Application of Derivatives PYQs - Last 5 Years

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Practice 227 MHT CET Mathematics questions from Application of Derivatives. Use the year-wise and type-wise breakdown to prioritize recent PYQs, then continue into the question list below.

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Last 5 Years Application of Derivatives Questions

Showing 50 of 227 filtered questions.

1Application Of Derivatives

A population $p(t)$ of 1000 bacteria introduced into a nutrient medium grows according to the relation $\mathrm{p}(\mathrm{t})=1000+\frac{1000 \mathrm{t}}{100+\mathrm{t}^2}$. The maximum size of this bacterial population is

2Application Of Derivatives

In the mean value theorem, $f^{\prime}(c)=\frac{f(b)-f(a)}{b-a}$, if $\mathrm{a}=0, \mathrm{~b}=\frac{1}{2}$ and $\mathrm{f}(x)=x(x-1)(x-2)$, then the value of $c$ is

3Application Of Derivatives

The position of a point in time $t$ is given by $x=\mathrm{a}+\mathrm{bt}-\mathrm{ct}^2, y=\mathrm{at}+\mathrm{bt}^2$. It's resultant acceleration at time $t$ in seconds is given by

4Application Of Derivatives

If two curves $x^2-4 y^2=2$ and $8 x^2=40-\mathrm{m} y^2$ are orthogonal to each other then $\mathrm{m}=$

5Application Of Derivatives

The equation of tangent to the curve $y=\cos (x+y)$ where $-2 \pi \leq x \leq 2 \pi$ and which is parallel to the line $x+2 y=0$, is

6Application Of Derivatives

If Mean value theorem holds for the function $\mathrm{f}(x)=(x-1)(x-2)(x-3), x \in[0,4]$ then the values of $c$ as per the theorem are

7Application Of Derivatives

If the curves $y^2=6 x, 9 x^2+\mathrm{b} y^2=16$ intersect each other at right angles, then the value of $b$ is

8Application Of Derivatives

The maximum value of $\frac{\log x}{x}$ is

9Application Of Derivatives

If $\mathrm{f}(x)=\frac{\log x}{x}(x>0)$, then it is increasing in

10Application Of Derivatives

Let C be a curve given by $y(x)=1+\sqrt{4 x-3}$, $x>\frac{3}{4}$. If P is a point on C , such that the tangent at P has slope $\frac{2}{3}$, then a point through which the normal at P passes, is

11Application Of Derivatives

A poster is to be printed on a rectangular sheet of paper of area $18 \mathrm{~m}^2$. The margins at the top and bottom of 75 cm each and at the sides 50 cm each are to be left. Then the dimensions i.e. height and breadth of the sheet so th...

12Application Of Derivatives

The function $\mathrm{f}(x)=2 x^3-6 x+5$ is an increasing function, if

13Application Of Derivatives

A square plate is contracting at the uniform rate $3 \mathrm{~cm}^2 / \mathrm{sec}$, then the rate at which the perimeter is decreasing, when the side of the square is 15 cm , is

14Application Of Derivatives

The equation of the tangent to the curve $x=\operatorname{acos}^3 \theta, y=\operatorname{asin}^3 \theta$ at $\theta=\frac{\pi}{4}$ is

15Application Of Derivatives

A wire of length 2 units is cut into two parts, which are bent respectively to form a square of side $x$ units and a circle of radius of r units. If the sum of the areas of square and the circle so formed is minimum, then

16Application Of Derivatives

If sum of two numbers is 3 , then the maximum value of the product of first number and square of the second number is

17Application Of Derivatives

The sum of intercepts on coordinate axes made by tangent to the curve $\sqrt{x}+\sqrt{y}=\sqrt{a}$ is

18Application Of Derivatives

If $8 \mathrm{f}(x)+6 \mathrm{f}\left(\frac{1}{x}\right)=x+5$ and $y=x^2 \mathrm{f}(x)$, then $\frac{\mathrm{d} y}{\mathrm{~d} x}$ at $x=-1$ is

19Application Of Derivatives

The function $f(x)=\frac{\log _e(\pi+x)}{\log _e(e+x)}$ is

20Application Of Derivatives

The equation of the tangent to the curve $y=1-\mathrm{e}^{\frac{x}{3}}$ at the point of intersection with Y -axis is

21Application Of Derivatives

A ladder 5 m long rests against a vertical wall. If its top slides downwards at the rate of $10 \mathrm{~cm} / \mathrm{sec}$., then the foot of the ladder is sliding at the rate of _________ $\mathrm{m} / \mathrm{sec}$., when it is 4 m away...

