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

If $\mathrm{f}(x)=x^3-10 x^2+200 x-10$, then

2Application Of Derivatives

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

3Application Of Derivatives

The Number of values of C that satisfy the conclusion of Rolle's theorem in case of following function $\mathrm{f}(x)=\sin 2 \pi x, x \in[-1,1]$ is

4Application Of Derivatives

After $t$ seconds, the acceleration of a particle, which starts from rest and moves in a straight line is $\left(8-\frac{\mathrm{t}}{5}\right) \mathrm{cm} / \mathrm{s}^2$, then velocity of the particle at the instant, when the acceleration ...

5Application Of Derivatives

The function $\mathrm{f}(x)=2 x^3-9 x^2+12 x+2$ is decreasing in

6Application Of Derivatives

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

7Application Of Derivatives

If equation of normal to the curve $x=\sqrt{t}$, $y=\mathrm{t}-\frac{1}{\sqrt{\mathrm{t}}}$ at $\mathrm{t}=4$ is

8Application Of Derivatives

If $\mathrm{f}(x)=x^3-6 x^2+9 x+3$ is monotonically decreasing function, then $x$ lies in

9Application 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

10Application Of Derivatives

The distance ' $s$ ' in meters covered by a body in $t$ seconds is given by $s=3 t^2-8 t+5$. The body will stop after

11Application Of Derivatives

If the normal to the curve $y=\mathrm{f}(x)$ at the point $(3,4)$ makes an angle of $\left(\frac{3 \pi}{4}\right)$ with the positive $X$-axis, then the value of $f^{\prime}(3)$ is

12Application Of Derivatives

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

13Application 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 values of $a, b, c$ respectively, are

14Application Of Derivatives

If $y=\mathrm{a} \log x+\mathrm{b} x^2+x$ has its extreme values at $x=-1$ and $x=2$, then the value of $\left(\frac{a}{b}+\frac{b}{a}\right)$ is

15Application Of Derivatives

Water is running in a hemispherical bowl of radius 180 cm at the rate of 108 cubic decimeters per minute. How fast the water level is rising when depth of the water level in the bowl is 120 cm ? ( 1 decimeter $=10 \mathrm{~cm}$)

16Application Of Derivatives

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

17Application Of Derivatives

A point moves along the arc of parabola $y=2 x^2$. Its abscissa increases uniformly at the rate of 2 units $/ \mathrm{sec}$. At the instant, the point is passing through ( 1,2 ), its distance from origin is increasing at the rate of

18Application Of Derivatives

The volume of a ball is increasing at the rate of $4 \pi \mathrm{cc} / \mathrm{sec}$. The rate of increase of the radius, when the volume is $288 \pi \mathrm{cc}$, is

19Application Of Derivatives

If $y=4 x-5$ is a tangent to the curve $y^2=p x^3+q$ at $(2,3)$, then the values of $p$ and $q$ are respectively

20Application Of Derivatives

Let $\mathrm{f}(x)=(x-1)(x-2)(x-3), x \in[0,4]$. Values of C will be __________ if L.M.V.T. (Lagrange's Mean Value Theorem) can be applied.

21Application Of Derivatives

A spherical raindrop evaporates at a rate proportional to its surface area. If originally its radius is \(3 \mathrm{~mm}\) and 1 hour later it reduces to \(2 \mathrm{~mm}\), then the expression for the radius \(R\) of the raindrop at any ti...

22Application Of Derivatives

An object is moving in the clockwise direction around the unit circle \(x^2+y^2=1\). As it passes through the point \(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\), its \(y\)-co-ordinate is decreasing at the rate of 3 units per sec. The ra...

23Application Of Derivatives

The maximum value of xy when x + 2y = 8 is

24Application Of Derivatives

The value of \(\alpha\), so that the volume of the parallelopiped formed by \(\hat{i}+\alpha \hat{j}+\hat{k}, \hat{j}+\alpha \hat{k}\) and \(\alpha \hat{i}+\hat{k}\) becomes maximum, is

25Application Of Derivatives

The value of \(c\) of Lagrange's mean value theorem for \(f(x)=\sqrt{25-x^2}\) on \([1,5]\) is

26Application Of Derivatives

Let \(\mathrm{f}(0)=-3\) and \(\mathrm{f}^{\prime}(x) \leq 5\) for all real values of \(x\). The \(\mathrm{f}(2)\) can have possible maximum value as

27Application Of Derivatives

A water tank has a shape of inverted right circular cone whose semi-vertical angle is \(\tan ^{-1}\left(\frac{1}{2}\right)\). Water is poured into it at constant rate of 5 cubic meter/minute. The rate in meter/ minute at which level of wate...

