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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 101-150 of 559

  1. If the absolute maximum value of the function $f(x)=(x^2-2x+7)e^{(4x^3-12x^2-180x+31)}$ in the interval $[-3,0]$ is $f(\alpha)$, then :
  2. Let $f:\mathbf{R}\to\mathbf{R}$ be a function defined by $f(x)=(x-3)^{n_1}(x-5)^{n_2}$, $n_1,n_2\in\mathbf{N}$. Then, which of the following is NOT true?
  3. Let $f$ be any function continuous on $[a,b]$ and twice differentiable on $(a,b)$. If for all $x\in(a,b)$, $f'(x)>0$ and $f''(x)<0$, then for any $c\in(a,b)$ …
  4. Let $f(x)=\begin{cases}x^3-x^2+10x-7, & x\le1\\ -2x+\log_2(b^2-4), & x>1\end{cases}$. Then the set of all values of $b$, for which $f(x)$ has maximum value at $x=1$, is :
  5. If the angle made by the tangent at the point $(x_0,y_0)$ on the curve $x=12(t+\sin t\cos t)$, $y=12(1+\sin t)^{2}$, $0<t<\frac{\pi}{2}$, with the positive $x$-axis is …
  6. Let a function $f:\mathbb{R}\to\mathbb{R}$ be defined as : $f(x)=\begin{cases}\int_0^x(5-|t-3|)\,dt, & x>4\\ x^2+bx, & x\le4\end{cases}$ where $b\in\mathbb{R}$. If $f$ …
  7. Let $f:\mathbf{R}\to\mathbf{R}$ be defined as $f(x)=x^{3}+x-5$. If $g(x)$ is a function such that $f(g(x))=x$, $\forall x\in\mathbf{R}$, then $g'(63)$ is equal to :
  8. A spherical iron ball of $10$ cm radius is coated with a layer of ice of uniform thickness that melts at a rate of $50\ \mathrm{cm}^3/\mathrm{min}$. When the thickness …
  9. Let $P$ and $Q$ be any points on the curves $(x-1)^2+(y+1)^2=1$ and $y=x^2$, respectively. The distance between $P$ and $Q$ is minimum for some value of the abscissa of …
  10. Let a function $f:[0,5]\to\mathbf{R}$ be continuous, $f(1)=3$ and $F$ be defined as : $F(x)=\int_1^xt^2g(t)\,dt$, where $g(t)=\int_1^tf(u)\,du$. Then for the function …
  11. If the minimum value of $f(x)=\frac{5x^2}{2}+\frac{\alpha}{x^5}$, $x>0$, is $14$, then the value of $\alpha$ is equal to :
  12. Let $P(h,k)$ be a point on the curve $y=x^2+7x+2$, nearest to the line, $y=3x-3$. Then the equation of the normal to the curve at $P$ is :
  13. If $xy^{4}$ attains maximum value at the point $(x,y)$ on the line passing through the points $(50+\alpha,0)$ and $(0,50+\alpha)$, $\alpha>0$, then $(x,y)$ also lies on …
  14. Let $f(x)=3^{(x^2-2)^3+4}$, $x\in\mathbf{R}$. Then which of the following statements are true ? $P$ : $x=0$ is a point of local minima of $f$. $Q$ : $x=\sqrt2$ is a …
  15. A wire of length $22$ m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, the length of the …
  16. Let $S$ be the set of all the natural numbers, for which the line $\frac{x}{a}+\frac{y}{b}=2$ is a tangent to the curve …
  17. Let $f:(-1,\infty)\to\mathbf{R}$ be defined by $f(0)=1$ and $f(x)=\frac1x\log_e(1+x)$, $x\ne0$. Then the function $f$ :
  18. Let $f(x)=2+|x|-|x-1|+|x+1|$, $x\in\mathbf{R}$. Consider (S1) : $f'\left(-\frac32\right)+f'\left(-\frac12\right)+f'\left(\frac12\right)+f'\left(\frac32\right)=2$ ; (S2) …
  19. Let $f:\mathbf{R}\to\mathbf{R}$ be a function defined by : $f(x)=\begin{cases}\max_{t\le x}\{t^{3}-3t\}; & x\le2\\ x^{2}+2x-6; & 2 5\end{cases}$ where $[t]$ is the …
  20. If $c$ is a point at which Rolle's theorem holds for the function, $f(x)=\log_e\left(\frac{x^2+\alpha}{7x}\right)$ in the interval $[3,4]$, where $\alpha\in\mathbf{R}$ …
  21. Let $m$ and $M$ respectively be the minimum and the maximum values of $f(x)=\sin^{-1}2x+\sin2x+\cos^{-1}2x+\cos2x$, $x\in\left[0,\frac{\pi}{8}\right]$. Then $m+M$ is …
  22. The minimum value of the twice differentiable function $f(x)=\int_0^xe^{x-t}f'(t)\,dt-(x^2-x+1)e^x$, $x\in\mathbf{R}$, is :
  23. The function $f(x)=xe^{x(1-x)}$, $x\in\mathbb{R}$, is :
  24. The length of the perpendicular from the origin, on the normal to the curve, $x^2+2xy-3y^2=0$ at the point $(2,2)$ is :
  25. The lengths of the sides of a triangle are $10+x^{2}$, $10+x^{2}$ and $20-2x^{2}$. If for $x=k$, the area of the triangle is maximum, then $3k^{2}$ is equal to :
  26. For all twice differentiable functions $f:\mathbf{R}\to\mathbf{R}$, with $f(0)=f(1)=f'(0)=0$,
  27. The minimum value of $\alpha$ for which the equation $\frac{4}{\sin x}+\frac{1}{1-\sin x}=\alpha$ has at least one solution in $\left(0,\frac{\pi}{2}\right)$ is …
  28. The number of points, at which the function $f(x)=|2x+1|-3|x+2|+\left|x^{2}+x-2\right|$, $x\in\mathbf{R}$ is not differentiable, is $\_\_\_\_\_\_$ .
