Application of Derivatives PYQs - Last 5 Years

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Practice 48 COMEDK 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 48 of 48 filtered questions.

1Application Of Derivatives

If $\mathbf{3 ~ c m} / \mathbf{s}$ is the rate at which the side of an equilateral triangle increases, then the rate of change of area, when the side is $\mathbf{1 2 ~ c m}$ is:

2Application Of Derivatives

If the function $f(x)=x^4-31 x^2+\boldsymbol{a} x+5$ has a turning point at $x=1$, then the value of ' $\boldsymbol{a}$ ' is $\_\_\_\_$ and the function attains a $\_\_\_\_$ at $x=1$

3Application Of Derivatives

The behaviour of the function $f(x)=\sin \left(2 x+\frac{\pi}{4}\right)$ on $\left(\frac{3 \pi}{8}, \frac{5 \pi}{8}\right)$ is:

4Application Of Derivatives

A square plate is contracting at a uniform rate of $2 \mathrm{~cm}^2 / \mathrm{min}$. The rate at which the perimeter is decreasing when the side of the square is 16 cm is:

5Application Of Derivatives

If $f(x)=x^3+\frac{3}{2} x^2+3 x+3$, then $f(x)$ is

6Application Of Derivatives

A movie screen on a wall is $\mathbf{2 0}$ feet high and $\mathbf{1 0}$ feet above the floor. What is the maximum viewing angle $\boldsymbol{\theta}$ (in radians) that can be achieved by positioning yourself at the optimal distance from the...

7Application Of Derivatives

An open hemispherical storage tank has radius 13 m . Oil flows into the tank such that the depth ' $\boldsymbol{h}$ ' of oil in the tank changes at the rate of $3 \mathrm{~m} / \mathrm{hr}$. When the depth $\boldsymbol{h}=1 \mathrm{~m}$, th...

8Application Of Derivatives

The absolute maximum and minimum values of the function $f(x)=\sin x+\sqrt{3} \cos x$ in $[0, \pi]$ are

9Application Of Derivatives

The function $f(x)=e^{a x}+e^{-a x}, x \in \mathbb{R}$ and $a<0$, is strictly decreasing for all values of ' $x$ ', where

10Application Of Derivatives

Oil from a conical funnel is dripping at the rate of $5 \mathrm{~cm}^3 / \mathrm{s}$. If the radius and height of the funnel are 10 cm and 20 cm respectively, then the rate at which the oil level drops when it is 5 cm from the top is

11Application Of Derivatives

The least value of ' $a$ ' such that the function $x^2+a x+1$ is increasing on $[1,2]$ is

12Application Of Derivatives

A spherical snowball is melting such that its volume is decreasing at the rate of $1 \mathrm{~cm}^3 / \mathrm{min}$. The rate at which the diameter is decreasing when the diameter is 10 cm is

13Application Of Derivatives

A solid S is made from a cylinder surmounted by a hemisphere on top with both its circular faces sharing a common centre. The radius of cylinder and radius of hemisphere are $x \mathrm{~cm}$. The height of the cylinder is $(20-4 x) \mathrm{...

14Application Of Derivatives

The curve $a x^3+b x^2+c x+d$ has a point of minima at $x=1$, then

15Application Of Derivatives

The function $y=\frac{\log x}{x^3}$ is strictly increasing function for

16Application Of Derivatives

Let $f(x)=x \sqrt{4 a x-x^2}, a>0$ then $f^{\prime}(x)$ at $x=2 a$ is :

17Application Of Derivatives

If the length of the diagonal of a square is increasing at the rate of $0.1 \mathrm{~cm} / \mathrm{sec}$.

What is the rate of increase of its area when the side is $\frac{15}{\sqrt{2}} \mathrm{~cm}$ ?

18Application Of Derivatives

If a quadratic function in $x$ has the value 19 when $x=1$ and has a maximum value 20 when $x=2$, then the function is

19Application Of Derivatives

A man is moving away from a tower 41.6 m high at a rate of $2 \mathrm{~m} / \mathrm{s}$. If the eyelevel of the man is 1.6 m above the ground, then the rate at which the angle of elevation of the top of the tower changes, when he is at a di...

