JEE MAIN 2022 Previous Year Questions (PYQs) – Page 10 of 31

The function $f(x) = x e^{\,x(1-x)}, \, x \in \mathbb{R}$ is :

Let f : N $\to$ R be a function such that $f(x + y) = 2f(x)f(y)$ for natural numbers x and y. If f(1) = 2, then the value of $\alpha$ for which

$\sum\limits_{k = 1}^{10} {f(\alpha + k) = {{512} \over 3}({2^{20}} - 1)} $

holds, is :

The sum of the absolute maximum and absolute minimum values of the function $$f(x) = \tan^{-1}(\sin x - \cos x)$$ in the interval $[0,\pi]$ is :

Let $A$ be a $3 \times 3$ real matrix such that $A\begin{pmatrix}1 \\ 1 \\ 0\end{pmatrix} = \begin{pmatrix}1 \\ 1 \\ 0\end{pmatrix}, \quad A\begin{pmatrix}1 \\ 0 \\ 1\end{pmatrix} = \begin{pmatrix}-1 \\ 0 \\ 1\end{pmatrix}, \quad A\begin{pmatrix}0 \\ 0 \\ 1\end{pmatrix} = \begin{pmatrix}1 \\ 1 \\ 2\end{pmatrix}.$ If $X = (x_1, x_2, x_3)^T$ and $I$ is an identity matrix of order $3$, then the system $(A - 2I)X = \begin{pmatrix}4 \\ 1 \\ 1\end{pmatrix}$ has:

Let $x(t) = 2\sqrt{2}\cos t \sqrt{\sin 2t}$ and $y(t) = 2\sqrt{2}\sin t \sqrt{\sin 2t}, \; t \in (0,\tfrac{\pi}{2}).$ Then $\dfrac{1+\left(\tfrac{dy}{dx}\right)^2}{\tfrac{d^2y}{dx^2}}$ at $t=\tfrac{\pi}{4}$ is equal to :

Let f : R $\to$ R be defined as $f(x) = {x^3} + x - 5$. If g(x) is a function such that $f(g(x)) = x,\forall 'x' \in R$, then g'(63) is equal to ________________.

Let $I_n(x) = \int_0^x \dfrac{1}{(t^2+5)^n} \, dt, \; n = 1, 2, 3, \dots$ Then :

If ${1 \over {2\,.\,{3^{10}}}} + {1 \over {{2^2}\,.\,{3^9}}} + \,\,.....\,\, + \,\,{1 \over {{2^{10}}\,.\,3}} = {K \over {{2^{10}}\,.\,{3^{10}}}}$, then the remainder when K is divided by 6 is :

$ \text{The area enclosed by the curves } y=\log_{e}(x+e^{2}),; x=\log_{e}!\left(\dfrac{2}{y}\right) \text{ and } x=\log_{e}2,\ \text{above the line } y=1,\ \text{is:} $

Let f(x) be a polynomial function such that $f(x) + f'(x) + f''(x) = {x^5} + 64$. Then, the value of $\mathop {\lim }\limits_{x \to 1} {{f(x)} \over {x - 1}}$ is equal to: