NTA JEE Main 28th June 2022 Shift 2

The probability that a randomly chosen one-one function from the set $$\{a, b, c, d\}$$ to the set $$\{1, 2, 3, 4, 5\}$$ satisfies $$f(a) + 2f(b) - f(c) = f(d)$$ is

Let $$f, g : \mathbf{R} \to \mathbf{R}$$ be functions defined by
$$f(x) = \begin{cases} [x] & , x < 0 \\ |1-x| & , x \geq 0 \end{cases}$$ and
$$g(x) = \begin{cases} e^x - x & , x < 0 \\ (x-1)^2 - 1 & , x \geq 0 \end{cases}$$
where $$[x]$$ denote the greatest integer less than or equal to $$x$$. Then, the function fog is discontinuous at exactly

Let $$f : \mathbf{R} \to \mathbf{R}$$ be a differentiable function such that $$f\left(\frac{\pi}{4}\right) = \sqrt{2}$$, $$f\left(\frac{\pi}{2}\right) = 0$$ and $$f'\left(\frac{\pi}{2}\right) = 1$$ and let $$g(x) = \int_x^{\pi} (f'(t) \sec t + \tan t \sec t \, f(t)) dt$$ for $$x \in \left[\frac{\pi}{4}, \frac{\pi}{2}\right)$$. Then $$\lim_{x \to \left(\frac{\pi}{2}\right)^-} g(x)$$ is equal to

Let $$f : \mathbf{R} \to \mathbf{R}$$ be continuous function satisfying $$f(x) + f(x+k) = n$$, for all $$x \in \mathbf{R}$$ where $$k > 0$$ and $$n$$ is a positive integer. If $$I_1 = \int_0^{4nk} f(x) dx$$ and $$I_2 = \int_{-k}^{3k} f(x) dx$$, then

The area of the bounded region enclosed by the curve $$y = 3 - \left|x - \frac{1}{2}\right| - |x + 1|$$ and the $$x$$-axis is

Let $$x = x(y)$$ be the solution of the differential equation $$2ye^{x/y^2} dx + \left(y^2 - 4xe^{x/y^2}\right) dy = 0$$ such that $$x(1) = 0$$. Then, $$x(e)$$ is equal to

Let the slope of the tangent to a curve $$y = f(x)$$ at $$(x, y)$$ be given by $$2 \tan x(\cos x - y)$$. If the curve passes through the point $$\left(\frac{\pi}{4}, 0\right)$$, then the value of $$\int_0^{\pi/2} y \, dx$$ is equal to

Let $$\vec{a} = \alpha \hat{i} + 2\hat{j} - \hat{k}$$ and $$\vec{b} = -2\hat{i} + \alpha \hat{j} + \hat{k}$$, where $$\alpha \in \mathbf{R}$$. If the area of the parallelogram whose adjacent sides are represented by the vectors $$\vec{a}$$ and $$\vec{b}$$ is $$\sqrt{15(\alpha^2 + 4)}$$, then the value of $$2|\vec{a}|^2 + (\vec{a} \cdot \vec{b})|\vec{b}|^2$$ is equal to

Let $$\vec{a}$$ be a vector which is perpendicular to the vector $$3\hat{i} + \frac{1}{2}\hat{j} + 2\hat{k}$$. If $$\vec{a} \times (2\hat{i} + \hat{k}) = 2\hat{i} - 13\hat{j} - 4\hat{k}$$, then the projection of the vector $$\vec{a}$$ on the vector $$2\hat{i} + 2\hat{j} + \hat{k}$$ is

Let the plane $$ax + by + cz = d$$ pass through $$(2, 3, -5)$$ and is perpendicular to the planes $$2x + y - 5z = 10$$ and $$3x + 5y - 7z = 12$$.
If $$a, b, c, d$$ are integers $$d > 0$$ and $$\gcd(|a|, |b|, |c|, d) = 1$$ then the value of $$a + 7b + c + 20d$$ is equal to