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 501-550 of 559

  1. Match the statements/expressions in Column I with the open intervals in Column II. $$\begin{array}{ll}\textbf{Column I}&\textbf{Column II}\\ (A)\ \text{Interval …
  2. In the following $[x]$ denotes the greatest integer less than or equal to $x$. Match the functions in Column I with the properties in Column II and indicate your answer …
  3. Match the statements/expressions given in Column I with the values given in Column II. $$\begin{array}{ll}\textbf{Column I} & \textbf{Column II}\\ \text{(A) The number …
  4. Prove that the minimum value of $\dfrac{(a+x)(b+x)}{c+x}$, $a,b>c$, $x>-c$, is $\left(\sqrt{a-c}+\sqrt{b-c}\right)^{2}$.
  5. If $ax^{2}+\dfrac{b}{x}\ge c$ for all positive $x$, where $a>0$ and $b>0$, show that $27ab^{2}\ge4c^{3}$.
  6. Show that $1+x\ln\left(x+\sqrt{x^{2}+1}\right)\ge\sqrt{1+x^{2}}$ for all $x\ge0$.
  7. If $(a+bx)e^{y/x}=x$, then prove that $x^{3}\dfrac{d^{2}y}{dx^{2}}=\left(x\dfrac{dy}{dx}-y\right)^{2}$.
  8. Show that $2\sin x+\tan x\ge3x$ where $0\le x<\dfrac{\pi}{2}$.
  9. If $x=\sec\theta-\cos\theta$ and $y=\sec^{n}\theta-\cos^{n}\theta$, then show that $\left(x^{2}+4\right)\left(\dfrac{dy}{dx}\right)^{2}=n^{2}\left(y^{2}+4\right)$.
  10. If $y=\dfrac{ax^{2}}{(x-a)(x-b)(x-c)}+\dfrac{bx}{(x-b)(x-c)}+\dfrac{c}{x-c}+1$, prove that …
  11. If $P(1)=0$ and $\dfrac{dP(x)}{dx}>P(x)$ for all $x\ge1$ then prove that $P(x)>0$ for all $x>1$.
  12. Using Rolle's theorem, prove that there is at least one root in $\left(45^{1/100},\,46\right)$ of the polynomial $P(x)=51x^{101}-2323x^{100}-45x+1035$.
  13. Using the relation $2(1-\cos x)<x^2$, $x\neq0$ or otherwise, prove that $\sin(\tan x)\ge x$, $\forall x\in\left[0,\dfrac{\pi}{4}\right]$.
  14. Prove that for $x\in\left[0,\dfrac{\pi}{2}\right]$, $\sin x+2x\ge\dfrac{3x(x+1)}{\pi}$. Explain the identity, if any, used in the proof.
  15. A spherical balloon is filled with $4500\pi$ cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of $72\pi$ cubic meters per …
  16. Let $a,b\in\mathbf{R}$ be such that the function $f$ given by $f(x)=\ln|x|+bx^2+ax$, $x\ne0$ has extreme values at $x=-1$ and $x=2$. Statement 1 : $f$ has local maximum …
  17. For all $x$ in $[0,1]$, let the second derivative $f''(x)$ of a function $f(x)$ exist and satisfy $|f''(x)|<1$. If $f(0)=f(1)$, then show that $|f'(x)|<1$ for all $x$ in …
  18. If $f(x)$ and $g(x)$ are differentiable functions for $0\le x\le1$ such that $f(0)=2$, $g(0)=0$, $f(1)=6$, $g(1)=2$, then show that there exists $c$ satisfying $0<c<1$ …
  19. Suppose $p(x)=a_0+a_1x+a_2x^{2}+\cdots+a_nx^{n}$. If $|p(x)|\le\left|e^{x-1}-1\right|$ for all $x\ge0$, prove that $|a_1+2a_2+\cdots+na_n|\le1$.
  20. $f(x)$ is a differentiable function and $g(x)$ is a twice differentiable function such that $|f(x)|\le1$ and $f'(x)=g(x)$. If $f^{2}(0)+g^{2}(0)=9$, prove that there …
  21. Let $f:\mathbb{R}\to\mathbb{R}$ be twice differentiable and satisfy $f(x+y)=f(x)+f(y)$. If $f'(0)=1$, then which of the following is true?
  22. Equation of a tangent to the curve, $y=2\displaystyle\int_{0}^{x}|t|\,dt$ which is parallel to the line, $2x-y=13$ is
  23. If $f$ be a twice differentiable function satisfying $f(x)=x^{4}+x^{2}f''(1)+f''(2)$, then the modulus of the minimum value of $f(x)$ is equal to $\_\_\_\_\_\_\_\_$ .
