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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 151-200 of 559

  1. Let $f:(0,\infty)\to(0,\infty)$ be a differentiable function such that $f(1)=e$ and $\lim_{t\to x}\frac{t^2f^2(x)-x^2f^2(t)}{t-x}=0$. If $f(x)=1$, then $x$ is equal to :
  2. Let $A(4,-2)$, $B(1,1)$ and $C(9,-3)$ be the vertices of a triangle $ABC$. Then the maximum area of the parallelogram $AFDE$, formed with vertices $D,E$ and $F$ on the …
  3. Let $a\in\mathbb{Z}$ and $[t]$ be the greatest integer $\le t$. Then the number of points, where the function $f(x)=[a+13\sin x]$, $x\in(0,\pi)$ is not differentiable …
  4. Let $f$ be a twice differentiable function on $(1,6)$. If $f(2)=8$, $f'(2)=5$, $f'(x)\ge1$ and $f''(x)\ge4$, for all $x\in(1,6)$, then :
  5. The set of all real values of $\lambda$ for which the function $f(x)=(1-\cos^2x)\cdot(\lambda+\sin x)$, $x\in\left(-\frac\pi2,\frac\pi2\right)$, has exactly one maxima …
  6. Let $f:(-2,2)\to\mathbb{R}$ be defined by $f(x)=\begin{cases}x[x], & -2<x<0\\ (x-1)[x], & 0\le x<2\end{cases}$ where $[x]$ denotes the greatest integer function. If $m$ …
  7. Let $m$ and $n$ be the number of points at which the function $f(x)=\max\{x,x^{3},x^{5},\ldots,x^{21}\}$, $x\in\mathbb{R}$, is not differentiable and not continuous …
  8. Let $f:\mathbf{R}\to\mathbf{R}$ be a function defined as $f(x)=\begin{cases}3\left(1-\frac{|x|}{2}\right) & \text{if } |x|\le2\\ 0 & \text{if } |x|>2\end{cases}$ Let …
  9. Let $f(x)=2^{x}-x^{2}$, $x\in\mathbb{R}$. If $m$ and $n$ are respectively the number of points at which the curves $y=f(x)$ and $y=f'(x)$ intersect the $x$-axis, then …
  10. Let $f:[-1,1]\to\mathbf{R}$ be defined as $f(x)=ax^{2}+bx+c$ for all $x\in[-1,1]$, where $a,b,c\in\mathbf{R}$ such that $f(-1)=2$, $f'(-1)=1$ and for $x\in(-1,1)$ the …
  11. The number of points, where the curve $y=x^{5}-20x^{3}+50x+2$ crosses the $x$-axis, is $\_\_\_\_\_\_$ .
  12. The local minimum value of $\frac{(5+x)(2+x)}{(1+x)}$, $(x\neq-1)$, is $\_\_\_\_\_\_$ .
  13. Let the set of all values of $p$, for which $f(x)=\left(p^{2}-6p+8\right)\left(\sin^{2}2x-\cos^{2}2x\right)+2(2-p)x+7$ does not have any critical point, be the interval …
  14. Let $f(x)=|(x-1)(x^{2}-2x-3)|+x-3$, $x\in\mathbf{R}$. If $m$ and $M$ are respectively the number of points of local minimum and local maximum of $f$ in the interval …
  15. Suppose $f(x)$ is a polynomial of degree four, having critical points at $-1,0,1$. If $T=\{x\in\mathbf{R}\mid f(x)=f(0)\}$, then the sum of squares of all the elements …
  16. A function $f$ is defined on $[-3,3]$ as $f(x)=\begin{cases}\min\{|x|,\ 2-x^{2}\}, & -2\le x\le2\\ [|x|], & 2<|x|\le3\end{cases}$ where $[x]$ denotes the greatest …
  17. A wire of length $36$ m is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two …
  18. If the surface area of a cube is increasing at a rate of $3.6\ \mathrm{cm}^2/\mathrm{sec}$, retaining its shape; then the rate of change of its volume (in …
  19. If the function $f(x)=\begin{cases}\frac{1}{|x|}, & |x|\ge2\\ ax^{2}+2b, & |x|<2\end{cases}$ is differentiable on $\mathbf{R}$, then $48(a+b)$ is equal to $\_\_\_\_\_\_$ …
  20. If $f(x)=\sin\left(\cos^{-1}\left(\frac{1-2^{2x}}{1+2^{2x}}\right)\right)$ and its first derivative with respect to $x$ is $-\frac{b}{a}\log_e2$ when $x=1$, where $a$ …
  21. Let $f(x)=\begin{cases}e^{x-1}, & x<0\\ x^{2}-5x+6, & x\ge0\end{cases}$ and $g(x)=f(|x|)+|f(x)|$. If the number of points where $g$ is not continuous and not …
  22. If the tangent to the curve, $y=f(x)=x\log_ex$, ($x>0$) at a point $(c,f(c))$ is parallel to the line-segment joining the points $(1,0)$ and $(e,e)$, then $c$ is equal …
  23. Let $k$ and $m$ be positive real numbers such that the function $f(x)=\begin{cases}3x^{2}+k\sqrt{x+1}, & 0 0$. Then $\frac{8f'(8)}{f'\left(\frac{1}{8}\right)}$ is equal …
  24. Let $P(x)$ be a real polynomial of degree $3$ which vanishes at $x=-3$. Let $P(x)$ have local minima at $x=1$, local maxima at $x=-1$ and $\int_{-1}^{1}P(x)\,dx=18$ …
  25. Let $f(t)=\int_{0}^{t}e^{x^{3}}\left(\frac{x^{8}}{\left(x^{6}+2x^{3}+2\right)^{2}}\right)dx$. If $f(1)+f'(1)=\alpha e-\frac{1}{6}$, then the value of $150\alpha$ is …
  26. Let $f(x)$ be a polynomial of degree $3$ such that $f(-1)=10$, $f(1)=-6$, $f(x)$ has a critical point at $x=-1$ and $f'(x)$ has a critical point at $x=1$. Then $f(x)$ …
  27. An air duct is made of a thick metallic sheet. The duct is open from front and back. Its cross-section is a trapezium of base angles $60^\circ$ and area $1.5$ m$^2$. Its …
  28. A water tank has the shape of a right circular cone with axis vertical and vertex downwards. Its semi-vertical angle is $\tan^{-1}\frac34$. Water is poured in it at a …
  29. Let $f$ and $g$ be twice differentiable even functions on $(-2,2)$ such that $f\left(\frac{1}{4}\right)=0$, $f\left(\frac{1}{2}\right)=0$, $f(1)=1$ and …
  30. Let $f:\mathbf{R}\to\mathbf{R}$ satisfy $f(x+y)=2^{x}f(y)+4^{y}f(x)$, $\forall x,y\in\mathbf{R}$. If $f(2)=3$, then $14\cdot\frac{f'(4)}{f'(2)}$ is equal to …
