NTA JEE Main 27th June 2022 Shift 2

Let $$\alpha, \beta$$ be the roots of the equation $$x^2 - 4\lambda x + 5 = 0$$ and $$\alpha, \gamma$$ be the roots of the equation $$x^2 - (3\sqrt{2} + 2\sqrt{3})x + 7 + 3\lambda\sqrt{3} = 0$$. If $$\beta + \gamma = 3\sqrt{2}$$, then $$(\alpha + 2\beta + \gamma)^2$$ is equal to ______

If the sum of the coefficients of all the positive powers of $$x$$, in the binomial expansion of $$\left(x^n + \frac{2}{x^5}\right)^7$$ is $$939$$, then the sum of all the possible integral values of $$n$$ is ______

Let a circle $$C$$ of radius $$5$$ lie below the $$x$$-axis. The line $$L_1 = 4x + 3y + 2$$ passes through the centre $$P$$ of the circle $$C$$ and intersects the line $$L_2 : 3x - 4y - 11 = 0$$ at $$Q$$. The line $$L_2$$ touches $$C$$ at the point $$Q$$. Then the distance of $$P$$ from the line $$5x - 12y + 51 = 0$$ is ______

Let $$[t]$$ denote the greatest integer $$\leq t$$ and $$\{t\}$$ denote the fractional part of $$t$$. Then integral value of $$\alpha$$ for which the left hand limit of the function $$f(x) = [1+x] + \frac{\alpha^{2[x]+\{x\}}+[x]-1}{2[x]+\{x\}}$$ at $$x = 0$$ is equal to $$\alpha - \frac{4}{3}$$ is ______

Let $$A$$ be a matrix of order $$2 \times 2$$, whose entries are from the set $$\{0, 1, 2, 3, 4, 5\}$$. If the sum of all the entries of $$A$$ is a prime number $$p, 2 < p < 8$$, then the number of such matrices $$A$$ is ______

Let $$S = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$$. Define $$f : S \to S$$ as $$f(n) = \begin{cases} 2n, & \text{if } n = 1,2,3,4,5 \\ 2n-11 & \text{if } n = 6,7,8,9,10 \end{cases}$$
Let $$g : S \geq S$$ be a function such that $$fog(n) = \begin{cases} n+1, & \text{if } n \text{ is odd} \\ n-1, & \text{if } n \text{ is even} \end{cases}$$, then
$$g(10)(g(1) + g(2) + g(3) + g(4) + g(5))$$ is equal to ______

If $$y(x) = (x^x)^x, x > 0$$ then $$\frac{d^2x}{dy^2} + 20$$ at $$x = 1$$ is equal to ______

If the area of the region $$\left\{(x,y) : x^{\frac{2}{3}} + y^{\frac{2}{3}} \leq 1, x + y \geq 0, y \geq 0\right\}$$ is $$A$$, then $$\frac{256A}{\pi}$$ is ______

Let $$y = y(x)$$ be the solution of the differential equation
$$(1 - x^2)dy = \left(xy + (x^3 + 2)\sqrt{1-x^2}\right)dx, -1 < x < 1$$
and $$y(0) = 0$$. If $$\int_{-\frac{1}{2}}^{\frac{1}{2}} \sqrt{1-x^2} y(x)dx = k$$ then $$k^{-1}$$ is equal to ______

Let $$S = \{E, E_2 \ldots E_8\}$$ be a sample space of a random experiment such that $$P(E_n) = \frac{n}{36}$$ for every $$n = 1, 2 \ldots 8$$. Then the number of elements in the set $$\{A \subset S : P(A) \geq \frac{4}{5}\}$$ is ______