NTA JEE Mains 30th Jan 2024 Shift 1

Let $$\alpha, \beta \in \mathbb{R}$$ be roots of equation $$x^2 - 70x + \lambda = 0$$, where $$\frac{\lambda}{2}, \frac{\lambda}{3} \notin \mathbb{Z}$$. If $$\lambda$$ assumes the minimum possible value, then $$\frac{(\sqrt{\alpha - 1} + \sqrt{\beta - 1})(\lambda + 35)}{|\alpha - \beta|}$$ is equal to :

Let $$\alpha = 1^2 + 4^2 + 8^2 + 13^2 + 19^2 + 26^2 + \ldots$$ upto $$10$$ terms and $$\beta = \sum_{n=1}^{10} n^4$$. If $$4\alpha - \beta = 55k + 40$$, then $$k$$ is equal to _______.

Number of integral terms in the expansion of $$\left\{7^{(1/2)} + 11^{(1/6)}\right\}^{824}$$ is equal to ______.

Let the latus rectum of the hyperbola $$\frac{x^2}{9} - \frac{y^2}{b^2} = 1$$ subtend an angle of $$\frac{\pi}{3}$$ at the centre of the hyperbola. If $$b^2$$ is equal to $$\frac{l}{m}(1 + \sqrt{n})$$, where $$l$$ and $$m$$ are co-prime numbers, then $$l^2 + m^2 + n^2$$ is equal to __________.

A group of $$40$$ students appeared in an examination of $$3$$ subjects - Mathematics, Physics & Chemistry. It was found that all students passed in at least one of the subjects, $$20$$ students passed in Mathematics, $$25$$ students passed in Physics, $$16$$ students passed in Chemistry, at most $$11$$ students passed in both Mathematics and Physics, at most $$15$$ students passed in both Physics and Chemistry, at most $$15$$ students passed in both Mathematics and Chemistry. The maximum number of students passed in all the three subjects is _____.

Let $$A = \{1, 2, 3, \ldots, 7\}$$ and let $$P(A)$$ denote the power set of $$A$$. If the number of functions $$f : A \rightarrow P(A)$$ such that $$a \in f(a), \forall a \in A$$ is $$m^n$$, $$m$$ and $$n \in \mathbb{N}$$ and $$m$$ is least, then $$m + n$$ is equal to ______.

If the function $$f(x) = \begin{cases} \frac{1}{|x|}, & |x| \geq 2 \\ ax^2 + 2b, & |x| < 2 \end{cases}$$ is differentiable on $$\mathbb{R}$$, then $$48(a + b)$$ is equal to _______.

The value $$9\int_0^9 \left[\sqrt{\frac{10x}{x+1}}\right] dx$$, where $$[t]$$ denotes the greatest integer less than or equal to $$t$$, is _____.

Let $$y = y(x)$$ be the solution of the differential equation $$(1 - x^2)dy = \left[xy + (x^3 + 2)\sqrt{3(1 - x^2)}\right]dx$$, $$-1 < x < 1$$, $$y(0) = 0$$. If $$y\left(\frac{1}{2}\right) = \frac{m}{n}$$, $$m$$ and $$n$$ are coprime numbers, then $$m + n$$ is equal to __________.

If $$d_1$$ is the shortest distance between the lines $$x + 1 = 2y = -12z$$, $$x = y + 2 = 6z - 6$$ and $$d_2$$ is the shortest distance between the lines $$\frac{x-1}{2} = \frac{y+8}{-7} = \frac{z-4}{5}$$, $$\frac{x-1}{2} = \frac{y-2}{1} = \frac{z-6}{-3}$$, then the value of $$\frac{32\sqrt{3} \, d_1}{d_2}$$ is :