NTA JEE Main 27th July 2022 Shift 2

$$\frac{2^3 - 1^3}{1 \times 7} + \frac{4^3 - 3^3 + 2^3 - 1^3}{2 \times 11} + \frac{6^3 - 5^3 + 4^3 - 3^3 + 2^3 - 1^3}{3 \times 15} + \ldots + \frac{30^3 - 29^3 + 28^3 - 27^3 + \ldots + 2^3 - 1^3}{15 \times 63}$$ is equal to ______.

Let for the $$9^{th}$$ term in the binomial expansion of $$(3 + 6x)^n$$, in the increasing powers of $$6x$$, to be the greatest for $$x = \frac{3}{2}$$, the least value of $$n$$ is $$n_0$$. If $$k$$ is the ratio of the coefficient of $$x^6$$ to the coefficient of $$x^3$$, then $$k + n_0$$ is equal to

A common tangent T to the curves $$C_1: \frac{x^2}{4} + \frac{y^2}{9} = 1$$ and $$C_2: \frac{x^2}{42} - \frac{y^2}{143} = 1$$ does not pass through the fourth quadrant. If T touches $$C_1$$ at $$(x_1, y_1)$$ and $$C_2$$ at $$(x_2, y_2)$$, then $$|2x_1 + x_2|$$ is equal to _______.

Consider a matrix $$A = \begin{pmatrix} \alpha & \beta & \gamma \\ \alpha^2 & \beta^2 & \gamma^2 \\ \beta+\gamma & \gamma+\alpha & \alpha+\beta \end{pmatrix}$$, where $$\alpha, \beta, \gamma$$ are three distinct natural numbers. If $$\frac{\det(\text{adj}(\text{adj}(\text{adj}(\text{adj} A))))}{(\alpha-\beta)^{16}(\beta-\gamma)^{16}(\gamma-\alpha)^{16}} = 2^{32} \times 3^{16}$$, then the number of such 3-tuples $$(\alpha, \beta, \gamma)$$ is ______.

The number of functions $$f$$, from the set $$A = \{x \in \mathbb{N}: x^2 - 10x + 9 \leq 0\}$$ to the set $$B = \{n^2 : n \in \mathbb{N}\}$$ such that $$f(x) \leq (x-3)^2 + 1$$, for every $$x \in A$$, is _______.

For the curve $$C: (x^2 + y^2 - 3) + (x^2 - y^2 - 1)^{5} = 0$$, the value of $$3y' - y^3 y''$$, at the point $$(\alpha, \alpha)$$, $$\alpha > 0$$ on C, is equal to ________.

A water tank has the shape of a right circular cone with axis vertical and vertex downwards. Its semivertical angle is $$\tan^{-1}\frac{3}{4}$$. Water is poured in it at a constant rate of 6 cubic meter per hour. The rate (in square meter per hour), at which the wet curved surface area of the tank is increasing, when the depth of water in the tank is 4 meters, is _______.

Let $$f(x) = \min\{[x-1], [x-2], \ldots, [x-10]\}$$ where $$[t]$$ denotes the greatest integer $$\leq t$$. Then $$\int_0^{10} f(x)dx + \int_0^{10} (f(x))^2 dx + \int_0^{10} |f(x)| dx$$ is equal to _______.

Let $$f$$ be a differentiable function satisfying $$f(x) = \frac{2}{\sqrt{3}} \int_0^{\sqrt{3}} f\left(\frac{\lambda^2 x}{3}\right) d\lambda$$, $$x > 0$$ and $$f(1) = \sqrt{3}$$. If $$y = f(x)$$ passes through the point $$(\alpha, 6)$$, then $$\alpha$$ is equal to _______.

Let $$\vec{a}, \vec{b}, \vec{c}$$ be three non-coplanar vectors such that $$\vec{a} \times \vec{b} = 4\vec{c}$$, $$\vec{b} \times \vec{c} = 9\vec{a}$$ and $$\vec{c} \times \vec{a} = \alpha\vec{b}$$, $$\alpha > 0$$. If $$|\vec{a}| + |\vec{b}| + |\vec{c}| = {36}$$, then $$\alpha$$ is equal to _______.