Skip to main content
Yukti Vault

Calculus

Differentiation and its Application: JEE Main & Advanced PYQs

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

Practice this chapter

Questions 51-100 of 559

  1. Let $a>0$. If the function $f(x)=6x^{3}-45ax^{2}+108a^{2}x+1$ attains its local maximum and minimum values at the points $x_1$ and $x_2$ respectively such that …
  2. For the differentiable function $f:\mathbb{R}-\{0\}\to\mathbb{R}$, let $3f(x)+2f\left(\frac{1}{x}\right)=\frac{1}{x}-10$, then …
  3. Let a variable line of slope $m>0$ passing through the point $(4,-9)$ intersect the coordinate axes at the points $A$ and $B$. The minimum value of the sum of the …
  4. Let $f:\mathbb{R}-\{0\}\to\mathbb{R}$ be a function satisfying $f\left(\frac{x}{y}\right)=\frac{f(x)}{f(y)}$ for all $x,y$, $f(y)\neq0$. If $f'(1)=2024$, then
  5. If $\log_ey=3\sin^{-1}x$, then $\left(1-x^{2}\right)y''-xy'$ at $x=\frac{1}{2}$ is equal to
  6. Let $f$ be a non-negative function in $[0,1]$ and twice differentiable in $(0,1)$. If $\int_{0}^{x}\sqrt{1-\left(f'(t)\right)^{2}}\,dt=\int_{0}^{x}f(t)\,dt$ …
  7. The interval in which the function $f(x)=x^{x}$, $x>0$, is strictly increasing is
  8. The maximum area of a triangle whose one vertex is at $(0,0)$ and the other two vertices lie on the curve $y=-2x^{2}+54$ at points $(x,y)$ and $(-x,y)$, where $y>0$, is :
  9. Let $[x]$ denote the greatest integer function and $f(x)=\max\{1+x+[x],\ 2+x,\ x+2[x]\}$, $0\le x\le2$. Let $m$ be the number of points in $[0,2]$, where $f$ is not …
  10. The surface area of a balloon of spherical shape being inflated, increases at a constant rate. If initially, the radius of balloon is $3$ units and after $5$ seconds, it …
  11. Let $f:[2,4]\to\mathbb{R}$ be a differentiable function such that $(x\log_ex)f'(x)+(\log_ex)f(x)+f(x)\ge1$, $x\in[2,4]$ with $f(2)=\frac{1}{2}$ and $f(4)=\frac{1}{4}$. …
  12. Let $\lambda^{*}$ be the largest value of $\lambda$ for which the function $f_\lambda(x)=4\lambda x^{3}-36\lambda x^{2}+36x+48$ is increasing for all $x\in\mathbf{R}$. …
  13. Suppose for a differentiable function $h$, $h(0)=0$, $h(1)=1$ and $h'(0)=h'(1)=2$. If $g(x)=h\left(e^{x}\right)e^{h(x)}$, then $g'(0)$ is equal to :
  14. Consider the function $f:\left[\frac{1}{2},1\right]\to\mathbb{R}$ defined by $f(x)=4\sqrt{2}x^{3}-3\sqrt{2}x-1$. Consider the statements (I) The curve $y=f(x)$ …
  15. For the function $f(x)=4\log_e(x-1)-2x^{2}+4x+5$, $x>1$, which one of the following is NOT correct ?
  16. $\max_{0\le x\le\pi}\left\{x-2\sin x\cos x+\frac{1}{3}\sin3x\right\}=$
  17. Let $f(x)=\left|2x^{2}+5|x|-3\right|$, $x\in\mathbf{R}$. If $m$ and $n$ denote the number of points where $f$ is not continuous and not differentiable respectively, then …
  18. If $y=\tan^{-1}\left(\sec x^{3}-\tan x^{3}\right)$, $\frac{\pi}{2}<x^{3}<\frac{3\pi}{2}$, then
  19. Let $f$ be any continuous function on $[0,2]$ and twice differentiable on $(0,2)$. If $f(0)=0$, $f(1)=1$ and $f(2)=2$, then :
  20. Let $f:\mathbf{R}\to\mathbf{R}$ be defined as : $f(x)=\begin{cases}\frac{a-b\cos2x}{x^{2}}; & x 1\end{cases}$ If $f$ is continuous everywhere in $\mathbf{R}$ and $m$ is …
  21. The number of critical points of the function $f(x)=(x-2)^{2/3}(2x+1)$ is
  22. If the function $f(x)=\left(\frac{1}{x}\right)^{2x}$; $x>0$ attains the maximum value at $x=\frac{1}{e}$ then :
  23. If $2x^{y}+3y^{x}=20$, then $\frac{dy}{dx}$ at $(2,2)$ is equal to:
  24. Let $g:\mathbf{R}\to\mathbf{R}$ be a non constant twice differentiable function such that $g'\left(\frac{1}{2}\right)=g'\left(\frac{3}{2}\right)$. If a real valued …
  25. If $y(x)=\begin{vmatrix}\sin x&\cos x&\sin x+\cos x+1\\27&28&27\\1&1&1\end{vmatrix}$, $x\in\mathbb{R}$, then $\frac{d^{2}y}{dx^{2}}+y$ is equal to
