NTA JEE Main 28th June 2022 Shift 1

Let $$f : \mathbb{R} \to \mathbb{R}$$ be defined as $$f(x) = \begin{cases} [e^x], & x < 0 \\ ae^x + [x-1], & 0 \leq x < 1 \\ b + [\sin(\pi x)], & 1 \leq x < 2 \\ [e^{-x}] - c, & x \geq 2 \end{cases}$$
where $$a, b, c \in \mathbb{R}$$ and $$[t]$$ denotes greatest integer less than or equal to $$t$$. Then, which of the following statements is true?

The number of real solutions of $$x^7 + 5x^3 + 3x + 1 = 0$$ is equal to ______

Let $$[t]$$ denote the greatest integer less than or equal to $$t$$. Then the value of the integral $$\int_0^1 [-8x^2 + 6x - 1] dx$$ is equal to

The area of the region $$S = \{(x,y) : y^2 \leq 8x, y \geq \sqrt{2}x, x \geq 1\}$$ is

Let the solution curve $$y = y(x)$$ of the differential equation, $$\left[\frac{x}{\sqrt{x^2-y^2}} + e^{y/x}\right]x\frac{dy}{dx} = x + \left[\frac{x}{\sqrt{x^2-y^2}} + e^{y/x}\right]y$$ pass through the points $$(1, 0)$$ and $$(2\alpha, \alpha), \alpha > 0$$. Then $$\alpha$$ is equal to

Let $$y = y(x)$$ be the solution of the differential equation $$x(1 - x^2)\frac{dy}{dx} + (3x^2y - y - 4x^3) = 0, x > 1$$ with $$y(2) = -2$$. Then $$y(3)$$ is equal to

If two distinct point $$Q$$, $$R$$ lie on the line of intersection of the planes $$-x + 2y - z = 0$$ and $$3x - 5y + 2z = 0$$ and $$PQ = PR = \sqrt{18}$$ where the point $$P$$ is $$(1, -2, 3)$$, then the area of the triangle $$PQR$$ is equal to

The acute angle between the planes $$P_1$$ and $$P_2$$, when $$P_1$$ and $$P_2$$ are the planes passing through the intersection of the planes $$5x + 8y + 13z - 29 = 0$$ and $$8x - 7y + z - 20 = 0$$ and the points $$(2, 1, 3)$$ and $$(0, 1, 2)$$, respectively, is

Let the plane $$P : \vec{r} \cdot \vec{a} = d$$ contain the line of intersection of two planes $$\vec{r} \cdot (\hat{i} + 3\hat{j} - \hat{k}) = 6$$ and $$\vec{r} \cdot (-6\hat{i} + 5\hat{j} - \hat{k}) = 7$$. If the plane $$P$$ passes through the point $$(2, 3, \frac{1}{2})$$, then the value of $$\frac{|13\vec{a}|^2}{d^2}$$ is equal to

The probability, that in a randomly selected 3-digit number at least two digits are odd, is