22Application Of Derivatives

A triangular park is enclosed on two sides by a fence and on the third side a straight river bank. The two sides having fence are of same length $x$. The maximum area (in sq. units) enclosed by the park is

23Application Of Derivatives

The equation of the normal to the curve $x=\theta+\sin \theta, y=1+\cos \theta$ at $\theta=\frac{\pi}{2}$ is

24Application Of Derivatives

A bullet is shot horizontally and its distance S cm at time t second is given by $\mathrm{S}=1200 \mathrm{t}-15 \mathrm{t}^2$, then the distance covered by the bullet when it comes to the rest, is

25Application Of Derivatives

If the half life of substance is 5 years, then the total amount of the substance left after 15 years, when initial amount is 64 gms is

26Application Of Derivatives

A stone is dropped into a quiet lake and waves move in circles at speed of $8 \mathrm{~cm} / \mathrm{sec}$. At the instant when the radius of the circular wave is 12 cm . how fast is the enclosed area increasing?

27Application Of Derivatives

The minimum value of the function $\mathrm{f}(x)=2 x^3-15 x^2+36 x-48$ on the set $\mathrm{A}=\left\{x \mid x^2+20 \leqslant 9 x\right\}$ is

28Application Of Derivatives

The value of c for which Rolle's theorem for the function $\mathrm{f}(x)=x^3-3 x^2+2 x$ in the interval $[0,2]$ are

29Application Of Derivatives

The curve $y=a x^3+b x^2+c x+5$ touches the $x$-axis at $(-2,0)$ and cuts the $y$-axis at a point Q where its gradient is 3 , then the value of $\mathrm{a}+\mathrm{b}+\mathrm{c}$ is

30Application Of Derivatives

If $y=a \log x+b x^2+x$ has its extreme value at $x=-1$ and $x=2$, then the value of $a+b$ is

31Application Of Derivatives

If $\mathrm{f}(x)=x^3+b x^2+c x+d$ and $0< b^2< c$, then in $(-\infty, \infty)$

32Application Of Derivatives

If Rolle's theorem holds for the function $\mathrm{f}(x)=x^3+\mathrm{bx}{ }^2+\mathrm{ax}+5$ on $[1,3]$ with $\mathrm{c}=2+\frac{1}{\sqrt{3}}$, then the values of $a$ and $b$ respectively are

33Application Of Derivatives

The length of the longest interval, in which the function $3 \sin x-4 \sin ^3 x$ is increasing, is

34Application Of Derivatives

The normal to the curve, $y(x-2)(x-3)=x+6$ at the point, where the curve intersects the Y-axis, passes through the point

35Application Of Derivatives

The equation of normal to the curve $x=\theta+\sin \theta, y=1+\cos \theta$ at $\theta=\frac{\pi}{2}$ is

36Application Of Derivatives

The co-ordinates of a point on the curve $y=x \log x$ at which the normal is parallel to the line $2 x-2 y=3$ are

37Application Of Derivatives

The approximate value of $\sqrt[3]{0.026}$ is

38Application Of Derivatives

The maximum value of the function
\(f(x)=3 x^3-18 x^2+27 x-40\)
on the set $\mathrm{S}=\left\{x \in \mathbb{R} / x^2+30 \leq 11 x\right\}$ is

39Application Of Derivatives

The value of C for which Mean value Theorem holds for the function $\mathrm{f}(x)=\log _e x$ on the interval $[1,3]$ is

40Application Of Derivatives

If $\mathrm{f}(1)=1, \mathrm{f}^{\prime}(1)=3$, then the derivative of $\mathrm{f}(\mathrm{f}(\mathrm{f}(x)))+(\mathrm{f}(x))^2$ at $x=1$ is

41Application Of Derivatives

The abscissa of the point on the curve $y=\mathrm{a}\left(\mathrm{e}^{\frac{x}{a}}+\mathrm{e}^{-\frac{x}{a}}\right)$ where the tangent is parallel to the X -axis is

42Application Of Derivatives

If $y=a \log x+b x^2+x$ has its extremum values at $x=-1$ and $x=2$, then

43Application Of Derivatives

If $\theta$ denotes the acute angle between the curves $y=10-x^2$ and $y=2+x^2$, at a point of the intersection, then $|\tan \theta|$ is equal to

44Application Of Derivatives

The set of all points, for which $f(x)=x^2 e^{-x}$ strictly increases, is

45Application Of Derivatives

If $x=-1$ and $x=2$ are extreme points of $f(x)=\alpha \log |x|+\beta x^2+x$, then

46Application Of Derivatives

The equation of the normal to the curve $y=x \log x$ parallel to $2 x-2 y+3=0$ is

47Application Of Derivatives

Water is being poured at the rate of $36 \mathrm{~m}^3 / \mathrm{min}$ into a cylindrical vessel, whose circular base is of radius 3 meters. Then the water level in the cylinder is rising at the rate of

48Application Of Derivatives

The rate of change of the volume of a sphere with respect to its surface area, when its radius is 2 cm , is _________ $\mathrm{cm}^3 / \mathrm{cm}^2$.

49Application Of Derivatives

A ladder 5 m in length is leaning against a wall. The bottom of the ladder is pulled along the ground away from the wall, at the rate of $2 \mathrm{~m} / \mathrm{sec}$. How fast is the height on the wall decreasing when the foot of the ladd...

50Application Of Derivatives

Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq.m) of the flowerbed is