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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 51-100 of 191

  1. Let $f:\mathbb R-\left\{-\frac12\right\}\to\mathbb R$ and $g:\mathbb R-\left\{-\frac52\right\}\to\mathbb R$ be defined as $f(x)=\frac{2x+3}{2x+1}$ and …
  2. If the function $f:R-\{1,-1\}\to A$ defined by $f(x)=\frac{x^{2}}{1-x^{2}}$, is surjective, then $A$ is equal to
  3. The graph of the function $y=f(x)$ is symmetrical about the line $x=2$, then
  4. If $g(x)=x^{2}+x-1$ and $(g\circ f)(x)=4x^{2}-10x+5$, then $f\left(\frac54\right)$ is equal to
  5. Let $f$ be an odd function defined on the set of real numbers such that for $x\ge0$, $f(x)=3\sin x+4\cos x$. Then $f(x)$ at $x=-\frac{11\pi}{6}$ is equal to :
  6. Let $f(x)=\begin{cases}\dfrac{ax^2+2ax+3}{4x^2+4x-3}, & x\ne-\frac32,\frac12\\ b, & x=-\frac32,\frac12\end{cases}$ be continuous at $x=-\frac32$. If …
  7. Domain of definition of the function $f(x)=\frac{3}{4-x^{2}}+\log_{10}\left(x^{3}-x\right)$, is
  8. Let $f:\mathbb R\to\mathbb R$ be a function defined by $f(x)=\log_{\sqrt m}\left\{\sqrt2(\sin x-\cos x)+m-2\right\}$, for some $m$, such that the range of $f$ is …
  9. The domain of $f(x)=\frac{\log_{(x+1)}(x-2)}{e^{2\log_e x}-(2x+3)}$, $x\in R$ is
  10. If $f(x)=\log_e\frac{1-x}{1+x},\ |x|<1$, then $f\left(\frac{2x}{1+x^{2}}\right)$ is equal to
  11. The largest interval lying in $\left(-\frac\pi2,\frac\pi2\right)$ for which the function $f(x)=4^{-x^{2}}+\cos^{-1}\left(\frac x2-1\right)+\log(\cos x)$, is defined, is
  12. Consider a function $f:\mathbb N\to\mathbb R$, satisfying $f(1)+2f(2)+3f(3)+\ldots+xf(x)=x(x+1)f(x)$; $x\ge2$ with $f(1)=1$. Then $\frac1{f(2022)}+\frac1{f(2028)}$ is …
  13. If the domain of the function $f(x)=\log_{(10x^2-17x+7)}\left(18x^2-11x+1\right)$ is $(-\infty,a)\cup(b,c)\cup(d,\infty)-\{e\}$, then $90(a+b+c+d+e)$ equals :
  14. For $x\in R-\{0,1\}$, let $f_1(x)=\frac1x$, $f_2(x)=1-x$ and $f_3(x)=\frac1{1-x}$ be three given functions. If a function, $J(x)$ satisfies …
  15. Let the domain of the function $f(x)=\log_3\log_5\left(7-\log_2\left(x^2-10x+85\right)\right)+\sin^{-1}\left(\left|\frac{3x-7}{17-x}\right|\right)$ be $(\alpha,\beta]$. …
  16. Let $a,b,c\in\mathbb R$. If $f(x)=ax^2+bx+c$ is such that $a+b+c=3$ and $f(x+y)=f(x)+f(y)+xy$, $\forall x,y\in\mathbb R$, then $\sum_{n=1}^{10}f(n)$ is equal to :
  17. Let $\sum_{k=1}^{10}f(a+k)=16\left(2^{10}-1\right)$, where the function $f$ satisfies $f(x+y)=f(x)f(y)$ for all natural numbers $x,y$ and $f(1)=2$. Then the natural …
  18. Let $f:R\to R$ be a function such that $f(x)=\frac{x^{2}+2x+1}{x^{2}+1}$. Then
  19. Let $f(x)=-1+|x-2|$, and $g(x)=1-|x|$; then the set of all points where $f\circ g$ is discontinuous is :
  20. Let $f(x)=2x^n+\lambda$, $\lambda\in\mathbb R$, $n\in\mathbb N$, and $f(4)=133$, $f(5)=255$. Then the sum of all the positive integer divisors of $(f(3)-f(2))$ is :
  21. Let $f:R\to R$ be defined by $f(x)=\frac{x}{1+x^{2}}$, $x\in R$. Then the range of $f$ is
  22. Let $f:N\to Y$ be a function defined as $f(x)=4x+3$ where $Y=\{y\in N,\ y=4x+3\text{ for some }x\in N\}$. Show that $f$ is invertible and its inverse is
  23. Let $f(x)=\frac{x^{2}-x}{x^{2}+2x}$, $x\neq0,-2$. Then $\frac{d}{dx}\left[f^{-1}(x)\right]$ (wherever it is defined) is equal to :
  24. A real valued function $f(x)$ satisfies the functional equation $f(x-y)=f(x)f(y)-f(a-x)f(a+y)$ where $a$ is a given constant and $f(0)=1$, $f(2a-x)$ is equal to