  31. The number of real roots of the equation $e^{4x}-e^{3x}-4e^{2x}-e^x+1=0$ is equal to ________.
  32. For the curve $C:(x^2+y^2-3)+(x^2-y^2-1)^5=0$, the value of $3y'-y^3y''$, at the point $(\alpha,\alpha)$, $\alpha>0$, on $C$, is equal to ________.
  33. Let the normal at a point $P$ on the curve $y^2-3x^2+y+10=0$ intersect the $y$-axis at $\left(0,\frac32\right)$. If $m$ is the slope of the tangent at $P$ to the curve …
  34. Let $M$ and $N$ be the number of points on the curve $y^5-9xy+2x=0$, where the tangents to the curve are parallel to $x$-axis and $y$-axis, respectively. Then the value …
  35. Let $f(x)=\begin{cases}|4x^2-8x+5|, & \text{if } 8x^2-6x+1\ge0\\ [4x^2-8x+5], & \text{if } 8x^2-6x+1<0\end{cases}$ where $[\alpha]$ denotes the greatest integer less …
  36. If the tangent to the curve, $y=e^x$ at a point $(c,e^c)$ and the normal to the parabola, $y^2=4x$ at the point $(1,2)$ intersect at the same point on the $x$-axis, then …
  37. Suppose a differentiable function $f(x)$ satisfies the identity $f(x+y)=f(x)+f(y)+xy^2+x^2y$, for all real $x$ and $y$. If $\lim_{x\to0}\frac{f(x)}x=1$, then $f'(3)$ is …
  38. Let $f:[0,3]\to\mathbf{R}$ be defined by $f(x)=\min\{x-[x],1+[x]-x\}$ where $[x]$ is the greatest integer less than or equal to $x$. Let $P$ denote the set containing …
  39. Let $f:[0,1]\to\mathbf{R}$ be a twice differentiable function in $(0,1)$ such that $f(0)=3$ and $f(1)=5$. If the line $y=2x+3$ intersects the graph of $f$ at only two …
  40. Let $l$ be a line which is normal to the curve $y=2x^{2}+x+2$ at a point $P$ on the curve. If the point $Q(6,4)$ lies on the line $l$ and $O$ is origin, then the area of …
  41. The number of integral values of $b$ for which $2x^3-3x^2-12x+b=0$ has three distinct real roots is ________.
  42. If the tangent to the curve $y=x^3-x^2+x$ at the point $(a,b)$ is also tangent to the curve $y=5x^2+2x-25$ at the point $(2,-1)$, then $|2a+9b|$ is equal to ________.
  43. If $y(x)=\left(x^{x}\right)^{x}$, $x>0$, then $\frac{d^{2}x}{dy^{2}}+20$ at $x=1$ is equal to $\_\_\_\_\_\_$ .
  44. Let $f:R\to R$ be a function defined by $f(x)=\min\{x+1,|x|+1\}$, then which of the following is true?
  45. Let $f:R\to R$ be a continuous function defined by $f(x)=\frac{1}{e^{x}+2e^{-x}}$. Statement - 1: $f(c)=\frac13$, for some $c\in R$. Statement - 2 …
  46. Given $P(x)=x^{4}+ax^{3}+bx^{2}+cx+d$ such that $x=0$ is the only real root of $P'(x)=0$. If $P(-1)<P(1)$, then in the interval $[-1,1]$ :
  47. If $p$ and $q$ are positive real numbers such that $p^{2}+q^{2}=1$, then the maximum value of $(p+q)$ is
  48. Let $f:R\to R$ be defined by $f(x)=\begin{cases}k-2x,&\text{if }x\le-1\\2x+3,&\text{if }x>-1\end{cases}$ If $f$ has a local minimum at $x=-1$, then a possible value of …
  49. The function $f(x)=\tan^{-1}(\sin x+\cos x)$ is an increasing function in
  50. For $x\in\left(0,\frac{5\pi}{2}\right)$, define $f(x)=\int_{0}^{x}\sqrt t\sin t\,dt$. Then $f$ has

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