28Application 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

29Application Of Derivatives

The range of values of \(x\) for which \(f(x)=x^3+6 x^2-36 x+7\) is increasing in

30Application Of Derivatives

The equation \(x^3+x-1=0\) has

31Application Of Derivatives

If slope of a tangent to the curve \(x y+a x+b y=0\) at the point \((1,1)\) on it is 2, then a - b is

32Application Of Derivatives

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

33Application Of Derivatives

Let \(\mathrm{B} \equiv(0,3)\) and \(\mathrm{C} \equiv(4,0)\). The point \(\mathrm{A}\) is moving on the line \(y=2 x\) at the rate of 2 units/second. The area of \(\triangle \mathrm{ABC}\) is increasing at the rate of

34Application Of Derivatives

Let the curve be represented by \(x=2(\cos t+t \sin t), y=2(\sin t-t \cos t)\). Then normal at any point '\(t\)' of the curve is at a distance of ______ units from the origin.

35Application Of Derivatives

Let \(x_0\) be the point of local minima of \(\mathrm{f}(x)=\overline{\mathrm{a}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{c}})\) where $$\overline{\mathrm{a}}=x \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}, \overline{...

36Application Of Derivatives

The diagonal of a square is changing at the rate of \(0.5 \mathrm{~cm} / \mathrm{sec}\). Then the rate of change of area when the area is \(400 \mathrm{~cm}^2\) is equal to

37Application Of Derivatives

If \(y=4 x-5\) is a tangent to the curve \(y^2=\mathrm{p} x^3+\mathrm{q}\) at \((2,3)\), then \(\mathrm{p}-\mathrm{q}\) is

38Application Of Derivatives

The function \(\mathrm{f}(x)=x^3-6 x^2+9 x+2\) has maximum value when \(x\) is

39Application Of Derivatives

Values of \(c\) as per Rolle's theorem for \(f(x)=\sin x+\cos x+6\) on \([0,2 \pi]\) are

40Application Of Derivatives

The slope of the normal to the curve \(x=\sqrt{t}\) and \(y=t-\frac{1}{\sqrt{t}}\) at \(t=4\) is

41Application Of Derivatives

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

42Application Of Derivatives

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

43Application Of Derivatives

Water is running in a hemispherical bowl of radius \(180 \mathrm{~cm}\) at the rate of 108 cubic decimeters per minute. How fast the water level is rising when depth of the water level in the bowl is \(120 \mathrm{~cm}\) ? (1 decimeter $$=1...

44Application Of Derivatives

\(A(1,-3), B(4,3)\) are two points on the curve \(y=x-\frac{4}{x}\). The points on the curve, the tangents at which are parallel to the chord \(A B\), are

45Application Of Derivatives

If slope of the tangent to the curve \(x y+a x+b y=0\) at the point \((1,1)\) on it is 2, then the value of \(3 a+b\) is

46Application Of Derivatives

Slope of the tangent to the curve \(y=2 e^x \sin \left(\frac{\pi}{4}-\frac{x}{2}\right) \cos \left(\frac{\pi}{4}-\frac{x}{2}\right)\), where \(0 \leq x \leq 2 \pi\) is minimum at \(x=\)

47Application Of Derivatives

A ladder 5 meters long rests against a vertical wall. If its top slides downwards at the rate of \(10 \mathrm{~cm} / \mathrm{s}\), then the angle between the ladder and the floor is decreasing at the rate of ________ rad./s when it's lower ...

48Application Of Derivatives

The function \(\mathrm{f}(x)=\sin ^4 x+\cos ^4 x\) is increasing in

49Application Of Derivatives

The angle between the tangents to the curves \(y=2 x^2\) and \(x=2 y^2\) at \((1,1)\) is

50Application Of Derivatives

A tank with a rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 4 meter and volume is 36 cubic meters. If building of the tank costs ₹ 100 per square meter for the base and ₹ 50 per square met...