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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.

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Questions 451-500 of 559

  1. Let $f(x)$ be a non-constant twice differentiable function defined on $(-\infty,\infty)$ such that $f(x)=f(1-x)$ and $f'\left(\frac14\right)=0$. Then,
  2. Let $f(x)=2+\cos x$ for all real $x$. STATEMENT-1: For each real $t$, there exists a point $c$ in $[t,t+\pi]$ such that $f'(c)=0$. because STATEMENT-2: $f(t)=f(t+2\pi)$ …
  3. $\frac{d^2x}{dy^2}$ equals
  4. The total number of local maxima and local minima of the function $$f(x)=\begin{cases}(2+x)^3, & -3<x\le-1\\ x^{2/3}, & -1<x<2\end{cases}$$ is
  5. Let $g(x)=\log f(x)$ where $f(x)$ is a twice differentiable positive function on $(0,\infty)$ such that $f(x+1)=x\,f(x)$. Then, for $N=1,2,3,\ldots$ …
  6. The minimum value of the sum of real numbers $a^{-5}$, $a^{-4}$, $3a^{-3}$, $1$, $a^8$ and $a^{10}$ with $a>0$ is
  7. Let $f:\mathbb{R}\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ be respectively given by $f(x)=|x|+1$ and $g(x)=x^2+1$. Define $h:\mathbb{R}\to\mathbb{R}$ by …
  8. The slope of the tangent to the curve $(y-x^5)^2=x(1+x^2)^2$ at the point $(1,3)$ is
  9. Let $f:\mathbb R\to\mathbb R$ be differentiable at $c\in\mathbb R$ and $f(c)=0$. If $g(x)=|f(x)|$, then at $x=c$, $g$ is :
  10. The equation of a normal to the curve, $\sin y=x\sin\left(\frac\pi3+y\right)$ at $x=0$, is :
  11. Let $f:R\to R$ be a function such that $|f(x)|\le x^{2}$, for all $x\in R$. Then, at $x=0$, $f$ is
  12. Statement-1: The equation $x\log x=2-x$ is satisfied by at least one value of $x$ lying between $1$ and $2$. Statement-2: The function $f(x)=x\log x$ is an increasing …
  13. The maximum value of the function $f(x)=2x^3-15x^2+36x-48$ on the set $A=\{x\mid x^2+20\le 9x\}$ is
  14. Let $p(x)$ be a polynomial of degree 4 having extremum at $x=1,2$ and $$\lim_{x\to0}\left(1+\frac{p(x)}{x^2}\right)=2.$$ Then the value of $p(2)$ is
  15. For the function $$f(x)=x\cos\frac1x,\quad x\ge1,$$
  16. Let f, g and h be real-valued functions defined on the interval $[0,1]$ by $f(x)=e^{x^2}+e^{-x^2}$, $g(x)=xe^{x^2}+e^{-x^2}$ and $h(x)=x^2e^{x^2}+e^{-x^2}$. If a, b and …
  17. Let $f$ be a function defined on $\mathbf{R}$ (the set of all real numbers) such that $f'(x)=2010(x-2009)(x-2010)^2(x-2011)^3(x-2012)^4$, for all $x\in\mathbf{R}$. If …
  18. Let $f(x)=\begin{cases} x^2\left|\cos\dfrac{\pi}{x}\right|, & x\neq 0\\ 0, & x=0\end{cases}$, $x\in\mathbb{R}$, then $f$ is
  19. Let $f:\mathbb{R}\to\mathbb{R}$ be defined as $f(x)=|x|+|x^2-1|$. The total number of points at which $f$ attains either a local maximum or a local minimum is
  20. Let $p(x)$ be a real polynomial of least degree which has a local maximum at $x=1$ and a local minimum at $x=3$. If $p(1)=6$ and $p(3)=2$, then $p'(0)$ is
  21. If $f(x) = \int_0^x e^{t^2}(t-2)(t-3)\,dt$ for all $x \in (0, \infty)$, then
  22. A rectangular sheet of fixed perimeter with sides having their lengths in the ratio $8:15$ is converted into an open rectangular box by folding after removing squares of …
  23. The function $f(x)=2|x|+|x+2|-||x+2|-2|x||$ has a local minimum or a local maximum at $x=$
  24. A cylindrical container is to be made from certain solid material with the following constraints: It has a fixed inner volume of $V$ mm$^3$, has a 2 mm thick solid wall …
  25. Let $g:\mathbb{R}\to\mathbb{R}$ be a differentiable function with $g(0)=0$, $g'(0)=0$ and $g'(1)\ne0$. Let $$f(x)=\begin{cases}\frac{x}{|x|}g(x), & x\ne0\\ 0, & …
  26. The least value of $\alpha\in\mathbb{R}$ for which $4\alpha x^2+\frac{1}{x}\ge1$, for all $x>0$, is
  27. Let $f:\mathbb{R}\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ be two non-constant differentiable functions. If $$f'(x)=\left(e^{(f(x)-g(x))}\right)g'(x)\quad\text{for …
  28. Consider all rectangles lying in the region $$\left\{(x,y)\in\mathbb{R}\times\mathbb{R}: 0\le x\le\frac\pi2 \text{ and } 0\le y\le 2\sin(2x)\right\}$$ and having one …
  29. Let $f:(0,\pi)\to\mathbb{R}$ be a twice differentiable function such that $$\lim_{t\to x}\frac{f(x)\sin t-f(t)\sin x}{t-x}=\sin^2 x\text{ for all }x\in(0,\pi).$$ If …
  30. Let the function $f:(0,\pi)\to\mathbb{R}$ be defined by $$f(\theta)=(\sin\theta+\cos\theta)^2+(\sin\theta-\cos\theta)^4.$$ Suppose the function $f$ has a local minimum …
  31. A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is $1$ cm, the ice-cream melts at the rate …
  32. The sum of all local minimum values of the function $f(x)=\begin{cases}1-2x, & x 2\end{cases}$ is
  33. The line $y=x$ meets $y=ke^x$ for $k\leq0$ at
  34. The positive value of $k$ for which $ke^x-x=0$ has only one root is
  35. For $k>0$, the set of all values of $k$ for which $ke^x-x=0$ has two distinct roots is
  36. The correct statement(s) is(are)
  37. Which of the following is true?
  38. The value of $2m_1+3n_1+m_1n_1$ is ___ .
  39. If $f(-10\sqrt2)=2\sqrt2$, then $f''(-10\sqrt2)=$
  40. Which of the following options is the only CORRECT combination?
  41. Which of the following is true?
  42. The value of $6m_2+4n_2+8m_2n_2$ is ___.
  43. Which of the following options is the only CORRECT combination?
  44. Which of the following is true?
  45. Consider the statements: $P$: There exists some $x \in \mathbb{R}$ such that $f(x) + 2x = 2(1 + x^2)$ $Q$: There exists some $x \in \mathbb{R}$ such that …
  46. Let $$g(x)=\int_0^{e^x}\frac{f'(t)}{1+t^2}\,dt.$$ Which of the following is true?
  47. Which of the following options is the only INCORRECT combination?
  48. The function $f'(x)$ is
  49. $$\begin{array}{ll}\textbf{List I} & \textbf{List II}\\ \text{P. Let } y(x)=\cos(3\cos^{-1}x),\ x\in[-1,1],\ x\ne\pm\frac{\sqrt3}{2}. \text{ Then } …
  50. Two towns $A$ and $B$ are $60$ km apart. A school is to be built to serve $150$ students in town $A$ and $50$ students in town $B$. If the total distance to be travelled …

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