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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 101-150 of 191

  1. Let $f:(0,1)\to\mathbb{R}$ be defined by $$f(x)=\frac{b-x}{1-bx},$$ where $b$ is a constant such that $0<b<1$. Then
  2. Let $S=(0,1)\cup(1,2)\cup(3,4)$ and $T=\{0,1,2,3\}$. Then which of the following statements is(are) true?
  3. If $f(x)=|x-2|$ and $g(x)=f[f(x)]$, then $g'(x)=$ ______ for $x>20$.
  4. Let the function $f:\mathbb{R}\to\mathbb{R}$ be defined by $$f(x)=\frac{\sin x}{e^{\pi x}}\frac{\left(x^{2023}+2024x+2025\right)}{\left(x^2-x+3\right)}+\frac{2}{e^{\pi …
  5. Let $f(x)=\frac{2^{x+2}+16}{2^{2x+1}+2^{x+4}+32}$. Then the value of $8\left(f\left(\frac1{15}\right)+f\left(\frac2{15}\right)+\ldots+f\left(\frac{59}{15}\right)\right)$ …
  6. Let $A=\{x\in\mathbb R: x\text{ is not a positive integer}\}$. Define a function $f:A\to\mathbb R$ as $f(x)=\frac{2x}{x-1}$, then $f$ is :
  7. Let $f$ be a polynomial function such that …
  8. $y=10^{x}$ is the reflection of $\log_{10}x$ in the line whose equation is ______.
  9. Let $\mathbb{R}$ denote the set of all real numbers. Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that $f(x)>0$ for all $x\in\mathbb{R}$, and $f(x+y)=f(x)f(y)$ for …
  10. The number of points in $(-\infty,\infty)$, for which $x^2-x\sin x-\cos x=0$, is
  11. Let $f(x)=\begin{cases}-x^{3}+\dfrac{b^{3}-b^{2}+b-1}{b^{2}+3b+2}, & 0\le x<1\\ 2x-3, & 1\le x\le3\end{cases}$ Find all possible real values of $b$ such that $f(x)$ has …
  12. For $x\in R$, $x\neq0$, $x\neq1$, let $f_0(x)=\frac{1}{1-x}$ and $f_{n+1}(x)=f_0(f_n(x))$, $n=0,1,2,\ldots$. Then the value of …
  13. If $p,q,r$ are any real numbers, then
  14. Let $f$ and $g$ be increasing and decreasing functions, respectively from $[0,\infty)$ to $[0,\infty)$. Let $h(x)=f(g(x))$. If $h(0)=0$, then $h(x)-h(1)$
  15. Let $\mathbb{N}$ denote the set of all natural numbers, and $\mathbb{Z}$ denote the set of all integers. Consider the functions $f:\mathbb{N}\to\mathbb{Z}$ and …
  16. Let $f:R-\{2,6\}\to R$ be real valued function defined as $f(x)=\frac{x^{2}+2x+1}{x^{2}-8x+12}$. Then range of $f$ is
  17. If the domain of the function $\log_5\left(18x-x^{2}-77\right)$ is $(\alpha,\beta)$ and the domain of the function …
  18. Let $f:\mathbb R-\{0,1\}\to\mathbb R$ be a function such that $f(x)+f\left(\frac1{1-x}\right)=1+x$. Then $f(2)$ is equal to :
  19. Let $f:\mathbb{R}\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ be functions defined by $$f(x)=\begin{cases}x|x|\sin\left(\frac{1}{x}\right), & x\neq0,\ 0, & …
  20. Let $f(x)=\log_e\sin x$, $0<x<\pi$ and $g(x)=\sin^{-1}\left(e^{-x}\right)$, $x\ge0$. If $\alpha$ is a positive real number such that $a=(f\circ g)'(\alpha)$ and …
  21. If $f(x)=\frac{2^{2x}}{2^{2x}+2}$, $x\in R$, then …
  22. Let $E_1=\left\{x\in\mathbb{R}:x\neq1\text{ and }\frac{x}{x-1}>0\right\}$ and …
  23. If $f(x)=\frac{2^{x}}{2^{x}+\sqrt{2}}$, $x\in\mathbb{R}$, then $\sum_{k=1}^{81}f\left(\frac{k}{82}\right)$ is equal to
  24. The absolute minimum value, of the function $f(x)=\left|x^{2}-x+1\right|+\left[x^{2}-x+1\right]$, where $[t]$ denotes the greatest integer function, in the interval …
  25. For real $x$, the function $\dfrac{(x-a)(x-b)}{x-c}$ will assume all real values provided