  29. If $(a+\sqrt2b\cos x)(a-\sqrt2b\cos y)=a^2-b^2$, where $a>b>0$, then $\frac{dx}{dy}$ at $\left(\frac\pi4,\frac\pi4\right)$ is :
  30. If a rectangle is inscribed in an equilateral triangle of side length $2\sqrt{2}$ as shown in the figure, then the square of the largest area of such a rectangle is …
  31. Let $S=(-1,\infty)$ and $f:S\to\mathbb{R}$ be defined as $f(x)=\int_{-1}^{x}\left(e^{t}-1\right)^{11}(2t-1)^{5}(t-2)^{7}(t-3)^{12}(2t-10)^{61}dt$. Let $p=$ Sum of …
  32. If the function $f(x)=\begin{cases}k_1(x-\pi)^2-1, & x\le\pi\\ k_2\cos x, & x>\pi\end{cases}$ is twice differentiable, then the ordered pair $(k_1,k_2)$ is equal to :
  33. If the point $P$ on the curve, $4x^2+5y^2=20$ is farthest from the point $Q(0,-4)$, then $PQ^2$ is equal to :
  34. If '$R$' is the least value of '$a$' such that the function $f(x)=x^{2}+ax+1$ is increasing on $[1,2]$ and '$S$' is the greatest value of '$a$' such that the function …
  35. Let $f:\mathbf{R}\to\mathbf{R}$ satisfy the equation $f(x+y)=f(x)\cdot f(y)$ for all $x,y\in\mathbf{R}$ and $f(x)\neq0$ for any $x\in\mathbf{R}$. If the function $f$ is …
  36. Consider the triangles with vertices $A(2,1)$, $B(0,0)$ and $C(t,4)$, $t\in[0,4]$. If the maximum and the minimum perimeters of such triangles are obtained at $t=\alpha$ …
  37. If $y=\frac{(\sqrt{x}+1)\left(x^{2}-\sqrt{x}\right)}{x\sqrt{x}+x+\sqrt{x}}+\frac{1}{15}\left(3\cos^{2}x-5\right)\cos^{3}x$, then $96y'\left(\frac{\pi}{6}\right)$ is …
  38. Let $f(x)$ be a cubic polynomial with $f(1)=-10$, $f(-1)=6$, and has a local minima at $x=1$, and $f'(x)$ has a local minima at $x=-1$. Then $f(3)$ is equal to …
  39. The number of points in $(0,2\pi)$, where the function $f:\mathbf{R}\to\mathbf{R}$ given by $f(x)=|\sin x|+|\cos x|$ is NOT differentiable, is $\_\_\_\_\_\_$ .
  40. Let $f(x)$ be a polynomial of degree $6$ in $x$, in which the coefficient of $x^{6}$ is unity and it has extrema at $x=-1$ and $x=1$. If …
  41. If the curves $x=y^{4}$ and $xy=k$ cut at right angles, then $(4k)^{6}$ is equal to $\_\_\_\_\_\_$ .
  42. The number of points, where the curve $f(x)=e^{8x}-e^{6x}-3e^{4x}-e^{2x}+1$, $x\in\mathbb{R}$ cuts $x$-axis, is equal to $\_\_\_\_\_\_$ .
  43. Let the set of all positive values of $\lambda$, for which the point of local minimum of the function $\left(1+x\left(\lambda^{2}-x^{2}\right)\right)$ satisfies …
  44. Let for a differentiable function $f:(0,\infty)\to\mathbf{R}$, $f(x)-f(y)\ge\log_e\left(\frac{x}{y}\right)+x-y$, $\forall x,y\in(0,\infty)$. Then …
  45. Let $f:\mathbf{R}\to\mathbf{R}$ be a function defined by $f(x)=\max\{x,x^2\}$. Let $S$ denote the set of all points in $\mathbf{R}$, where $f$ is not differentiable. …
  46. Let $f(x)=x^{3}+x^{2}f'(1)+xf''(2)+f'''(3)$, $x\in\mathbf{R}$. Then $f'(10)$ is equal to $\_\_\_\_\_\_$ .
  47. The function $f(x)=\begin{cases}\frac\pi4+\tan^{-1}x, & |x|\le1\\ \frac12(|x|-1), & |x|>1\end{cases}$ is :
  48. The minimum value of $2^{\sin x}+2^{\cos x}$ is :
  49. If $y=y(x)$ is an implicit function of $x$ such that $\log_e(x+y)=4xy$, then $\frac{d^{2}y}{dx^{2}}$ at $x=0$ is equal to $\_\_\_\_\_\_$ .
  50. The function, $f(x)=(3x-7)x^{2/3}$, $x\in\mathbf{R}$, is increasing for all $x$ lying in :

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