JEE MAIN Function Previous Year Questions (PYQs) – Page 2 of 16

Let $f:\mathbb{R}-\{0,1\}\to\mathbb{R}$ be a function such that $f(x)+f\!\left(\frac{1}{1-x}\right)=1+x.$ Then $f(2)$ is equal to:

Let a relation $R$ on $\mathbb N\times\mathbb N$ be defined by $(x_1,y_1),R,(x_2,y_2)$ iff $x_1\le x_2$ or $y_1\le y_2$. Consider: (I) $R$ is reflexive but not symmetric. (II) $R$ is transitive. Which of the following is true?

Let $\mathbb{N}$ be the set of natural numbers and two functions $f$ and $g$ be defined as $f,g:\mathbb{N}\to\mathbb{N}$ such that $$ f(n)= \begin{cases} \dfrac{n+1}{2}, & \text{if $n$ is odd},\\[4pt] \dfrac{n}{2}, & \text{if $n$ is even}, \end{cases} \qquad g(n)=n-(-1)^n. $$ Then $f\circ g$ is –

$ \text{Let } f:\mathbb{R}\to\mathbb{R} \text{ be a function defined as } f(x)=a\sin\!\left(\frac{\pi\lfloor x\rfloor}{2}\right)+\lfloor 2-x\rfloor,\ a\in\mathbb{R}, \text{ where } \lfloor t\rfloor \text{ is the greatest integer } \le t. \text{ If } \lim_{x\to -1} f(x) \text{ exists, then the value of } \int_{0}^{4} f(x)\,dx \text{ is equal to:}$

If the domain of the function $f(x) = \sin^{-1}\left(\frac{5 - x}{3 + 2x}\right) + \log\left(\frac{1}{10 - x}\right)$ is $(-\infty, \alpha] \cup [\beta, \gamma) - {8}$, then $6(\alpha + \beta + \gamma + \delta)$ is equal to

Let a function $f: \mathbb{R} \rightarrow \mathbb{R}$ be defined as :

$f(x)= \begin{cases}\int\limits_{0}^{x}(5-|t-3|) d t, & x>4 \\ x^{2}+b x & , x \leq 4\end{cases}$

where $\mathrm{b} \in \mathbb{R}$. If $f$ is continuous at $x=4$, then which of the following statements is NOT true?

Let $f:\mathbb{R}-{0}\to\mathbb{R}$ be a function such that $f(x) - 6f\left(\frac{1}{x}\right) = \frac{35}{3x} - \frac{5}{2}.$ If $\displaystyle \lim_{x\to 0}\left(\frac{1}{x} + f(x)\right) = \beta,\ \alpha, \beta \in \mathbb{R},$ then $\alpha + 2\beta$ is equal to :

If for $p\ne q\ne 0$, the function $f(x)=\dfrac{\sqrt[7]{p(729+x)}-3}{\sqrt[3]{729+qx}-9}$ is continuous at $x=0$, then:

Let $f(x) = \dfrac{9x^2 + 16}{2^{2x+1} + 2^{x+4} + 32}$. Then the value of $8 \big( f\left(\dfrac{1}{15}\right) + f\left(\dfrac{2}{15}\right) + \dots + f\left(\dfrac{50}{15}\right) \big)$ is equal to

Let $f(x)=2+|x|-|x-1|+|x+1|,;x\in\mathbb{R}$. Consider (S1): $f'!\left(-\tfrac{3}{2}\right)+f'!\left(-\tfrac{1}{2}\right)+f'!\left(\tfrac{1}{2}\right)+f'!\left(\tfrac{3}{2}\right)=2$ (S2): $\displaystyle \int_{-2}^{2} f(x),dx = 12$