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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 1-50 of 559

  1. For the function $f(x)=e^{\sin|x|}-|x|$, $x\in\mathbf{R}$, consider the following statements : Statement I : $f$ is differentiable for all $x\in\mathbf{R}$. Statement II …
  2. Let $f:S\to S$ where $S=(0,\infty)$ be a twice differentiable function such that $f(x+1)=xf(x)$. If $g:S\to\mathbf{R}$ be defined as $g(x)=\log_ef(x)$, then the value of …
  3. A wire of length $20$ m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a regular hexagon. Then the length of the side (in …
  4. The equation of the tangent to the curve $y=\int_{x^{2}}^{x^{3}}\frac{dt}{\sqrt{1+t^{2}}}$ at the point whose abscissa is $1$, is
  5. If the tangent to the curve $y=x^{3}$ at the point $P(t,t^{3})$ meets the curve again at $Q$, then the ordinate of the point which divides $PQ$ internally in the ratio …
  6. The local maximum value of the function $f(x)=\left(\frac{2}{x}\right)^{x^{2}}$, $x>0$, is :
  7. The maximum slope of the curve $y=\frac{1}{2}x^{4}-5x^{3}+18x^{2}-19x$ occurs at the point :
  8. The minimum value of $f(x)=a^{a^{x}}+a^{1-a^{x}}$, where $a,x\in\mathbf{R}$ and $a>0$, is equal to :
  9. If $f(x)=\begin{cases}\frac{1}{|x|}, & |x|\ge1\\ ax^{2}+b, & |x|<1\end{cases}$ is differentiable at every point of the domain, then the values of $a$ and $b$ are …
  10. If $f(x)=\begin{cases}\int_{0}^{x}(5+|1-t|)dt, & x>2\\ 5x+1, & x\le2\end{cases}$, then
  11. If the curve $y=ax^{2}+bx+c$, $x\in\mathbf{R}$, passes through the point $(1,2)$ and the tangent line to this curve at origin is $y=x$, then the possible values of …
  12. A box open from top is made from a rectangular sheet of dimension $a\times b$ by cutting squares each of side $x$ from each of the four corners and folding up the flaps. …
  13. Let $f:[0,\infty)\to[0,\infty)$ be defined as $f(x)=\int_{0}^{x}[y]\,dy$ where $[x]$ is the greatest integer less than or equal to $x$. Which of the following is true ?
  14. Let $f(x)=3\sin^{4}x+10\sin^{3}x+6\sin^{2}x-3$, $x\in\left[-\frac{\pi}{6},\frac{\pi}{2}\right]$. Then, $f$ is :
  15. Let $f:\mathbf{R}\to\mathbf{R}$ be defined as $f(x)=\begin{cases}-55x, & \text{if } x 4\end{cases}$ Let $A=\{x\in\mathbf{R}:f$ is increasing$\}$. Then $A$ is equal to :
  16. The triangle of maximum area that can be inscribed in a given circle of radius '$r$' is :
  17. The sum of the intercepts on the coordinate axes made by the tangent at any point on the curve, $\sqrt{x}+\sqrt{y}=3$ is
  18. If the function $f:\mathbf{R}\to\mathbf{R}$ defined by $f(x)=\frac{x^{3}}{3}-(2a+3b)x^{2}+24abx+5$ has a local maxima at $-4$ and a local minima at $4$, then a value of …
  19. Let $f:\mathbf{R}\to\mathbf{R}$ be a differentiable function such that $f(0)=2$. If $\left|\frac{df}{dx}\right|\le3$, for all $x\in\mathbf{R}$, then $f(1)$ lies in the …
  20. Let $A=\begin{bmatrix}1&3&-1\\2&1&\alpha\\0&1&-1\end{bmatrix}$ be a singular matrix. Let $f(x)=\int_{0}^{x}\left(t^{2}+2t+3\right)dt$, $x\in[1,\alpha]$. If $M$ and $m$ …
  21. If Rolle's theorem holds for the function $f(x)=x^{3}-ax^{2}+bx-4$, $x\in[1,2]$ with $f'\left(\frac{4}{3}\right)=0$, then ordered pair $(a,b)$ is equal to :
  22. The equation of the normal to the curve $y=e^{-|x|}$ at a point, where the curve cuts the line $x=-1$ is
  23. Let $f(x)$ and $g(x)$ be twice differentiable functions satisfying $f''(x)=g''(x)$ for all $x\in\mathbf{R}$, $f'(1)=2g'(1)=4$ and $g(2)=3f(2)=9$. Then $f(25)-g(25)$ is …
  24. The function $f(x)=\frac{4x^{3}-3x^{2}}{6}-2\sin x+(2x-1)\cos x$ :
  25. Let $f(x)$ be a polynomial of degree $5$, and have extrema at $x=1$ and $x=-1$. If $\lim_{x\to 0}\left(\frac{f(x)}{x^{3}}\right)=-5$, then $f(2)-f(-2)$ is equal to :
