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Yukti Vault

Calculus

Functions: JEE Main & Advanced PYQs

191 previous year questions on Functions from JEE Main and JEE Advanced papers, each with its exam and year, an answer and a solution.

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Questions 1-50 of 191

  1. If the domain of the function $f(x)=\frac{\cos^{-1}\sqrt{x^{2}-x+1}}{\sqrt{\sin^{-1}\left(\frac{2x-1}{2}\right)}}$ is the interval $(\alpha,\beta]$, then $\alpha+\beta$ …
  2. The domain of the function $f(x)=\sin^{-1}\left(\frac{3x^{2}+x-1}{(x-1)^{2}}\right)+\cos^{-1}\left(\frac{x-1}{x+1}\right)$ is :
  3. If $[x]$ be the greatest integer less than or equal to $x$, then $\sum_{n=8}^{100}\left[\frac{(-1)^{n}n}{2}\right]$ is equal to :
  4. Let $f$ and $g$ be two functions defined by $f(x)=\begin{cases}x+1, & x<0\\ |x-1|, & x\ge0\end{cases}$ and $g(x)=\begin{cases}x+1, & x<0\\ 1, & x\ge0\end{cases}$. Then …
  5. Let $f,g:(1,\infty)\to\mathbb{R}$ be defined as $f(x)=\frac{2x+3}{5x+2}$ and $g(x)=\frac{2-3x}{1-x}$. If the range of the function $f\circ g:[2,4]\to\mathbb{R}$ is …
  6. The inverse of $y=5^{\log x}$ is :
  7. Let $f:\mathbf{R}\to\mathbf{R}$ be defined as $f(x)=\frac{2x^{2}-3x+2}{3x^{2}+x+3}$. Then $f$ is :
  8. For the function $f:[1,\infty)\to[1,\infty)$ defined by $f(x)=(x-1)^{4}+1$, among the two statements: (I) The set $S=\{x\in[1,\infty):f(x)=f^{-1}(x)\}$ contains exactly …
  9. Let $f(x)=\frac{x-1}{x+1}$, $x\in\mathbf{R}-\{0,-1,1\}$. If $f^{n+1}(x)=f(f^{n}(x))$ for all $n\in\mathbf{N}$, then $f^{6}(6)+f^{7}(7)$ is equal to :
  10. Let the domain of the function $f(x)=\log_2\log_4\log_6\left(3+4x-x^{2}\right)$ be $(a,b)$. If $\int_{0}^{b-a}\left[x^{2}\right]dx=p-\sqrt{q}-\sqrt{r}$ …
  11. If $f(x)=\frac{4x+3}{6x-4}$, $x\neq\frac{2}{3}$ and $(f\circ f)(x)=g(x)$, where $g:\mathbb{R}-\left\{\frac{2}{3}\right\}\to\mathbb{R}-\left\{\frac{2}{3}\right\}$, then …
  12. Let for some $\alpha\in\mathbb{R}$, $f:\mathbb{R}\to\mathbb{R}$ be a function satisfying $f(x+y)=f(x)+2y^{2}+y+\alpha xy$ for all $x,y\in\mathbb{R}$. If $f(0)=-1$ and …
  13. Let $f:\mathbf{R}\to\mathbf{R}$ be defined as $f(x)=x-1$ and $g:\mathbf{R}-\{1,-1\}\to\mathbf{R}$ be defined as $g(x)=\frac{x^{2}}{x^{2}-1}$. Then the function …
  14. The domain of the function $f(x)=\frac{1}{\sqrt{[x]^{2}-3[x]-10}}$ is (where $[x]$ denotes the greatest integer less than or equal to $x$)
  15. Let $f:\mathbf{R}\to\mathbf{R}$ be such that $f(xy)=f(x)f(y)$, for all $x,y\in\mathbf{R}$ and $f(0)\neq0$. Let $g:[1,\infty)\to\mathbf{R}$ be a differentiable function …