  26. A variable line $L$ passes through the point $(3,5)$ and intersects the positive coordinate axes at the points $A$ and $B$. The minimum area of the triangle $OAB$, where …
  27. Let the function $f(x)=\frac{x}{3}+\frac{3}{x}+3$, $x\neq0$ be strictly increasing in $(-\infty,\alpha_1)\cup(\alpha_2,\infty)$ and strictly decreasing in …
  28. Let $f:\mathbb{R}\to\mathbb{R}$ be a function defined by $f(x)=\big||x+2|-2|x|\big|$. If $m$ is the number of points of local minima and $n$ is the number of points of …
  29. Suppose $f(x)=\frac{\left(2^{x}+2^{-x}\right)\tan x\sqrt{\tan^{-1}\left(x^{2}-x+1\right)}}{\left(7x^{2}+3x+1\right)^{3}}$. Then the value of $f'(0)$ is equal to
  30. The sum of the absolute minimum and the absolute maximum values of the function $f(x)=\left|3x-x^{2}+2\right|-x$ in the interval $[-1,2]$ is :
  31. The curve $y(x)=ax^3+bx^2+cx+5$ touches the $x$-axis at the point $P(-2,0)$ and cuts the $y$-axis at the point $Q$, where $y'$ is equal to $3$. Then the local maximum …
  32. The value of $\log_e2\,\frac{d}{dx}\left(\log_{\cos x}\mathrm{cosec}\,x\right)$ at $x=\frac\pi4$ is
  33. Let $x(t)=2\sqrt2\cos t\sqrt{\sin2t}$ and $y(t)=2\sqrt2\sin t\sqrt{\sin2t}$, $t\in\left(0,\frac\pi2\right)$. Then …
  34. The number of points, where the function $f:\mathbf{R}\to\mathbf{R}$, $f(x)=|x-1|\cos|x-2|\sin|x-1|+(x-3)|x^2-5x+4|$, is NOT differentiable, is :
  35. Let $f:(a,b)\to\mathbf{R}$ be twice differentiable function such that $f(x)=\int_a^xg(t)\,dt$ for a differentiable function $g(x)$. If $f(x)=0$ has exactly five distinct …
  36. Let $f(x)=\min\{1,1+x\sin x\}$, $0\le x\le2\pi$. If $m$ is the number of points, where $f$ is not differentiable and $n$ is the number of points, where $f$ is not …
  37. If the tangent to the curve $y=x+\sin y$ at a point $(a,b)$ is parallel to the line joining $\left(0,\frac32\right)$ and $\left(\frac12,2\right)$, then :
  38. If $m$ and $n$ respectively are the number of local maximum and local minimum points of the function $f(x)=\int_{0}^{x^{2}}\frac{t^{2}-5t+4}{2+e^{t}}dt$, then the …
  39. The number of real roots of the equation, $e^{4x}+e^{3x}-4e^{2x}+e^x+1=0$ is :
  40. If $p(x)$ be a polynomial of degree three that has a local maximum value $8$ at $x=1$ and a local minimum value $4$ at $x=2$; then $p(0)$ is equal to :
  41. The equation of the normal to the curve $y=(1+x)^{2y}+\cos^2(\sin^{-1}x)$ at $x=0$ is :
  42. The sum of the absolute maximum and absolute minimum values of the function $f(x)=\tan^{-1}(\sin x-\cos x)$ in the interval $[0,\pi]$ is :
  43. Consider a cuboid of sides $2x,4x$ and $5x$ and a closed hemisphere of radius $r$. If the sum of their surface areas is a constant $k$, then the ratio $x:r$, for which …
  44. If $\cos^{-1}\left(\frac{y}{2}\right)=\log_e\left(\frac{x}{5}\right)^{5}$, $|y|<2$, then :
  45. Let $f(x)=x\cos^{-1}(-\sin|x|)$, $x\in\left[-\frac\pi2,\frac\pi2\right]$, then which of the following is true ?
  46. Let $f:[0,\infty)\to[0,3]$ be a function defined by $f(x)=\begin{cases}\max\{\sin t:0\le t\le x\}, & 0\le x\le\pi\\ 2+\cos x, & x>\pi\end{cases}$. Then which of the …
  47. Let $f$ and $g$ be differentiable functions on $\mathbf{R}$ such that $f\circ g$ is the identity function. If for some $a,b\in\mathbf{R}$, $g'(a)=5$ and $g(a)=b$, then …
  48. Let $f(x)=4x^{3}-11x^{2}+8x-5$, $x\in\mathbf{R}$. Then $f$ :
  49. Let $f$ be a differentiable function in $\left(0,\frac{\pi}{2}\right)$. If $\int_{\cos x}^{1}t^{2}f(t)\,dt=\sin^{3}x+\cos x$, then …
  50. Water is being filled at the rate of $1\ \text{cm}^{3}/\text{sec}$ in a right circular conical vessel (vertex downwards) of height $35$ cm and diameter $14$ cm. When the …

Other chapters