NTA JEE Main 11th April 2023 Shift 1

Let $$f(x) = x^2 - [x] + |-x + [x]|$$, where $$x \in \mathbb{R}$$ and $$[t]$$ denotes the greatest integer less than or equal to $$t$$. Then, $$f$$ is

Let $$f: [2, 4] \to \mathbb{R}$$ be a differentiable function such that $$x\log_e xf'(x) + \log_e xf(x) + f(x) \geq 1$$, $$x \in [2, 4]$$ with $$f(2) = \frac{1}{2}$$ and $$f(4) = \frac{1}{2}$$.
Consider the following two statements:
(A) $$f(x) \leq 1$$, for all $$x \in [2, 4]$$
(B) $$f(x) \geq 1/8$$, for all $$x \in [2, 4]$$
Then,

The value of the integral $$\int_{-\log_e 2}^{\log_e 2} e^x \log_e e^x + \sqrt{1 + e^{2x}} \, dx$$ is equal to

Area of the region $$\{(x, y): x^2 + (y-2)^2 \leq 4, x^2 \geq 2y\}$$ is

Let $$y = y(x)$$ be a solution curve of the differential equation, $$(1 - x^2y^2)dx = ydx + xdy$$. If the line $$x = 1$$ intersects the curve $$y = y(x)$$ at $$y = 2$$ and the line $$x = 2$$ intersects the curve $$y = y(x)$$ at $$y = \alpha$$, then a value of $$\alpha$$ is

For any vector $$\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$$, with $$10a_i < 1$$, $$i = 1, 2, 3$$, consider the following statements:
$$A: \max(a_1, a_2, a_3) \leq \vec{a}$$
$$B: |\vec{a}| \leq 3\max a_1, a_2, a_3$$

Let $$\vec{a}$$ be a non-zero vector parallel to the line of intersection of the two planes described by $$\hat{i} + \hat{j}, \hat{i} + \hat{k}$$ and $$\hat{i} - \hat{j}, \hat{j} - \hat{k}$$. If $$\theta$$ is the angle between the vector $$\vec{a}$$ and the vector $$\vec{b} = 2\hat{i} - 2\hat{j} + \hat{k}$$ and $$\vec{a} \cdot \vec{b} = 6$$, then the ordered pair $$(\theta, |\vec{a} \times \vec{b}|)$$ is equal to

Let $$(\alpha, \beta, \gamma)$$ be the image of point $$P(2, 3, 5)$$ in the plane $$2x + y - 3z = 6$$. Then $$\alpha + \beta + \gamma$$ is equal to

If the equation of the plane that contains the point $$(-2, 3, 5)$$ and is perpendicular to each of the planes $$2x + 4y + 5z = 8$$ and $$3x - 2y + 3z = 5$$ is $$\alpha x + \beta y + \gamma z + 97 = 0$$ then $$\alpha + \beta + \gamma =$$

Let $$S = M = a_{ij}$$, $$a_{ij} \in \{0, 1, 2\}$$, $$1 \leq i, j \leq 2$$ be a sample space and $$A = \{M \in S: M \text{ is invertible}\}$$ be an even. Then $$P(A)$$ is equal to