Application of Derivatives PYQs - Last 10 Years

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Last 10 Years Application of Derivatives Questions

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1Application Of Derivatives

There is a possible error of 0.02 cm in measuring the base diameter of a right circular cone as 14 cm . If the semi-vertical angle of the cone is $45^{\circ}$, then the approximate error in its volume is (in $\mathrm{cu} . \mathrm{cm}$ )

2Application Of Derivatives

A rod of length 41 m with an end $A$ on the floor and another end $B$ on the wall perpendicular to the floor is sliding away horizontally from the wall at the rate of $3 \mathrm{fit} / \mathrm{min}$. When the end $B$ is at the height of 9 f...

3Application Of Derivatives

The real valued function $f(x)=\frac{x^2}{2}-\log \left(x^2+x+1\right)$ is

4Application Of Derivatives

If $x$ and $y$ are two positive real numbers such that $x y=4$, then the minimum value of $\left(\sqrt{x}+\frac{y^2}{2}\right)$ is

5Application Of Derivatives

If $x=2 \sqrt{2} \sqrt{\cos 2 \theta}$ and $y=2 \sqrt{2} \sqrt{\sin 2 \theta}, 0<\theta<\frac{\pi}{4}$, then the value of $\frac{d y}{d x}$ at $\theta=22 \frac{1}{2}^{\circ}$ is

6Application Of Derivatives

If the extreme values of the function $f(x)=(2 \sqrt{6}+1) \cos x+(2 \sqrt{2}-\sqrt{3}) \sin x-6$ are $m$ and $M$ then $\sqrt{\left|M^2-m^2\right|}=$

7Application Of Derivatives

If the curves $y^2=12 x-3$ and $y^2=12-k x$ cut each other orthogonally, then the length of the sub-tangent at $(1, b)$ on the curve $y^2=12-k x$ is

8Application Of Derivatives

The height of a cone with semi-vertical angle $\frac{\pi}{3}$ is increasing at the rate of 2 units $/ \mathrm{min}$. The rate at which the radius of the cone is to be decreased so as to have a fixed volume always is

9Application Of Derivatives

$f(x)=x^2-2(4 k-1) x+g(k)>0, \forall x \in R$ and for $k \in(a, b)$. If $g(k)=15 k^2-2 k-7$, then

10Application Of Derivatives

For the curve $\frac{x^n}{a^n}+\frac{y^n}{b^n}=2,(n \in N$ and $n>1)$ the line $\frac{x}{a}+\frac{y}{b}=2$ is

11Application Of Derivatives

If local maximum of $f(x)=\frac{a x+b}{(x-1)(x-4)}$ exists at $(2,-1)$, then $a+b=$

12Application Of Derivatives

If the percentage error in the radius of a circle is 3 , then the percentage error in its area is

13Application Of Derivatives

The function $f(x)=2 x^3-9 a x^2+12 a^2 x+1$ where $a>0$ attains its local maximum and local minimum at $p$ and $q$ respectively. If $p^2=q$, then $a=$

14Application Of Derivatives

Consider all functions given in List I in the interval [1,3]. The list II has the value of ' $c$ ' obtained by applying Lagrange's mean value theorem on the function of List I . Match the function and values of ' c '
$$ \begin{array}{llll} ...

15Application Of Derivatives

If a balloon lying at an altitude of 30 m from an observed at a particular instant is moving horizontally. At the rate of $1 \mathrm{~m} / \mathrm{s}$ away from him, then the rate at which the balloon is moving away directly from the observ...

16Application Of Derivatives

The approximate value of $\sqrt{6560}$ is

17Application Of Derivatives

If the normal drawn at the point $P$ on the curve $y^2=x^3-x+1$ makes equal intercepts on the coordinate axes, then the equation of the tangent drawn to the curve at $P$ is

18Application Of Derivatives

If the point $P\left(x_1, y_1\right)$ lying on the curve $y=x^2-x+1$ is the closest point to the line $y=x-3$, then the perpendicular distance from $P$ to the line $3 x+4 y-2=0$ is

19Application Of Derivatives

A man of 5 feet height is walking away from a light fixed at a height of 15 feet at the rate of of $K$ miles/hour. If the rate of increase of his shadow is $\frac{11}{5}$ feet $/ \mathrm{sec}$, then $K=($ Take 1 mile $=5280$ feet $)$

20Application Of Derivatives

If the tangent and the normal drawn to the curve $x y^2+x^2 y=12$ at the point $(1,3)$ meet the X -axis in $T$ and $N$ respectively, then $T N=$

21Application Of Derivatives

There is a possible error of 0.03 cm in a scale of length 1 foot with which the height of a closed right circular cylinder and the diameter of a sphere are measured as 3.5 feet each. If the radii of both cylinder and sphere are same, then t...

22Application Of Derivatives

A real valued function $f(x)=\left|x^2-3 x+2\right|+2 x-3$ is defined on $[-2,1]$. If $m$ and $M$ are absolute minimum and absolute maximum values of $f$ respectively, then $M-4 m=$

23Application Of Derivatives

If $f(x)=x+\log \left(\frac{x-1}{x+1}\right)$ is a well-defined real valued function, then $f$ is

24Application Of Derivatives

If the rate of change of the slope of the tangent drawn to the curve $y=x^3-2 x^2+3 x-2$ at the point $(2,4)$ is $k$ times the rate of change of its abscissa, then $k=$

25Application Of Derivatives

The acute angle between the curves $y=3 x^2-2 x-1$ and $y=x^3-1$ at their point of intersection which lies in the first quadrant is

26Application Of Derivatives

$P(5,2)$ is a point on the curve $y=f(x)$ and $\frac{7}{2}$ is the slope of the tangent to the curve at $P$. The area of the triangle (in sq. units) formed by the tangent and the normal to the curve at $P$ with $X$-axis is

27Application Of Derivatives

If a particle is moving in a straight line so that after $t$ seconds its distance $S$ (in cms) from a fixed point on the line is given by $S=f(t)=t^3-5 t^2+8 t$, then the acceleration of the particle at $t=5 \mathrm{sec}$ is (in $\mathrm{cm...