  16. Let $A=\{1,2,3,\ldots,10\}$ and $f:A\to A$ be defined as $f(k)=\begin{cases}k+1 & \text{if } k \text{ is odd}\\ k & \text{if } k \text{ is even}\end{cases}$ Then the …
  17. If the function $f:(-\infty,-1]\to(a,b]$ defined by $f(x)=e^{x^{3}-3x+1}$ is one-one and onto, then the distance of the point $P(2b+4,a+2)$ from the line $x+e^{-3}y=4$ …
  18. If $f(x)=\begin{cases}2+2x, & -1\le x<0\\ 1-\frac{x}{3}, & 0\le x\le3\end{cases}$; $g(x)=\begin{cases}-x, & -3\le x\le0\\ x, & 0<x\le1\end{cases}$, then range of …
  19. If the domain of the function $f(x)=\sin^{-1}\left(\frac{x-1}{2x+3}\right)$ is $\mathbf{R}-(\alpha,\beta)$, then $12\alpha\beta$ is equal to :
  20. If the domain of the function $f(x)=\log_e\left(\frac{2x+3}{4x^{2}+x-3}\right)+\cos^{-1}\left(\frac{2x-1}{x+2}\right)$ is $(\alpha,\beta]$, then the value of …
  21. Let $f(x)=\frac{1}{7-\sin5x}$ be a function defined on $\mathbf{R}$. Then the range of the function $f(x)$ is equal to :
  22. If the domain of the function $f(x)=\frac{\sqrt{x^{2}-25}}{(4-x^{2})}+\log_{10}\left(x^{2}+2x-15\right)$ is $(-\infty,\alpha)\cup[\beta,\infty)$, then …
  23. Let $f(x)=\sin^{-1}x$ and $g(x)=\frac{x^{2}-x-2}{2x^{2}-x-6}$. If $g(2)=\lim_{x\to2}g(x)$, then the domain of the function $f\circ g$ is :
  24. Let $f,g:\mathbf{N}\to\mathbf{N}$ such that $f(n+1)=f(n)+f(1)\ \forall n\in\mathbf{N}$ and $g$ be any arbitrary function. Which of the following statements is NOT true ?
  25. Let $f:\mathbf{R}\to\mathbf{R}$ and $g:\mathbf{R}\to\mathbf{R}$ be two functions defined by $f(x)=\log_e\left(x^{2}+1\right)-e^{-x}+1$ and …
  26. A function $f(x)$ is given by $f(x)=\frac{5^{x}}{5^{x}+5}$, then the sum of the series …
  27. Let $f:\mathbf{N}\to\mathbf{N}$ be a function such that $f(m+n)=f(m)+f(n)$ for every $m,n\in\mathbf{N}$. If $f(6)=18$, then $f(2)\cdot f(3)$ is equal to :
  28. The real valued function $f(x)=\frac{\operatorname{cosec}^{-1}x}{\sqrt{x-[x]}}$, where $[x]$ denotes the greatest integer less than or equal to $x$, is defined for all …
  29. If $f(x)=\frac{(\tan1^{\circ})x+\log_e(123)}{x\log_e(1234)-(\tan1^{\circ})}$, $x>0$, then the least value of $f(f(x))+f\left(f\left(\frac{4}{x}\right)\right)$ is
  30. Consider functions $f:A\to B$ and $g:B\to C$ ($A,B,C\subseteq\mathbf{R}$) such that $(g\circ f)^{-1}$ exists, then :
  31. Let $f:\mathbf{R}\to\mathbf{R}$ be defined as $f(x)=2x-1$ and $g:\mathbf{R}-\{1\}\to\mathbf{R}$ be defined as $g(x)=\frac{x-\frac{1}{2}}{x-1}$. Then the composition …
  32. If the domain of the function $f(x)=\cos^{-1}\left(\frac{2-|x|}{4}\right)+\left\{\log_e(3-x)\right\}^{-1}$ is $[-\alpha,\beta)-\{\gamma\}$, then $\alpha+\beta+\gamma$ is …
