NTA JEE Main 31st August 2021 Shift 2

Let $$f : N \rightarrow N$$ be a function such that $$f(m+n) = f(m) + f(n)$$ for every $$m, n \in N$$. If $$f(6) = 18$$ then $$f(2) \cdot f(3)$$ is equal to:

The domain of the function, $$f(x) = \sin^{-1}\frac{3x^2+x-1}{(x-1)^2} + \cos^{-1}\frac{x-1}{x+1}$$ is:

An angle of intersection of the curves, $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ and $$x^2 + y^2 = ab$$, $$a \gt b$$, is:

Let $$f$$ be any continuous function on $$[0, 2]$$ and twice differentiable on $$(0, 2)$$. If $$f(0) = 0$$, $$f(1) = 1$$ and $$f(2) = 2$$, then:

If $$x$$ is the greatest integer $$\leq x$$, then $$\pi^2 \int_0^2 \sin\frac{\pi x}{2} x - x^{[x]} dx$$ is equal to:

If $$y\frac{dy}{dx} = x\frac{y^2}{x^2} + \frac{\phi\frac{y^2}{x^2}}{\phi'\frac{y^2}{x^2}}$$, $$x > 0$$, $$\phi > 0$$, and $$y(1) = -1$$, then $$\phi\frac{y^2}{4}$$ is equal to:

If $$\frac{dy}{dx} = \frac{2^x y + 2^y \cdot 2^x}{2^x + 2^x + y\log_e 2}$$, $$y(0) = 0$$, then for $$y = 1$$, the value of $$x$$ lies in the interval:

Let $$\vec{a}$$, $$\vec{b}$$, $$\vec{c}$$ be three vectors mutually perpendicular to each other and have same magnitude. If a vector $$\vec{r}$$ satisfies $$\vec{a} \times \{\vec{r} - \vec{b} \times \vec{a}\} + \vec{b} \times \{\vec{r} - \vec{c} \times \vec{b}\} + \vec{c} \times \{\vec{r} - \vec{a} \times \vec{c}\} = \vec{0}$$, then $$\vec{r}$$ is equal to:

The distance of the point $$(-1, 2, -2)$$ from the line of intersection of the planes $$2x + 3y + 2z = 0$$ and $$x - 2y + z = 0$$ is:

Let $$S = \{1, 2, 3, 4, 5, 6\}$$. Then the probability that a randomly chosen onto function $$g$$ from $$S$$ to $$S$$ satisfies $$g(3) = 2g(1)$$ is: