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

The tangent to the curve $y^2 - xy + 9 = 0$ is vertical when

2Application Of Derivatives

If the function $f(x) = ax^3 + bx^2 + 11x - 6$, defined on $[1, 3]$, satisfies all the conditions of Rolle's theorem for $c = 2 + \dfrac{1}{\sqrt{3}}$, then

3Application Of Derivatives

The approximate value of $\frac{1}{(2.002)^2}$ is

4Application Of Derivatives

The minimum value of the slope of the tangent to curve $y=x^3-3 x^2+2 x+93$ is

5Application Of Derivatives

A spherical balloon is filled with $4500 \pi$ cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of $72 \pi$ cubic meters per minute, then the rate (in meters per minute) at which the radius of the bal...

6Application Of Derivatives

The area of the triangle formed by the co-ordinate axes and a tangent to the curve $x y=\mathrm{a}^2$ at the point $\left(x_1, y_1\right)$ is _______ sq. units (where a, $x_1$ and $y_1$ are non-zero)

7Application Of Derivatives

If $y=\alpha \log x+\beta x^3-x$ has extreme values at $x=-1$ and $x=1$, then $\alpha$ and $\beta$ are respectively

8Application Of Derivatives

The equation of the tangent to the curve $\left(1+x^2\right) y=2-x$, where it crosses the X -axis, is

9Application Of Derivatives

A manufacturer sells $x$ items at a price of rupees $\left(6-\frac{x}{40}\right)$ each. The cost price of $x$ items is ₹ $\left(\frac{x}{5}+193\right)$. The maximum profit in ₹ __________ is

10Application Of Derivatives

The function $\mathrm{f}(x)=[x(x-2)]^2$ is increasing in the set

11Application Of Derivatives

The minimum value of $a x+b y$ where $x y=c^2$ is

12Application Of Derivatives

The equation of the tangent to the curve $y=\mathrm{be}^{-x / \mathrm{a}}$ at the point where it crosses the Y axis is

13Application Of Derivatives

If $x$ and $y$ are sides of two squares such that $y=x-x^2$, then the rate of change of area of the second square with respect to that of the first square is

14Application Of Derivatives

The rate of change of the volume of a sphere with respect to its surface area, when the radius is 5 m is

15Application Of Derivatives

$\mathrm{f}(x)=\frac{x}{2}+\frac{2}{x}, x \neq 0$ is strictly decreasing in

16Application Of Derivatives

The angle $\theta$, at which the curves $y=3^x$ and $y=7^x$ intersect, is given by

17Application Of Derivatives

The function $\mathrm{f}(x)=x^3-6 x^2+\mathrm{ax}+\mathrm{b}$ satisfies the conditions of Rolle's theorem in $[1,3]$. Then the values of $a$ and $b$ are respectively

18Application Of Derivatives

If $\mathrm{f}(x)=\log (1+x)-\frac{2 x}{2+x}$ then $\mathrm{f}(x)$ is increasing in

19Application Of Derivatives

The length of the perpendicular drawn from the origin on the normal to the curve $x^2+2 x y-3 y^2=0$ at the point $(2,2)$ is

20Application Of Derivatives

The maximum value of $x^{2 / 3}+(x-2)^{2 / 3}$ is

21Application Of Derivatives

A particle moves along a curve $y=\frac{2 x^3-1}{3}$. The points on the curve at which the $y$ co-ordinate is changing 18 times the $x$ co-ordinate are

22Application Of Derivatives

The equation of motion of the particle is $\mathrm{s}=\mathrm{at}^2+\mathrm{bt}+\mathrm{c}$. If the displacement after 1 second is 20 m , velocity after 2 seconds is $30 \mathrm{~m} /$ seconds and the acceleration is $10 \mathrm{~m} /$ seco...

23Application Of Derivatives

The point on the curve $4 y^2-4 y+2 x-1=0$ at which the tangent becomes parallel to Y -axis is

24Application Of Derivatives

The length and breadth of a rectangle are $x_{x \mathrm{~cm}}$ and $y \mathrm{~cm}$ respectively. If the length decreases at the rate of $5 \mathrm{~cm} /$ minute and the breadth increases at the rate of $3 \mathrm{~cm} /$ minute, then the ...

25Application Of Derivatives

The combined equation of the tangent and normal to the curve $x y=15$ at the point $(5,3)$ is________

26Application Of Derivatives

The sum of two nonzero numbers is 4 . The minimum value of the sum of their reciprocals is

27Application Of Derivatives

If the line $a x+b y+c=0$ is normal to the curve $x y=1$, then

28Application Of Derivatives

The approximate value of $\sqrt[3]{64 \cdot 04}$ is

29Application Of Derivatives

If $x$ is real, then the difference between the greatest and least values of $\frac{x^2-x+1}{x^2+x+1}$ is

30Application Of Derivatives

If $\mathrm{f}(x)=x \cdot \mathrm{e}^{x(1-x)}$, then $\mathrm{f}(x)$ is

31Application Of Derivatives

An open tank with a square bottom is to contain 4000 cubic cm . of liquid. The dimensions of the tank so that the surface area of the tank is minimum, is

32Application Of Derivatives

The normal to the curve $x=9(1+\cos \theta)$, $y=9 \sin \theta$ at $\theta$ always passes through the fixed point

33Application Of Derivatives

The function $\mathrm{f}(x)=\sin ^4 x+\cos ^4 x$ increases if

34Application Of Derivatives

Let $f$ be a function which is continuous and differentiable for all $x$. If $\mathrm{f}(1)=1$ and $\mathrm{f}^{\prime}(x) \leq 5$ for all $x$ in $[1,5]$, then the maximum value of $\mathrm{f}(5)$ is

35Application Of Derivatives

The function $x^5-5 x^4+5 x^3-10$ has a maximum, when $x$ is equal to

36Application Of Derivatives

The function f defined by $\mathrm{f}(x)=(x+2) \mathrm{e}^{-x}$ is

37Application Of Derivatives

The radius of the base of a cone is increasing at the rate $3 \mathrm{~cm} /$ minute and the altitude is decreasing at the rate $4 \mathrm{~cm} /$ minute . The rate at which the lateral surface area is changing, when the radius is 7 cm and ...

38Application Of Derivatives

If the function $\mathrm{f}(x)=x(x+3) \mathrm{e}^{-\frac{x}{2}}$ satisfies all the conditions of Rolle's theorem in $[-3,0]$, then c is

39Application Of Derivatives

A wire of length 8 units is cut into two parts which are bent respectively in the form of a square and a circle. The least value of the sum of the areas so formed is

40Application Of Derivatives

If $\mathrm{f}(x)=\frac{\mathrm{k} \sin x+2 \cos x}{\sin x+\cos x}$ is strictly increasing for all real values of $x$, then

41Application Of Derivatives

Let $x$ be the length of each of the equal sides of an isosceles triangle and $\theta$ be the angle between these sides. If $x$ is increasing at the rate $\frac{1}{12} \mathrm{~m} /$ hour and $\theta$ is increasing at the rate $\frac{\pi}{1...

42Application Of Derivatives

The abscissae of the points of the curve $y=x^3$ are in the interval $[-2,2]$, where the slope of the tangents can be obtained by mean value theorem for the interval $[-2,2]$ are

43Application Of Derivatives

The shortest distance between the line $y-x=1$ and the curve $x=y^2$ is

44Application Of Derivatives

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

45Application Of Derivatives

20 is divided into two parts so that the product of the cube of one part and the square of the other part is maximum, then these two parts are

46Application Of Derivatives

A manufacturer produces $x$ items per week at a total cost of ₹ $\left(x^2+78 x+2500\right)$. The price per unit is given by $8 x=600-\mathrm{p}$ where ' p ' is the price of each unit. Then the maximum profit obtained is

47Application Of Derivatives

If $2 \mathrm{f}(x)+3 \mathrm{f}\left(\frac{1}{x}\right)=x^2+1, x \neq 0$ and $y=5 x^2 \mathrm{f}(x)$, then $y$ is strictly increasing in

48Application Of Derivatives

If the curve $y=a x^2-6 x+b$ passes through $(0,4)$ and has its tangent parallel to the X-axis at $x=\frac{3}{2}$, then the values of $a$ and $b$ respectively are

49Application Of Derivatives

The angle between the curves $x y=6$ and $x^2 y=12$ is

50Application Of Derivatives

By dropping a stone in a quiet lake, a wave in the form of circle is generated. The radius of the circular wave increases at the rate of $2.1 \mathrm{~cm} / \mathrm{sec}$. Then the rate of increase of the enclosed circular region, when the ...