  25. If $a\in R$ and the equation $-3(x-[x])^{2}+2(x-[x])+a^{2}=0$ (where $[x]$ denotes the greater integer $\le x$) has no integral solution, then all possible values of $a$ …
  26. Let $f:(0,1)\to R$ be a function defined by $f(x)=\frac{1}{1-e^{-x}}$, and $g(x)=\left(f(-x)-f(x)\right)$. Consider two statements (I) $g$ is an increasing function in …
  27. If the domain of the function $f(x)=\sin^{-1}\left(\frac{5-x}{3+2x}\right)+\frac{1}{\log_e(10-x)}$ is $(-\infty,\alpha]\cup[\beta,\gamma)-\{\delta\}$, then …
  28. Let $f:A\to B$ be a function defined as $f(x)=\frac{x-1}{x-2}$, where $A=\mathbb R-\{2\}$ and $B=\mathbb R-\{1\}$. Then $f$ is
  29. The domain of the definition of the function $f(x)=\frac1{4-x^{2}}+\log_{10}\left(x^{3}-x\right)$ is :
  30. If the domain of the function $f(x)=\sin^{-1}\left(\frac{1}{x^2-2x-2}\right)$ is $(-\infty,\alpha]\cup[\beta,\gamma]\cup[\delta,\infty)$, then …
  31. If $f(x)=x^{2}-x+5$, $x>\frac12$, and $g(x)$ is its inverse function, then $g'(7)$ equals :
  32. If $f:R\to S$, defined by $f(x)=\sin x-\sqrt3\cos x+1$, is onto, then the interval of $S$ is
  33. If $f(x)+2f\left(\frac1x\right)=3x$, $x\neq0$, and $S=\{x\in\mathbb R: f(x)=f(-x)\}$, then $S$
  34. Let $f$ be a composite function of $x$ defined by $f(u)=\frac{1}{u^{2}+u-2}$, $u(x)=\frac{1}{x-1}$. Then the number of points $x$ where $f$ is discontinuous is :
  35. The domain of the function $f(x)=\frac{1}{\sqrt{|x|-x}}$ is
  36. The function $f(x)=|\sin4x|+|\cos2x|$ is a periodic function with a fundamental period
  37. Given below are two statements : Statement I : The function $f:\mathbf{R}\to\mathbf{R}$ defined by $f(x)=\frac{x}{1+|x|}$ is one-one. Statement II : The function …
  38. If $f:R\to R$ satisfies $f(x+y)=f(x)+f(y)$, for all $x,y\in R$ and $f(1)=7$, then $\sum_{r=1}^{n}f(r)$ is
  39. The function $f(x)=\log\left(x+\sqrt{x^{2}+1}\right)$, is
  40. Let $f(x)=a^x$ $(a>0)$ be written as $f(x)=f_1(x)+f_2(x)$, where $f_1(x)$ is an even function and $f_2(x)$ is an odd function. Then $f_1(x+y)+f_1(x-y)$ equals :
  41. Let $f$ be a function such that $3f(x)+2f\left(\frac{m}{19x}\right)=5x$, $x\ne0$, where $m=\sum_{i=1}^{9}i^2$. Then $f(5)-f(2)$ is equal to :
  42. Let $f:R\to R$ be defined by $f(x)=\frac{|x|-1}{|x|+1}$, then $f$ is
  43. If $g(x)=3x^2+2x-3$, $f(0)=-3$ and $4g(f(x))=3x^2-32x+72$, then $f(g(2))$ is equal to :
  44. If $f(x)=\left(\frac35\right)^{x}+\left(\frac45\right)^{x}-1,\ x\in R$, then the equation $f(x)=0$ has :
  45. The function $f:N\to I$ defined by $f(x)=x-5\left[\frac x5\right]$, where $N$ is the set of natural numbers and $[x]$ denotes the greatest integer less than or equal to …
  46. A function $f$ from the set of natural numbers to integers defined by …
  47. Let $[\cdot]$ denote the greatest integer function. If the domain of the function $f(x)=\sin^{-1}\left(\frac{x+[x]}{3}\right)$ is $[\alpha,\beta)$, then …
  48. Let $[\cdot]$ denote the greatest integer function. If the domain of the function $f(x)=\cos^{-1}\left(\frac{4x+2[x]}{3}\right)$ is $[\alpha,\beta]$, then …
  49. Let $[x]$ denote the greatest integer $\le x$, where $x\in R$. If the domain of the real valued function $f(x)=\sqrt{\frac{|[x]|-2}{|[x]|-3}}$ is …
  50. Let $f(x)=2^{10}x+1$ and $g(x)=3^{10}x-1$. If $(f\circ g)(x)=x$, then $x$ is equal to :

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