NTA JEE Mains 8th April 2024 Shift 2

Let $$f(x) = \begin{cases} -a & \text{if } -a \leq x \leq 0 \\ x + a & \text{if } 0 < x \leq a \end{cases}$$ where $$a > 0$$ and $$g(x) = (f(|x|) - |f(x)|)/2$$. Then the function $$g : [-a, a] \rightarrow [-a, a]$$ is :

For $$a, b > 0$$, let $$f(x) = \begin{cases} \frac{\tan((a+1)x) + b\tan x}{x}, & x < 0 \\ 3, & x = 0 \\ \frac{\sqrt{ax + b^2x^2} - \sqrt{ax}}{b\sqrt{ax}\sqrt{x}}, & x > 0 \end{cases}$$ be a continuous function at $$x = 0$$. Then $$\frac{b}{a}$$ is equal to :

If the function $$f(x) = 2x^3 - 9ax^2 + 12a^2x + 1$$, $$a > 0$$ has a local maximum at $$x = \alpha$$ and a local minimum at $$x = \alpha^2$$, then $$\alpha$$ and $$\alpha^2$$ are the roots of the equation :

Let $$\int_{\alpha}^{\log_e 4} \frac{dx}{\sqrt{e^x - 1}} = \frac{\pi}{6}$$. Then $$e^{\alpha}$$ and $$e^{-\alpha}$$ are the roots of the equation :

The area of the region in the first quadrant inside the circle $$x^2 + y^2 = 8$$ and outside the parabola $$y^2 = 2x$$ is equal to :

Let $$y = y(x)$$ be the solution curve of the differential equation $$\sec y \frac{dy}{dx} + 2x\sin y = x^3\cos y$$, $$y(1) = 0$$. Then $$y(\sqrt{3})$$ is equal to :

Let $$\vec{a} = 4\hat{i} - \hat{j} + \hat{k}$$, $$\vec{b} = 11\hat{i} - \hat{j} + \hat{k}$$ and $$\vec{c}$$ be a vector such that $$(\vec{a} + \vec{b}) \times \vec{c} = \vec{c} \times (-2\vec{a} + 3\vec{b})$$. If $$(2\vec{a} + 3\vec{b}) \cdot \vec{c} = 1670$$, then $$|\vec{c}|^2$$ is equal to :

Let $$\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}$$, $$\vec{b} = 2\hat{i} + 3\hat{j} - 5\hat{k}$$ and $$\vec{c} = 3\hat{i} - \hat{j} + \lambda\hat{k}$$ be three vectors. Let $$\vec{r}$$ be a unit vector along $$\vec{b} + \vec{c}$$. If $$\vec{r} \cdot \vec{a} = 3$$, then $$3\lambda$$ is equal to :

If the shortest distance between the lines $$\frac{x - \lambda}{2} = \frac{y - 4}{3} = \frac{z - 3}{4}$$ and $$\frac{x - 2}{4} = \frac{y - 4}{6} = \frac{z - 7}{8}$$ is $$\frac{13}{\sqrt{29}}$$, then a value of $$\lambda$$ is :

There are three bags $$X, Y$$ and $$Z$$. Bag $$X$$ contains 5 one-rupee coins and 4 five-rupee coins; Bag $$Y$$ contains 4 one-rupee coins and 5 five-rupee coins and Bag $$Z$$ contains 3 one-rupee coins and 6 five-rupee coins. A bag is selected at random and a coin drawn from it at random is found to be a one-rupee coin. Then the probability, that it came from bag Y, is :