NTA JEE Main 25th July 2022 Shift 1

If the absolute maximum value of the function $$f(x) = (x^2 - 2x + 7)e^{(4x^3 - 12x^2 - 180x + 31)}$$ in the interval $$[-3, 0]$$ is $$f(\alpha)$$, then

The curve $$y(x) = ax^3 + bx^2 + cx + 5$$ touches the $$x$$-axis at the point $$P(-2, 0)$$ and cuts the $$y$$-axis at the point $$Q$$ where $$y'$$ is equal to $$3$$. Then the local maximum value of $$y(x)$$ is

For any real number $$x$$, let $$[x]$$ denote the largest integer less than or equal to $$x$$. Let $$f$$ be a real-valued function defined on the interval $$[-10, 10]$$ by
$$f(x) = \begin{cases} x - [x], & \text{if } [x] \text{ is odd} \\ 1 + [x] - x, & \text{if } [x] \text{ is even} \end{cases}$$
Then, the value of $$\dfrac{\pi^2}{10} \displaystyle\int_{-10}^{10} f(x) \cos \pi x \, dx$$ is

The area of the region given by $$A = \{(x, y) : x^2 \le y \le \min\{x + 2, 4 - 3x\}\}$$ is

The slope of the tangent to a curve $$C: y = y(x)$$ at any point $$[x, y)$$ on it is $$\dfrac{2e^{2x} - 6e^{-x} + 9}{2 + 9e^{-2x}}$$. If $$C$$ passes through the points $$\left(0, \dfrac{1}{2} + \dfrac{\pi}{2\sqrt{2}}\right)$$ and $$\left(\alpha, \dfrac{1}{2}e^{2\alpha}\right)$$ then $$e^{\alpha}$$ is equal to

The general solution of the differential equation $$(x - y^2)dx + y(5x + y^2)dy = 0$$ is

Let $$ABC$$ be a triangle such that $$\vec{BC} = \vec{a}$$, $$\vec{CA} = \vec{b}$$, $$\vec{AB} = \vec{c}$$, $$|\vec{a}| = 6\sqrt{2}$$, $$|\vec{b}| = 2\sqrt{3}$$ and $$\vec{b} \cdot \vec{c} = 12$$. Consider the statements:
$$S_1: |\vec{a} \times (\vec{b} + \vec{c})| \times |\vec{b} - \vec{c}| = 6(2\sqrt{2} - 1)$$
$$S_2: \angle ABC = \cos^{-1}\sqrt{\dfrac{2}{3}}$$
Then

Let $$P$$ be the plane containing the straight line $$\dfrac{x - 3}{9} = \dfrac{y + 4}{-1} = \dfrac{z - 7}{-5}$$ and perpendicular to the plane containing the straight lines $$\dfrac{x}{2} = \dfrac{y}{3} = \dfrac{z}{5}$$ and $$\dfrac{x}{3} = \dfrac{y}{7} = \dfrac{z}{8}$$. If $$d$$ is the distance of $$P$$ from the point $$(2, -5, 11)$$, then $$d^2$$ is equal to

If the sum and the product of mean and variance of a binomial distribution are $$24$$ and $$128$$ respectively, then the probability of one or two successes is

If the numbers appeared on the two throws of a fair six faced die are $$\alpha$$ and $$\beta$$, then the probability that $$x^2 + \alpha x + \beta > 0$$, for all $$x \in \mathbb{R}$$, is