  26. Let $f$ be a real valued function, defined on $\mathbf{R}-\{-1,1\}$ and given by $f(x)=3\log_e\left|\frac{x-1}{x+1}\right|-\frac{2}{x-1}$. Then in which of the following …
  27. Let $f$ be any function defined on $\mathbf{R}$ and let it satisfy the condition : $|f(x)-f(y)|\le\left|(x-y)^{2}\right|$, $\forall(x,y)\in\mathbf{R}$. If $f(0)=1$, then …
  28. Consider the function $f:\mathbf{R}\to\mathbf{R}$ defined by $f(x)=\begin{cases}\left(2-\sin\left(\frac{1}{x}\right)\right)|x|, & x\neq0\\ 0, & x=0\end{cases}$. Then $f$ …
  29. Let $f:\mathbf{R}\to\mathbf{R}$ be defined as $f(x)=\begin{cases}-\frac{4}{3}x^{3}+2x^{2}+3x, & x>0\\ 3xe^{x}, & x\le0\end{cases}$ Then $f$ is increasing function in the …
  30. $\max_{0\le x\le\pi}\left(16\sin\left(\frac{x}{2}\right)\cos^{3}\left(\frac{x}{2}\right)\right)$ is equal to:
  31. For the function $f(x)=(\cos x)-x+1$, $x\in\mathbb{R}$, between the following two statements (S1) $f(x)=0$ for only one value of $x$ in $[0,\pi]$. (S2) $f(x)$ is …
  32. If $f(x)=\begin{cases}x^{3}\sin\left(\frac{1}{x}\right), & x\neq0\\ 0, & x=0\end{cases}$, then
  33. If $5f(x)+4f\left(\frac{1}{x}\right)=x^{2}-2$, $\forall x\neq0$ and $y=9x^{2}f(x)$, then $y$ is strictly increasing in :
  34. Let $f(x)=(x+3)^{2}(x-2)^{3}$, $x\in[-4,4]$. If $M$ and $m$ are the maximum and minimum values of $f$, respectively in $[-4,4]$, then the value of $M-m$ is
  35. Let $f(x)=ax^{3}+bx^{2}+cx+41$ be such that $f(1)=40$, $f'(1)=2$ and $f''(1)=4$. Then $a^{2}+b^{2}+c^{2}$ is equal to :
  36. Let $x=-1$ and $x=2$ be the critical points of the function $f(x)=x^{3}+ax^{2}+b\log_e|x|+1$, $x\neq0$. Let $m$ and $M$ respectively be the absolute minimum and the …
  37. The sum of absolute maximum and absolute minimum values of the function $f(x)=\left|2x^{2}+3x-2\right|+\sin x\cos x$ in the interval $[0,1]$ is :
  38. Let $x,y>0$. If $x^{3}y^{2}=2^{15}$, then the least value of $3x+2y$ is
  39. Let $f(x)=\begin{cases}\frac{\sin(x-[x])}{x-[x]}, & x\in(-2,-1)\\ \max\{2x,3[|x|]\}, & |x|<1\\ 1, & \text{otherwise}\end{cases}$ where $[t]$ denotes greatest integer …
  40. Consider the function $f:(0,\infty)\to\mathbb{R}$ defined by $f(x)=e^{-|\log_ex|}$. If $m$ and $n$ be respectively the number of points at which $f$ is not continuous …
  41. If the function $f(x)=2x^{3}-9ax^{2}+12a^{2}x+1$, $a>0$ has a local maximum at $x=\alpha$ and a local minimum at $x=\alpha^{2}$, then $\alpha$ and $\alpha^{2}$ are the …
  42. Let a variable line passing through the centre of the circle $x^{2}+y^{2}-16x-4y=0$, meet the positive co-ordinate axes at the points $A$ and $B$. Then the minimum value …
  43. Let $f:\mathbb{R}\to\mathbb{R}$ be a polynomial function of degree four having extreme values at $x=4$ and $x=5$. If $\lim_{x\to0}\frac{f(x)}{x^{2}}=5$, then $f(2)$ is …
  44. Let $a,b,c$ and $d$ be positive real numbers such that $a+b+c+d=11$. If the maximum value of $a^{5}b^{3}c^{2}d$ is $3750\beta$, then the value of $\beta$ is
  45. Let $f(x)=4\cos^{3}x+3\sqrt{3}\cos^{2}x-10$. The number of points of local maxima of $f$ in interval $(0,2\pi)$ is
  46. If the tangent at the point $(x_1,y_1)$ on the curve $y=x^{3}+3x^{2}+5$ passes through the origin, then $(x_1,y_1)$ does NOT lie on the curve :
  47. The function $f(x)=\left|x^{2}-2x-3\right|\cdot e^{\left|9x^{2}-12x+4\right|}$ is not differentiable at exactly :
  48. Let $f(x)=\frac{\sin x+\cos x-\sqrt{2}}{\sin x-\cos x}$, $x\in[0,\pi]-\left\{\frac{\pi}{4}\right\}$. Then $f\left(\frac{7\pi}{12}\right)f''\left(\frac{7\pi}{12}\right)$ …
  49. A square piece of tin of side $30$ cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. If the volume of …
  50. If the function $f(x)=2x^{3}-9ax^{2}+12a^{2}x+1$, where $a>0$, attains its local maximum and local minimum values at $p$ and $q$, respectively, such that $p^{2}=q$, then …

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