  24. The range of $a\in\mathbf{R}$ for which the function $f(x)=(4a-3)\left(x+\log_e5\right)+2(a-7)\cot\left(\dfrac{x}{2}\right)\sin^{2}\left(\dfrac{x}{2}\right)$ …
  25. If the normal to the curve $y(x)=\displaystyle\int_{0}^{x}\left(2t^{2}-15t+10\right)dt$ at a point $(a,b)$ is parallel to the line $x+3y=-5$, $a>1$, then the value of …
  26. Let $f(x)=x^{5}+2e^{x/4}$ for all $x\in\mathbf{R}$. Consider a function $g(x)$ such that $(g\circ f)(x)=x$ for all $x\in\mathbf{R}$. Then the value of $8g'(2)$ is :
  27. Let $f:\mathbb{R}\to\mathbb{R}$ be a thrice differentiable function such that $f(0)=0$, $f(1)=1$, $f(2)=-1$, $f(3)=2$ and $f(4)=-2$. Then, the minimum number of zeros of …
  28. Let a rectangle $ABCD$ of sides $2$ and $4$ be inscribed in another rectangle $PQRS$ such that the vertices of the rectangle $ABCD$ lie on the sides of the rectangle …
  29. For the function $f(x)=\sin x+3x-\dfrac{2}{\pi}\left(x^{2}+x\right)$, where $x\in\left[0,\dfrac{\pi}{2}\right]$, consider the following two statements : (I) $f$ is …
  30. Let $f(x)=x^{5}+2x^{3}+3x+1$, $x\in\mathbf{R}$, and $g(x)$ be a function such that $g(f(x))=x$ for all $x\in\mathbf{R}$. Then $\dfrac{g(7)}{g'(7)}$ is equal to :
  31. Let $f$ be a differentiable function in the interval $(0,\infty)$ such that $f(1)=1$ and $\displaystyle\lim_{t\to x}\dfrac{t^{2}f(x)-x^{2}f(t)}{t-x}=1$ for each $x>0$. …
  32. Let the maximum and minimum values of $\left(\sqrt{8x-x^{2}-12}-4\right)^{2}+(x-7)^{2}$, $x\in\mathbf{R}$ be $M$ and $m$, respectively. Then $M^{2}-m^{2}$ is equal to …
  33. If $y(\theta)=\dfrac{2\cos\theta+\cos2\theta}{\cos3\theta+4\cos2\theta+5\cos\theta+2}$, then at $\theta=\dfrac{\pi}{2}$, $y''+y'+y$ is equal to :
  34. The derivative of $\tan^{-1}\left(\dfrac{\sqrt{1+x^{2}}-1}{x}\right)$ with respect to $\tan^{-1}\left(\dfrac{2x\sqrt{1-x^{2}}}{1-2x^{2}}\right)$ at $x=\dfrac{1}{2}$ is :
  35. Which of the following points lies on the tangent to the curve $x^{4}e^{y}+2\sqrt{y+1}=3$ at the point $(1,0)$?
  36. If $x=1$ is a critical point of the function $f(x)=\left(3x^{2}+ax-2-a\right)e^{x}$, then :
  37. If the lines $x+y=a$ and $x-y=b$ touch the curve $y=x^{2}-3x+2$ at the points where the curve intersects the $x$-axis, then $\dfrac{a}{b}$ is equal to $\_\_\_\_\_\_\_\_$ …
  38. The position of a moving car at time $t$ is given by $f(t)=at^{2}+bt+c$, $t>0$, where $a,b$ and $c$ are real numbers greater than $1$. Then the average speed of the car …
  39. Let $AD$ and $BC$ be two vertical poles at $A$ and $B$ respectively on a horizontal ground. If $AD=8$ m, $BC=11$ m and $AB=10$ m; then the distance (in meters) of a …
  40. Let $f(x)=\displaystyle\int_{0}^{x^{2}}\dfrac{t^{2}-8t+15}{e^{t}}\,dt$, $x\in\mathbf{R}$. Then the numbers of local maximum and local minimum points of $f$ …
  41. If the set of all values of $a$, for which the equation $5x^{3}-15x-a=0$ has three distinct real roots, is the interval $(\alpha,\beta)$, then $\beta-2\alpha$ is equal …
  42. Let $(2,3)$ be the largest open interval in which the function $f(x)=2\log_e(x-2)-x^{2}+ax+1$ is strictly increasing and $(b,c)$ be the largest open interval, in which …
  43. The normal to the curve $x=a(1+\cos\theta)$, $y=a\sin\theta$ at $\theta$ always passes through the fixed point
  44. The normal to the curve $x=a(\cos\theta+\theta\sin\theta)$, $y=a(\sin\theta-\theta\cos\theta)$ at any point $\theta$ is such that
  45. If the equation $a_nx^{n}+a_{n-1}x^{n-1}+\cdots+a_1x=0$, $a_1\ne0$, $n\ge2$, has a positive root $x=\alpha$, then the equation …
  46. Let $f$ be differentiable for all $x$. If $f(1)=-2$ and $f'(x)\ge2$ for $x\in[1,6]$, then
  47. Angle between the tangents to the curve $y=x^{2}-5x+6$ at the points $(2,0)$ and $(3,0)$ is
  48. The normal to a curve at $P(x,y)$ meets the $x$-axis at $G$. If the distance of $G$ from the origin is twice the abscissa of $P$, then the curve is a :
  49. A value of $c$ for which the conclusion of Mean Value Theorem holds for the function $f(x)=\log_ex$ on the interval $[1,3]$ is
  50. The equation of the tangent to the curve $y=x+\frac{4}{x^{2}}$, that is parallel to the $x$-axis, is

Other chapters