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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 151-191 of 191

  1. Let $R=\{a,b,c,d,e\}$ and $S=\{1,2,3,4\}$. Total number of onto functions $f:R\to S$ such that $f(a)\neq1$, is equal to $\_\_\_\_\_\_$ .
  2. Let $c,k\in\mathbf{R}$. If $f(x)=(c+1)x^{2}+(1-c^{2})x+2k$ and $f(x+y)=f(x)+f(y)-xy$, for all $x,y\in\mathbf{R}$, then the value of $|2(f(1)+f(2)+f(3)+\ldots+f(20))|$ is …
  3. Let $f^1(x)=\frac{3x+2}{2x+3}$, $x\in R-\left\{-\frac32\right\}$. For $n\ge2$, define $f^n(x)=f^1\circ f^{n-1}(x)$. If $f^5(x)=\frac{ax+b}{bx+a}$, $\gcd(a,b)=1$, then …
  4. Let the domain of the function $f(x)=\cos^{-1}\left(\frac{4x+5}{3x-7}\right)$ be $[\alpha,\beta]$ and the domain of $g(x)=\log_2\left(2-6\log_{27}(2x+5)\right)$ be …
  5. Let $f:\mathbf{R}\to\mathbf{R}$ be a function defined by $f(x)=\frac{2e^{2x}}{e^{2x}+e}$. Then …
  6. Let $f:\mathbf{R}\to\mathbf{R}$ and $g:\mathbf{R}\to\mathbf{R}$ be defined as $f(x)=\begin{cases}x+a, & x<0\\ |x-1|, & x\ge0\end{cases}$ and …
  7. Let $f(x)$ and $g(x)$ be two real polynomials of degree $2$ and $1$ respectively. If $f(g(x))=8x^{2}-2x$, and $g(f(x))=4x^{2}+6x+1$, then the value of $f(2)+g(2)$ is …
  8. If $f(x)=|3-x|+|2+x|+|5-x|$, $x\in\mathbf{R}$, then the minimum value of $f(x)$ is $\_\_\_\_\_\_$ .
  9. If a function $f$ satisfies $f(m+n)=f(m)+f(n)$ for all $m,n\in\mathbf{N}$ and $f(1)=1$, then the largest natural number $\lambda$ such that …
  10. If domain of the function $\log_e\left(\frac{6x^{2}+5x+1}{2x-1}\right)+\cos^{-1}\left(\frac{2x^{2}-3x+4}{3x-5}\right)$ is $(\alpha,\beta)\cup(\gamma,\delta]$, then …
  11. If $a+\alpha=1$, $b+\beta=2$ and $af(x)+\alpha f\left(\frac{1}{x}\right)=bx+\frac{\beta}{x}$, $x\neq0$, then the value of the expression …
  12. Let $S=\{1,2,3,4\}$. Then the number of elements in the set $\{f:S\times S\to S:f$ is onto and $f(a,b)=f(b,a)\ge a\ \forall(a,b)\in S\times S\}$ is $\_\_\_\_\_\_$ .
  13. Let $A=\{0,1,2,3,4,5,6,7\}$. Then the number of bijective functions $f:A\to A$ such that $f(1)+f(2)=3-f(3)$ is equal to $\_\_\_\_\_\_$ .
  14. 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 $g:S\to S$ be …
  15. Let $f:\mathbf{R}\to\mathbf{R}$ be a function defined by $f(x)=\left(2\left(1-\frac{x^{25}}{2}\right)\left(2+x^{25}\right)\right)^{\frac{1}{50}}$. If the function …
  16. For some $a,b,c\in\mathbb N$, let $f(x)=ax-3$ and $g(x)=x^{b}+c,\ x\in\mathbb R$. If $(f\circ g)^{-1}(x)=\left(\frac{x-7}{2}\right)^{\frac13}$, then …
  17. Let $f(x)$ be a polynomial of degree $3$ such that $f(k)=-\frac2k$ for $k=2,3,4,5$. Then the value of $52-10f(10)$ is equal to ______.
