NTA JEE Main 29th January 2023 Shift 2

The value of the integral $$\int_{1/2}^{2} \frac{\tan^{-1}x}{x} dx$$ is equal to

The area of the region $$A = \{(x,y) : |\cos x - \sin x| \leq y \leq \sin x, 0 \leq x \leq \frac{\pi}{2}\}$$

Let $$y = y(x)$$ be the solution of the differential equation $$x \log_e x \frac{dy}{dx} + y = x^2 \log_e x$$, $$(x > 1)$$. If $$y(2) = 2$$, then $$y(e)$$ is equal to

If $$\vec{a} = \hat{i} + 2\hat{k}$$, $$\vec{b} = \hat{i} + \hat{j} + \hat{k}$$, $$\vec{c} = 7\hat{i} - 3\hat{j} + 4\hat{k}$$, $$\vec{r} \times \vec{b} + \vec{b} \times \vec{c} = \vec{0}$$ and $$\vec{r} \cdot \vec{a} = 0$$ then $$\vec{r} \cdot \vec{c}$$ is equal to:

Let $$\vec{a} = 4\hat{i} + 3\hat{j}$$ and $$\vec{b} = 3\hat{i} - 4\hat{j} + 5\hat{k}$$ and $$\vec{c}$$ is a vector such that $$\vec{c} \cdot (\vec{a} \times \vec{b}) + 25 = 0$$, $$\vec{c} \cdot (\hat{i} + \hat{j} + \hat{k}) = 4$$ and projection of $$\vec{c}$$ on $$\vec{a}$$ is $$1$$, then the projection of $$\vec{c}$$ on $$\vec{b}$$ equals:

Shortest distance between the lines $$\frac{x-1}{2} = \frac{y+8}{-7} = \frac{z-4}{5}$$ and $$\frac{x-1}{2} = \frac{y-2}{1} = \frac{z-6}{-3}$$ is:

The plane $$2x - y + z = 4$$ intersects the line segment joining the points $$A(a, -2, 4)$$ and $$B(2, b, -3)$$ at the point $$C$$ in the ratio $$2 : 1$$ and the distance of the point $$C$$ from the origin is $$\sqrt{5}$$. If $$ab < 0$$ and $$P$$ is the point $$(a-b, b, 2b-a)$$ then $$CP^2$$ is equal to:

If the lines $$\frac{x-1}{1} = \frac{y-2}{2} = \frac{z+3}{1}$$ and $$\frac{x-a}{2} = \frac{y+2}{3} = \frac{z-3}{1}$$ intersects at the point $$P$$, then the distance of the point $$P$$ from the plane $$z = a$$ is:

Let $$S = \{w_1, w_2, \ldots\}$$ be the sample space associated to a random experiment. Let $$P(w_n) = \frac{P(w_{n-1})}{2}$$, $$n \geq 2$$. Let $$A = \{2k + 3l; k, l \in \mathbb{N}\}$$ and $$B = \{w_n; n \in A\}$$. Then $$P(B)$$ is equal to

Let $$\alpha_1, \alpha_2, \ldots, \alpha_7$$ be the roots of the equation $$x^7 + 3x^5 - 13x^3 - 15x = 0$$ and $$|\alpha_1| \geq |\alpha_2| \geq \ldots \geq |\alpha_7|$$. Then, $$\alpha_1\alpha_2 - \alpha_3\alpha_4 + \alpha_5\alpha_6$$ is equal to ______.