac{x^2}{x^2 - 1}$$. Then the function $$fog$$ is:

Let $$f(x) = \min\{1, 1 + x\sin x\}, 0 \leq x \leq 2\pi$$. If $$m$$ is the number of points, where $$f$$ is not differentiable and $$n$$ is the number of points, where $$f$$ is not continuous, then the ordered pair $$(m, n)$$ is equal to

Consider a cuboid of sides $$2x, 4x$$ and $$5x$$ and a closed hemisphere of radius $$r$$. If the sum of their surface areas is constant $$k$$, then the ratio $$x : r$$, for which the sum of their volumes is maximum, is

If $$\int \frac{1}{x}\sqrt{\frac{1-x}{1+x}} dx = g(x) + c, g(1) = 0$$, then $$g\left(\frac{1}{2}\right)$$ is equal to

The area of the region bounded by $$y^2 = 8x$$ and $$y^2 = 16(3 - x)$$ is equal to

If $$y = y(x)$$ is the solution of the differential equation $$x\frac{dy}{dx} + 2y = xe^x, y(1) = 0$$ then the local maximum value of the function $$z(x) = x^2y(x) - e^x, x \in R$$ is

If $$\frac{dy}{dx} + e^x(x^2 - 2)y = (x^2 - 2x)(x^2 - 2)e^{2x}$$ and $$y(0) = 0$$, then the value of $$y(2)$$ is

Let $$\vec{a} = \hat{i} + \hat{j} + 2\hat{k}, \vec{b} = 2\hat{i} - 3\hat{j} + \hat{k}$$ and $$\vec{c} = \hat{i} - \hat{j} + \hat{k}$$ be the three given vectors. Let $$\vec{v}$$ be a vector in the plane of $$\vec{a}$$ and $$\vec{b}$$ whose projection on $$\vec{c}$$ is $$\frac{2}{\sqrt{3}}$$. If $$\vec{v} \cdot \hat{j} = 7$$, then $$\vec{v} \cdot (\hat{i} + \hat{k})$$ is equal to

If the plane $$2x + y - 5z = 0$$ is rotated about its line of intersection with the plane $$3x - y + 4z - 7 = 0$$ by an angle of $$\frac{\pi}{2}$$, then the plane after the rotation passes through the point

If the lines $$\vec{r} = (\hat{i} - \hat{j} + \hat{k}) + \lambda(3\hat{j} - \hat{k})$$ and $$\vec{r} = (\alpha\hat{i} - \hat{j}) + \mu(2\hat{i} - 3\hat{k})$$ are co-planar, then the distance of the plane containing these two lines from the point $$(\alpha, 0, 0)$$ is