  18. Let $f:\mathbb R\to\mathbb R$ be a differentiable function that satisfies the relation $f(x+y)=f(x)+f(y)-1$, $\forall x,y\in\mathbb R$. If $f'(0)=2$, then $|f(-2)|$ is …
  19. Let $A=\{1,2,3,5,8,9\}$. Then the number of possible functions $f:A\to A$ such that $f(m\cdot n)=f(m)\cdot f(n)$ for every $m,n\in A$ with $m\cdot n\in A$ is equal to
  20. If the domain of the function $f(x)=\sqrt{\log_{(0.6)}\left(\left|\frac{2x-5}{x^{2}-4}\right|\right)}$ is $(-\infty,a]\cup\{b\}\cup[c,d)\cup(e,\infty)$, then the value …
  21. The value of $n$ is ____________.
  22. For real $x$, let $f(x)=x^{3}+5x+1$, then
  23. The domain of the function $f(x)=\frac{\cos^{-1}\left(\frac{x^{2}-5x+6}{x^{2}-9}\right)}{\log_e\left(x^{2}-3x+2\right)}$ is :
  24. For $x\in\mathbb{R}$, two real valued functions $f(x)$ and $g(x)$ are such that, $g(x)=\sqrt{x}+1$ and $f\circ g(x)=x+3-\sqrt{x}$. Then $f(0)$ is equal to
  25. Let $g:\mathbf{N}\to\mathbf{N}$ be defined as $g(3n+1)=3n+2$, $g(3n+2)=3n+3$, $g(3n+3)=3n+1$, for all $n\ge0$. Then which of the following statements is true ?
  26. Let the domains of the functions $f(x)=\log_4\log_3\log_7\left(8-\log_2\left(x^{2}+4x+5\right)\right)$ and $g(x)=\sin^{-1}\left(\frac{7x+10}{x-2}\right)$ be …
  27. Match the statements given in Column I with the intervals/union of intervals given in Column II. Column I: (A) The set …
  28. The domain of the function $f(x)=\dfrac{\log_{2}(x+3)}{(x+1)(x+2)}$ is
  29. Let the sum of the maximum and the minimum values of the function $f(x)=\dfrac{2x^{2}-3x+8}{2x^{2}+3x+8}$ be $\dfrac{m}{n}$, where $\gcd(m,n)=1$. Then $m+n$ is equal to :
  30. If the domain of the function $\sin^{-1}\left(\dfrac{3x-22}{2x-19}\right)+\log_e\left(\dfrac{3x^{2}-8x+5}{x^{2}-3x-10}\right)$ is $(\alpha,\beta]$, then …
  31. Let $f(x)=3\sqrt{x-2}+\sqrt{4-x}$ be a real valued function. If $\alpha$ and $\beta$ are respectively the minimum and the maximum values of $f$, then …
  32. Consider the function $f:\mathbb{R}\to\mathbb{R}$ defined by $f(x)=\dfrac{2x}{\sqrt{1+9x^{2}}}$. If the composition of $f$ …
  33. Let $A=\{1,3,7,9,11\}$ and $B=\{2,4,5,7,8,10,12\}$. Then the total number of one-one maps $f:A\to B$, such that $f(1)+f(3)=14$, is :
  34. Let $f,g:\mathbf{R}\to\mathbf{R}$ be defined as : $f(x)=|x-1|$ and $g(x)=\begin{cases}e^{x}, & x\ge0\\ x+1, & x\le0.\end{cases}$ Then the function $f(g(x))$ is
  35. Let $f(x)$ be a real differentiable function such that $f(0)=1$ and $f(x+y)=f(x)f'(y)+f'(x)f(y)$ for all $x,y\in\mathbf{R}$. Then …
  36. Let $f(x)=\log_e x$ and $g(x)=\dfrac{x^{4}-2x^{3}+3x^{2}-2x+2}{2x^{2}-2x+1}$. Then the domain of $f\circ g$ is
  37. The function $f:(-\infty,\infty)\to(-\infty,1)$, defined by $f(x)=\dfrac{2^{x}-2^{-x}}{2^{x}+2^{-x}}$ is :
  38. Let $f:[0,3]\to A$ be defined by $f(x)=2x^{3}-15x^{2}+36x+7$ and $g:[0,\infty)\to B$ be defined by $g(x)=\dfrac{x^{2025}}{x^{2025}+1}$. If both the functions are onto …
  39. Let $f:\mathbf{R}-\{0\}\to(-\infty,1)$ be a polynomial of degree $2$, satisfying $f(x)f\left(\dfrac{1}{x}\right)=f(x)+f\left(\dfrac{1}{x}\right)$. If $f(K)=-2K$, then …
  40. The range of the function …
  41. Let $f:\mathbf{R}-\left\{\dfrac{\alpha}{6}\right\}\to\mathbf{R}$ be defined by $f(x)=\dfrac{5x+3}{6x-\alpha}$. Then the value of $\alpha$ for which $(f\circ f)(x)=x$ …

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