  26. For $x\in\left(0,\frac32\right)$, let $f(x)=\sqrt x$, $g(x)=\tan x$ and $h(x)=\frac{1-x^2}{1+x^2}$. If $\phi(x)=(h\circ f)\circ g(x)$, then $\phi\left(\frac\pi3\right)$ …
  27. Let $f(x)=x^2$ and $g(x)=\sin x$ for all $x\in\mathbb{R}$. Then the set of all $x$ satisfying $(f\circ g\circ g\circ f)(x)=(g\circ g\circ f)(x)$, where …
  28. Let the function $f:[0,1]\to\mathbb{R}$ be defined by $$f(x)=\frac{4^x}{4^x+2}.$$ Then the value of …
  29. Let $f:\mathbb{R}\to\mathbb{R}$ be a function defined by $$f(x)=\begin{cases}x^2\sin\left(\frac{\pi}{x^2}\right), & \text{if } x\ne0,\\ 0, & \text{if } x=0.\end{cases}$$ …
  30. If the function $f(x)=x^3+e^{x/2}$ and $g(x)=f^{-1}(x)$, then the value of $g'(1)$ is
  31. Let $\mathbb{R}$ denote the set of all real numbers. Let $f:\mathbb{R}\to\mathbb{R}$ and $g:\mathbb{R}\to(0,4)$ be functions defined by …
  32. Let $f_1:\mathbb{R}\to\mathbb{R}$, $f_2:[0,\infty)\to\mathbb{R}$, $f_3:\mathbb{R}\to\mathbb{R}$ and $f_4:\mathbb{R}\to[0,\infty)$ be defined by $$f_1(x)=\begin{cases}|x| …
  33. Let $f(x)=\sin\left(\frac{\pi}{6}\sin\left(\frac{\pi}{2}\sin x\right)\right)$ for all $x\in\mathbb{R}$ and $g(x)=\frac{\pi}{2}\sin x$ for all $x\in\mathbb{R}$. Let …
  34. Let $y=\sqrt{\dfrac{(x+1)(x-3)}{(x-2)}}$. Find all the real values of $x$ for which $y$ takes real values.
  35. Let $f$ be a real-valued function defined on the interval $(-1,1)$ such that $e^{-x}f(x)=2+\int_0^x\sqrt{t^4+1}\,dt$, for all $x\in(-1,1)$, and let $f^{-1}$ be the …
  36. If $\log_{0.3}(x-1)<\log_{0.09}(x-1)$, then $x$ lies in the interval
  37. Given $A=\left\{x:\dfrac{\pi}{6}\le x\le\dfrac{\pi}{3}\right\}$ and $f(x)=\cos x-x(1+x)$; find $f(A)$.
  38. State TRUE or FALSE: For $0<a<x$, the minimum value of the function $\log_a x+\log_x a$ is $2$.
  39. The range of the function $f(x)=\sqrt{3-x}+\sqrt{2+x}$ is
  40. The number of points of intersection of two curves $y=2\sin x$ and $y=5x^{2}+2x+3$ is
  41. Suppose $f:R\to(0,\infty)$ be a differentiable function such that $5f(x+y)=f(x)\cdot f(y)$, $\forall\,x,y\in R$. If $f(3)=320$, then $\sum_{n=0}^{5}f(n)$ is equal to:
  42. The function $f:[0,3]\to[1,29]$, defined by $f(x)=2x^3-15x^2+36x+1$, is
  43. Let $f: (-1, 1) \to \mathbb{R}$ be such that $f(\cos 4\theta) = \frac{2}{2 - \sec^2\theta}$ for …
  44. Let $\mathbb{R}$ denote the set of all real numbers. Let $a_i, b_i \in \mathbb{R}$ for $i \in \{1,2,3\}$. Define the functions $f:\mathbb{R}\to\mathbb{R}$ …
  45. The number of solutions of the equation $\sin\left(e^{x}\right)=5^{x}+5^{-x}$ is
  46. For $x\in R$, let $[x]$ denote the greatest integer $\le x$, then the sum of the series …
  47. Let $f:\mathbb{R}\to\mathbb{R}$ be a function defined by $f(x)=(2+3a)x^2+\left(\frac{a+2}{a-1}\right)x+b$, $a\neq 1$. If $f(x+y)=f(x)+f(y)+1-\frac{2}{7}xy$, then the …
  48. If the domain of the function $f(x)=\frac{[x]}{1+x^{2}}$, where $[x]$ is the greatest integer $\le x$, is $[2,6)$, then its range is
  49. If the function $f:\mathbb{R}\to\mathbb{R}$ is defined by $f(x)=|x|(x-\sin x)$, then which of the following statements is TRUE?
  50. Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that $f(x+y)=f(x)+f(y)$ for all $x,y\in\mathbb{R}$, and $g:\mathbb{R}\to(0,\infty)$ be a function such that …

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