20Application Of Derivatives

If the function $f(x)=\mu \sin x+\frac{1}{3} \sin 3 x$ has its derivative equal to zero at $x=\frac{\pi}{3}$, then the value of ' $\mu$ ' is

21Application Of Derivatives

Quadrilateral PQRS is inscribed inside a rectangle of dimensions $10 \mathrm{~cm} \times 8 \mathrm{~cm}$. The value of ' $x$ ', if the area of the quadrilateral is minimum is

22Application Of Derivatives

The least area of a circle circumscribing any right-angle triangle of area $\frac{9}{\pi}$ sq units is

23Application Of Derivatives

$x=a(\theta+\sin \theta)$ and $y=a(1-\cos \theta)$ represents the equation of a curve. If $\theta$ changes at a constant rate $k$ then the rate of change of the slope of the tangent to the curve at $\theta=\frac{\pi}{3}$ is

24Application Of Derivatives

In the interval $(0,1)$ the function $f(x)=x^2-x+1$ is

25Application Of Derivatives

The curve $4 y=3 x^4-2 x^2$ attains ----------- at the points $x=-\frac{1}{\sqrt{3}}$ and $x=\frac{1}{\sqrt{3}}$

26Application Of Derivatives

Let $A$ and $G$ denote the arithmetic mean and geometric mean of positive real numbers $5^x$ and $5^{1-x}$. Then the minimum value of the expression $5^x+5^{1-x}$ where $x \in R$ is

27Application Of Derivatives

The side of an equilateral triangle expands at the rate of \(\sqrt{3} \mathrm{~cm} / \mathrm{sec}\). When the side is \(12 \mathrm{~cm}\), the rate of increase of its area is

28Application Of Derivatives

If \(f(x)=2 x^3+9 x^2+\lambda x+20\) is a decreasing function of \(x\) in the largest possible interval \((-2,-1)\), then \(\lambda\) is equal to

29Application Of Derivatives

The most economical proportion of the height of a covered box of fixed volume whose base is a rectangle with one side three times as long as the other, is

30Application Of Derivatives

For a given curve \(y=2 x-x^2\), when \(x\) increases at the rate of 3 units/sec, then how does the slope of the curve change?

31Application Of Derivatives

\(\text { The point on the curve } x^2=x y \text { which is closest to }(0,5) \text { is }\)

32Application Of Derivatives

What is the nature of the function \(f(x)=x^3-3 x^2+4 x\) on real numbers?

33Application Of Derivatives

The side of a cube is equal to the diameter of a sphere. If the side and radius increase at the same rate then the ratio of the increase of their surface area is

34Application Of Derivatives

The turning point of the function \(y=\frac{a x-b}{(x-1)(x-4)}\) at the point \(P(2,-1)\) is

35Application Of Derivatives

\(\text { The rate of change of the volume of a sphere with respect to its surface area } \mathrm{S} \text { is }\)

36Application Of Derivatives

The dimensions of the largest rectangle of side \(x\) and \(y\) that can be inscribed in the right angled triangle of sides \(\mathrm{a}\) and \(\mathrm{b}\) is

37Application Of Derivatives

If \(f(x)=\log x+b x^2+a x, x \neq 0\) has extreme values (or turning points) at \(x=-1\) and \(x=2\) then the values of \(\mathrm{a}\) and \(\mathrm{b}\) are

38Application Of Derivatives

\(\text { The function } y=\tan x-x \text { is }\)

39Application Of Derivatives

If \((x-a)^2+(y-b)^2=c^2\), where \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) are some constants, \(c>0\) then \(\frac{\left[1+\left(\frac{d y}{d x}\right)^2\right]^{\frac{3}{2}}}{\frac{d^2 y}{d x^2}}\) is independent of

40Application Of Derivatives

\(\text { If } f(x)=\frac{a \sin x+b \cos x}{c \sin x+d \cos x} \text { is decreasing for all } x \text {, then }\)

41Application Of Derivatives

The function \(f(x)=\frac{x}{2}+\frac{2}{x}\) has a local minimum at

42Application Of Derivatives

The altitude of a cone is \(20 \mathrm{~cm}\) and its semi vertical angle is \(30^{\circ}\). If the semi vertical angle is increasing at the rate of \(2^0\) per second, then the radius of the base is increasing at the rate of

43Application Of Derivatives

If the volume of a sphere is increasing at a constant rate, then the rate at which its radius is increasing is

44Application Of Derivatives

\(f(x)=2 x-\tan ^{-1} x-\log (x+\sqrt{x^2+1}) \text { is monotonically increasing, when }\)

45Application Of Derivatives

Let \(f(x)=a+(x-4)^{\frac{4}{9}}\), then minima of \(f(x)\) is

46Application Of Derivatives

The slope of the tangent to the curve, \(y=x^2-x y\) at \(\left(1, \frac{1}{2}\right)\) is

47Application Of Derivatives

Let \(f(x) = a - {(x - 3)^{8/9}}\), then maxima of \(f(x)\) is

48Application Of Derivatives

If the tangent to the curve \(xy + ax + by = 0\) at (1, 1) is inclined at an angle \({\tan ^{ - 1}}2\) with X-axis, then