NTA JEE Main 25th July 2022 Shift 2

Let $$f(x)$$ be a quadratic polynomial with leading coefficient $$1$$ such that $$f(0) = p, p \neq 0$$, and $$f(1) = \dfrac{1}{3}$$. If the equations $$f(x) = 0$$ and $$fofofof(x) = 0$$ have a common real root, then $$f(-3)$$ is equal to ______.

If the circles $$x^2 + y^2 + 6x + 8y + 16 = 0$$ and $$x^2 + y^2 + 2(3 - \sqrt{3})x + 2(4 - \sqrt{6})y = k + 6\sqrt{3} + 8\sqrt{6}$$, $$k > 0$$, touch internally at the point $$P(\alpha, \beta)$$, then $$(\alpha + \sqrt{3})^2 + (\beta + \sqrt{6})^2$$ is equal to ______.

Let $$A = \{1, 2, 3, 4, 5, 6, 7\}$$. Define $$B = \{T \subseteq A :$$ either $$1 \notin T$$ or $$2 \in T\}$$ and $$C = \{T \subseteq A :$$ the sum of all the elements of $$T$$ is a prime number $$\}$$. Then the number of elements in the set $$B \cup C$$ is ______.

Let $$A = \begin{pmatrix} 1 & a & a \\ 0 & 1 & b \\ 0 & 0 & 1 \end{pmatrix}$$, $$a, b \in \mathbb{R}$$. If for some $$n \in \mathbb{N}$$, $$A^n = \begin{pmatrix} 1 & 48 & 2160 \\ 0 & 1 & 96 \\ 0 & 0 & 1 \end{pmatrix}$$ then $$n + a + b$$ is equal to ______.

Let $$x = \sin(2\tan^{-1}\alpha)$$ and $$y = \sin\left(\dfrac{1}{2}\tan^{-1}\dfrac{4}{3}\right)$$. If $$S = \{\alpha \in \mathbb{R} : y^2 = 1 - x\}$$, then $$\displaystyle\sum_{\alpha \in S} 16\alpha^3$$ is equal to ______.

The sum of the maximum and minimum values of the function $$f(x) = |5x - 7| + [x^2 + 2x]$$ in the interval $$\left[\dfrac{5}{4}, 2\right]$$, where $$[t]$$ is the greatest integer $$\le t$$, is ______.

Let the area enclosed by the $$x$$-axis, and the tangent and normal drawn to the curve $$4x^3 - 3xy^2 + 6x^2 - 5xy - 8y^2 + 9x + 14 = 0$$ at the point $$(-2, 3)$$ be $$A$$. Then $$8A$$ is equal to ______.

Let $$f$$ be a twice differentiable function on $$\mathbb{R}$$. If $$f'(0) = 4$$ and $$f(x) + \displaystyle\int_0^x (x-t)f'(t) dt = (e^{2x} + e^{-2x})\cos 2x + \dfrac{2}{a}x$$, then $$(2a+1)5a^2$$ is equal to ______.

Let $$a_n = \displaystyle\int_{-1}^{n} \left(1 + \dfrac{x}{2} + \dfrac{x^2}{3} + \ldots + \dfrac{x^{n-1}}{n}\right) dx$$ for every $$n \in \mathbb{N}$$. Then the sum of all the elements of the set $$\{n \in \mathbb{N} : a_n \in (2, 30)\}$$ is ______.

Let $$y = y(x)$$ be the solution of the differential equation $$\dfrac{dy}{dx} = \dfrac{4y^3 + 2yx^2}{3xy^2 + x^3}$$, $$y(1) = 1$$. If for some $$n \in \mathbb{N}$$, $$y(2) \in [n-1, n)$$, then $$n$$ is equal to ______.