NTA JEE Mains 27th Jan 2024 Shift 2

Let $$f: \mathbb{R} - \frac{-1}{2}\to \mathbb{R}$$ and $$g: \mathbb{R} - \frac{-5}{2} \to \mathbb{R}$$ be defined as $$f(x) = \frac{2x + 3}{2x + 1}$$ and $$g(x) = \frac{|x| + 1}{2x + 5}$$. Then the domain of the function fog is:

Consider the function $$f: (0, 2) \to \mathbb{R}$$ defined by $$f(x) = \frac{x}{2} + \frac{2}{x}$$ and the function $$g(x)$$ defined by $$g(x) = \begin{cases} \min\{f(t)\},\ 0 < t \leq x & \text{and } 0 < x \leq 1 \\ \frac{3}{2} + x, & 1 < x < 2 \end{cases}$$. Then

Let $$g(x) = 3f\left(\frac{x}{3}\right) + f(3 - x)$$ and $$f''(x) > 0$$ for all $$x \in (0, 3)$$. If g is decreasing in $$(0, \alpha)$$ and increasing in $$(\alpha, 3)$$, then $$8\alpha$$ is

The integral $$\int \frac{x^8 - x^2}{(x^{12} + 3x^6 + 1)\tan^{-1}\left(x^3 + \frac{1}{x^3}\right)} dx$$ is equal to :

For $$0 < a < 1$$, the value of the integral $$\int_0^{\pi} \frac{dx}{1 - 2a\cos x + a^2}$$ is :

If $$y = y(x)$$ is the solution curve of the differential equation $$(x^2 - 4)dy - (y^2 - 3y)dx = 0$$, $$x > 2$$, $$y(4) = \frac{3}{2}$$ and the slope of the curve is never zero, then the value of $$y(10)$$ equals :

The position vectors of the vertices A, B and C of a triangle are $$2\hat{i} - 3\hat{j} + 3\hat{k}$$, $$2\hat{i} + 2\hat{j} + 3\hat{k}$$ and $$-\hat{i} + \hat{j} + 3\hat{k}$$ respectively. Let $$l$$ denotes the length of the angle bisector AD of $$\angle BAC$$ where D is on the line segment BC, then $$2l^2$$ equals :

Let the position vectors of the vertices A, B and C of a triangle be $$2\hat{i} + 2\hat{j} + \hat{k}$$, $$\hat{i} + 2\hat{j} + 2\hat{k}$$ and $$2\hat{i} + \hat{j} + 2\hat{k}$$ respectively. Let $$l_1, l_2$$ and $$l_3$$ be the lengths of perpendiculars drawn from the ortho centre of the triangle on the sides AB, BC and CA respectively, then $$l_1^2 + l_2^2 + l_3^2$$ equals :

Let the image of the point $$(1, 0, 7)$$ in the line $$\frac{x}{1} = \frac{y - 1}{2} = \frac{z - 2}{3}$$ be the point $$(\alpha, \beta, \gamma)$$. Then which one of the following points lies on the line passing through $$(\alpha, \beta, \gamma)$$ and making angles $$\frac{2\pi}{3}$$ and $$\frac{3\pi}{4}$$ with y-axis and z-axis respectively and an acute angle with x-axis?

An urn contains 6 white and 9 black balls. Two successive draws of 4 balls are made without replacement. The probability, that the first draw gives all white balls and the second draw gives all black balls, is :