NTA JEE Mains 05th April 2024 Shift 2

Let $$f, g : \mathbb{R} \rightarrow \mathbb{R}$$ be defined as : $$f(x) = |x - 1|$$ and $$g(x) = \begin{cases} e^x, & x \geq 0 \\ x + 1, & x \leq 0 \end{cases}$$. Then the function $$f(g(x))$$ is

Let $$f : [-1, 2] \rightarrow \mathbb{R}$$ be given by $$f(x) = 2x^2 + x + [x^2] - [x]$$, where $$[t]$$ denotes the greatest integer less than or equal to $$t$$. The number of points, where $$f$$ is not continuous, is :

If $$y(\theta) = \frac{2\cos\theta + \cos 2\theta}{\cos 3\theta + 4\cos 2\theta + 5\cos\theta + 2}$$, then at $$\theta = \frac{\pi}{2}$$, $$y'' + y' + y$$ is equal to :

Let $$\beta(m, n) = \int_0^1 x^{m-1}(1-x)^{n-1} dx$$, $$m, n > 0$$. If $$\int_0^1 (1 - x^{10})^{20} dx = a \times \beta(b, c)$$, then $$100(a + b + c)$$ equals

The area enclosed between the curves $$y = x|x|$$ and $$y = x - |x|$$ is :

The differential equation of the family of circles passing through the origin and having centre at the line $$y = x$$ is :

Consider three vectors $$\vec{a}, \vec{b}, \vec{c}$$. Let $$|\vec{a}| = 2, |\vec{b}| = 3$$ and $$\vec{a} = \vec{b} \times \vec{c}$$. If $$\alpha \in \left[0, \frac{\pi}{3}\right]$$ is the angle between the vectors $$\vec{b}$$ and $$\vec{c}$$, then the minimum value of $$27|\vec{c} - \vec{a}|^2$$ is equal to:

Let $$\vec{a} = 2\hat{i} + 5\hat{j} - \hat{k}$$, $$\vec{b} = 2\hat{i} - 2\hat{j} + 2\hat{k}$$ and $$\vec{c}$$ be three vectors such that $$(\vec{c} + \hat{i}) \times (\vec{a} + \vec{b} + \hat{i}) = \vec{a} \times (\vec{c} + \hat{i})$$. If $$\vec{a} \cdot \vec{c} = -29$$, then $$\vec{c} \cdot (-2\hat{i} + \hat{j} + \hat{k})$$ is equal to:

Let $$(\alpha, \beta, \gamma)$$ be the image of the point $$(8, 5, 7)$$ in the line $$\frac{x-1}{2} = \frac{y+1}{3} = \frac{z-2}{5}$$. Then $$\alpha + \beta + \gamma$$ is equal to :

The coefficients $$a, b, c$$ in the quadratic equation $$ax^2 + bx + c = 0$$ are from the set $$\{1, 2, 3, 4, 5, 6\}$$. If the probability of this equation having one real root bigger than the other is $$p$$, then $$216p$$ equals :