NTA JEE Main 28th July 2022 Shift 1

Considering the principal values of the inverse trigonometric functions, the sum of all the solutions of the equation $$\cos^{-1}(x) - 2\sin^{-1}(x) = \cos^{-1}(2x)$$ is equal to

Let $$\alpha, \beta$$ and $$\gamma$$ be three positive real numbers. Let $$f(x) = \alpha x^5 + \beta x^3 + \gamma x$$, $$x \in \mathbb{R}$$ and $$g: \mathbb{R} \to \mathbb{R}$$ be such that $$g(f(x)) = x$$ for all $$x \in \mathbb{R}$$. If $$a_1, a_2, a_3, \ldots, a_n$$ be in arithmetic progression with mean zero, then the value of $$f\left(g\left(\frac{1}{n}\sum_{i=1}^{n} f(a_i)\right)\right)$$ is equal to

Considering only the principal values of the inverse trigonometric functions, the domain of the function $$f(x) = \cos^{-1}\left(\frac{x^2 - 4x + 2}{x^2 + 3}\right)$$ is

The minimum value of the twice differentiable function $$f(x) = \int_0^x e^{x-t} f'(t) dt - (x^2 - x + 1)e^x$$, $$x \in \mathbb{R}$$, is

Let the solution curve of the differential equation $$x dy = (\sqrt{x^2 + y^2} + y) dx$$, $$x > 0$$, intersect the line $$x = 1$$ at $$y = 0$$ and the line $$x = 2$$ at $$y = \alpha$$. Then the value of $$\alpha$$ is

If $$y = y(x)$$, $$x \in \left(0, \frac{\pi}{2}\right)$$ be the solution curve of the differential equation $$\sin^2(2x)\frac{dy}{dx} + (8\sin^2(2x) + 2\sin(4x))y = 2e^{-4x}(2\sin(2x) + \cos(2x))$$, with $$y\left(\frac{\pi}{4}\right) = e^{-\pi}$$, then $$y\left(\frac{\pi}{6}\right)$$ is equal to

Let the vectors $$\vec{a} = (1+t)\hat{i} + (1-t)\hat{j} + \hat{k}$$, $$\vec{b} = (1-t)\hat{i} + (1+t)\hat{j} + 2\hat{k}$$ and $$\vec{c} = t\hat{i} - t\hat{j} + \hat{k}$$, $$t \in \mathbb{R}$$ be such that for $$\alpha, \beta, \gamma \in \mathbb{R}$$, $$\alpha\vec{a} + \beta\vec{b} + \gamma\vec{c} = \vec{0} \Rightarrow \alpha = \beta = \gamma = 0$$. Then, the set of all values of $$t$$ is

Let a vector $$\vec{a}$$ has magnitude 9. Let a vector $$\vec{b}$$ be such that for every $$(x, y) \in \mathbb{R} \times \mathbb{R} - \{(0,0)\}$$, the vector $$x\vec{a} + y\vec{b}$$ is perpendicular to the vector $$6y\vec{a} - 18x\vec{b}$$. Then the value of $$|\vec{a} \times \vec{b}|$$ is equal to

The foot of the perpendicular from a point on the circle $$x^2 + y^2 = 1, z = 0$$ to the plane $$2x + 3y + z = 6$$ lies on which one of the following curves?

Out of 60% female and 40% male candidates appearing in an exam, 60% candidates qualify it. The number of females qualifying the exam is twice the number of males qualifying it. A candidate is randomly chosen from the qualified candidates. The probability, that the chosen candidate is a female, is