  33. Let $f(x)=\begin{cases}-a & \text{if } -a\le x\le0\\ x+a & \text{if } 0 0$ and $g(x)=(f(|x|)-|f(x)|)/2$. Then the function $g:[-a,a]\to[-a,a]$ is
  34. Let $[x]$ denote the greatest integer less than or equal to $x$. Then, the values of $x\in\mathbf{R}$ satisfying the equation …
  35. If the domain of the function $f(x)=\log_7\left(1-\log_4\left(x^{2}-9x+18\right)\right)$ is $(\alpha,\beta)\cup(\gamma,\delta)$, then $\alpha+\beta+\gamma+\delta$ is …
  36. Let $f$ be a function such that $f(x)+3f\left(\frac{24}{x}\right)=4x$, $x\neq0$. Then $f(3)+f(8)$ is equal to
  37. If the domain of the function $f(x)=\log_e\left(4x^{2}+11x+6\right)+\sin^{-1}(4x+3)+\cos^{-1}\left(\frac{10x+6}{3}\right)$ is $(\alpha,\beta]$, then $36|\alpha+\beta|$ …
  38. Let the range of the function $f(x)=\frac{1}{2+\sin3x+\cos3x}$, $x\in\mathbb{R}$ be $[a,b]$. If $\alpha$ and $\beta$ are respectively the A.M. and the G.M. of $a$ and …
  39. The function $f:\mathbf{N}-\{1\}\to\mathbf{N}$; defined by $f(n)=$ the highest prime factor of $n$, is :
  40. If the range of the function $f(x)=\frac{5-x}{x^{2}-3x+2}$, $x\neq1,2$, is $(-\infty,\alpha]\cup[\beta,\infty)$, then $\alpha^{2}+\beta^{2}$ is equal to :
  41. The function $f(x)=\frac{x^{2}+2x-15}{x^{2}-4x+9}$, $x\in\mathbb{R}$ is
  42. Let $f,g:\mathbf{R}\to\mathbf{R}$ be two real valued functions defined as $f(x)=\begin{cases}-|x+3|, & x<0\\ e^{x}, & x\ge0\end{cases}$ and …
  43. Let $f:\mathbf{N}\to\mathbf{R}$ 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 …
  44. Let $x$ denote the total number of one-one functions from a set $A$ with $3$ elements to a set $B$ with $5$ elements and $y$ denote the total number of one-one functions …
  45. Let $f:\mathbf{R}-\{3\}\to\mathbf{R}-\{1\}$ be defined by $f(x)=\frac{x-2}{x-3}$. Let $g:\mathbf{R}\to\mathbf{R}$ be given as $g(x)=2x-3$. Then, the sum of all the …
  46. Let $f:\mathbf{R}\to\mathbf{R}$ and $g:\mathbf{R}\to\mathbf{R}$ be defined as $f(x)=\begin{cases}\log_ex, & x>0\\ e^{-x}, & x\le0\end{cases}$ and …
  47. Let a function $f:\mathbf{N}\to\mathbf{N}$ be defined by …
  48. If the functions are defined as $f(x)=\sqrt{x}$ and $g(x)=\sqrt{1-x}$, then what is the common domain of the following functions : $f+g$, $f-g$, $f/g$, $g/f$, $g-f$ …
  49. If the domain of the function $f(x)=\log_e\left(\frac{2x-3}{5+4x}\right)+\sin^{-1}\left(\frac{4+3x}{2-x}\right)$ is $[\alpha,\beta)$, then $\alpha^{2}+4\beta$ is equal to
  50. If the domain of the function $f(x)=\frac{1}{\sqrt{10+3x-x^{2}}}+\frac{1}{\sqrt{x+|x|}}$ is $(a,b)$, then $(1+a)^{2}+b^{2}$ is equal to :

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