NTA JEE Main 29th July 2022 Shift 1

The number of points, where the function $$f: \mathbb{R} \to \mathbb{R}$$, $$f(x) = |x - 1|\cos|x - 2|\sin|x - 1| + (x - 3)|x^2 - 5x + 4|$$, is NOT differentiable, is

Let $$f(x) = 3(x^2 - 2)^3 + 4$$, $$x \in \mathbb{R}$$. Then which of the following statements are true?
P: $$x = 0$$ is a point of local minima of f
Q: $$x = \sqrt{2}$$ is a point of inflection of f
R: $$f'$$ is increasing for $$x > \sqrt{2}$$

The integral $$\int_0^{\pi/2} \frac{1}{3 + 2\sin x + \cos x} dx$$ is equal to:

If $$f(\alpha) = \int_1^\alpha \frac{\log_{10} t}{1+t} dt$$, $$\alpha > 0$$, then $$f(e^3) + f(e^{-3})$$ is equal to

The area of the region $$\{(x, y) : |x - 1| \leq y \leq \sqrt{5 - x^2}\}$$ is equal to

Let the solution curve $$y = y(x)$$ of the differential equation $$(1 + e^{2x})\left(\frac{dy}{dx} + y\right) = 1$$ pass through the point $$\left(0, \frac{\pi}{2}\right)$$. Then, $$\lim_{x \to \infty} e^x y(x)$$ is equal to

Let $$\vec{a} = 3\hat{i} + \hat{j}$$ and $$\vec{b} = \hat{i} + 2\hat{j} + \hat{k}$$. Let $$\vec{c}$$ be a vector satisfying $$\vec{a} \times (\vec{b} \times \vec{c}) = \vec{b} + \lambda\vec{c}$$. If $$\vec{b}$$ and $$\vec{c}$$ are non-parallel, then the value of $$\lambda$$ is

Let $$\hat{a}$$ and $$\hat{b}$$ be two unit vectors such that the angle between them is $$\frac{\pi}{4}$$. If $$\theta$$ is the angle between the vectors $$(\hat{a} + \hat{b})$$ and $$(\hat{a} + 2\hat{b} + 2(\hat{a} \times \hat{b}))$$ then the value of $$164\cos^2\theta$$ is equal to

If the foot of the perpendicular from the point $$A(-1, 4, 3)$$ on the plane $$P: 2x + my + nz = 4$$, is $$\left(-2, \frac{7}{2}, \frac{3}{2}\right)$$, then the distance of the point A from the plane P, measured parallel to a line with direction ratios 3, -1, -4, is equal to

Let $$S = \{1, 2, 3, \ldots, 2022\}$$. Then the probability, that a randomly chosen number n from the set S such that $$HCF(n, 2022) = 1$$, is