NTA JEE Mains 30th Jan 2024 Shift 1

Consider the system of linear equation $$x + y + z = 4\mu$$, $$x + 2y + 2\lambda z = 10\mu$$, $$x + 3y + 4\lambda^2 z = \mu^2 + 15$$, where $$\lambda, \mu \in \mathbb{R}$$. Which one of the following statements is NOT correct?

If the domain of the function $$f(x) = \cos^{-1}\left(\frac{2 - |x|}{4}\right) + (\log_e(3 - x))^{-1}$$ is $$[-\alpha, \beta) - \{\gamma\}$$, then $$\alpha + \beta + \gamma$$ is equal to :

Let $$g : \mathbb{R} \rightarrow \mathbb{R}$$ be a non constant twice differentiable such that $$g'\left(\frac{1}{2}\right) = g'\left(\frac{3}{2}\right)$$. If a real valued function $$f$$ is defined as $$f(x) = \frac{1}{2}[g(x) + g(2 - x)]$$, then

The value of $$\lim_{n \to \infty} \sum_{k=1}^{n} \frac{n^3}{(n^2 + k^2)(n^2 + 3k^2)}$$ is :

The area (in square units) of the region bounded by the parabola $$y^2 = 4(x - 2)$$ and the line $$y = 2x - 8$$.

Let $$y = y(x)$$ be the solution of the differential equation $$\sec x \, dy + \{2(1 - x)\tan x + x(2 - x)\}dx = 0$$ such that $$y(0) = 2$$. Then $$y(2)$$ is equal to :

Let $$A(2, 3, 5)$$ and $$C(-3, 4, -2)$$ be opposite vertices of a parallelogram $$ABCD$$ if the diagonal $$\vec{BD} = \hat{i} + 2\hat{j} + 3\hat{k}$$ then the area of the parallelogram is equal to

Let $$\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$$ and $$\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$$ be two vectors such that $$|\vec{a}| = 1$$; $$\vec{a} \cdot \vec{b} = 2$$ and $$|\vec{b}| = 4$$. If $$\vec{c} = 2(\vec{a} \times \vec{b}) - 3\vec{b}$$, then the angle between $$\vec{b}$$ and $$\vec{c}$$ is equal to :

Let $$(\alpha, \beta, \gamma)$$ be the foot of perpendicular from the point $$(1, 2, 3)$$ on the line $$\frac{x+3}{5} = \frac{y-1}{2} = \frac{z+4}{3}$$. then $$19(\alpha + \beta + \gamma)$$ is equal to :

Two integers $$x$$ and $$y$$ are chosen with replacement from the set $$\{0, 1, 2, 3, \ldots, 10\}$$. Then the probability that $$|x - y| > 5$$ is :