28Application Of Derivatives

If $f:[a, b] \rightarrow[c, d]$ is a continuous and strictly increasing function, then $\frac{d-c}{b-a}$ is

29Application Of Derivatives

The local maximum value $l$ and local minimum value $m$ of $f(x)=\frac{x^2+2 x+2}{x+1}$ in $R-\{-1\}$ exist at $\alpha, \beta$ respectively, then $\frac{l+m}{\alpha+\beta}=$

30Application Of Derivatives

The radius of a cone of height 9 units is changed from 2 units to 2.12 units. The exact change and approximate change in the volume of the cone are respectively

31Application Of Derivatives

If $f(x)=(2 x-1)(3 x+2)(4 x-3)$ is a real valued function defined on $\left[\frac{1}{2}, \frac{3}{4}\right]$, then the value(s) of $c$ as defined in the statement of Rolle's theorem

32Application Of Derivatives

The curve $y=x^3-2 x^2+3 x-4$ intersects the horizontal line $y=-2$ at the point $P(h, k)$. If the tangent drawn to this curve at $P$ meets the $X$-axis at $\left(x_1, y_1\right)$, then $x_1=$

33Application Of Derivatives

If the interval in which the real valued function $f(x)=\log \left(\frac{1+x}{1-x}\right)-2 x-\frac{x^3}{1-x^2}$ is decreasing in $(a, b)$, where $|b-a|$ is maximum, then $\frac{a}{b}=$

34Application Of Derivatives

If the slope of the tangent drawn at any point $(x, y)$ on the curve $y=f(x)$ is $\left(6 x^2+10 x-9\right)$ and $f(2)=0$, then $f(-2)=$

35Application Of Derivatives

The radius of a sphere is 7 cm . If an error of 0.08 sq cm is made in measuring it, then the approximate error (in cubic cm ) found in its volume is

36Application Of Derivatives

The vertical angle of a right circular cone is $60^{\circ}$. If water is being poured in to the cone at the rate of $\frac{1}{\sqrt{3}} \mathrm{~m}^3 / \mathrm{min}$, then the rate ( $\mathrm{m} / \mathrm{min}$ ) at which the radius of the ...

37Application Of Derivatives

A right circular cone is inscribed in a sphere of radius 3 units. If the volume of the cone is maximum, then semi-vertical angle of the cone is

38Application Of Derivatives

The approximate value of $\sqrt[3]{730}$ obtained by the application of derivatives is

39Application Of Derivatives

If $\theta$ is the acute angle between the curves $y^2=x$ and $x^2+y^2=2$, then $\tan \theta=$

40Application Of Derivatives

If $x=\cos 2 t+\log (\tan t)$ and $y=2 t+\cot 2 t$, then $\frac{d y}{d x}=$

41Application Of Derivatives

If $f(x)=k x^3-3 x^2-12 x+8$ is strictly decreasing for all $x \in R$, then

42Application Of Derivatives

If $4+3 x-7 x^2$ attains its maximum value $M$ at $x=\alpha$ and $5 x^2-2 x+1$ attains its minimum value $m$ at $x=\beta$, then $\frac{28(M-a)}{5(m+\beta)}=$

43Application Of Derivatives

If $m$ and $M$ are respectively the absolute minimum and absolute maximum values of a function $f(x)=2 x^{3}+9 x^{2}+12 x+1$ defined on $[-3,0]$, then $m+M=$

44Application Of Derivatives

$y=2 x^{3}-8 x^{2}+10 x-4$ is a function defined on [1,2]. If the tangent drawn at a point $(a, b)$ on the graph of this function is parallel to X-axis $a \in(1,2)$, then $a=$

45Application Of Derivatives

If Water is poured into a cylindrical tank of radius 3.5 ft at the rate of $1 \mathrm{cu} \mathrm{ft} / \mathrm{min}$, then the rate at which the level of the water in the tank increases (in $\mathrm{ft} / \mathrm{min}$ ) is

46Application Of Derivatives

The length of the normal drawn at $t=\frac{\pi}{4}$ on the curve $x=2(\cos 2 t+t \sin 2 t), y=4(\sin 2 t+t \cos 2 t)$ is

47Application Of Derivatives

For a given function $y=f(x), \delta y$ denote the actual error in $y$ corresponding to actual error $\delta x$ in $x$ and $d y$ denotes the approximately value of $\delta y$. If $y=f(x)=2 x^{2}-3 x+4$ and $\delta x=0.02$, then the value of...

48Application Of Derivatives

If $x$ and $y$ are two positive integers such that $x+y=24$ and $x^3 y^5$ is maximum, then $x^2+y^2=$

49Application Of Derivatives

The equation of the normal drawn to the curve $y^3=4 x^5$ at the point $(4,16)$ is

50Application Of Derivatives

A point $P$ is moving on the curve $x^3 y^4=2^7$. The $x$-coordinate of $P$ is decreasing at the rate of 8 units per second. When the point $P$ is at $(2,2)$, the $y$-coordinate of $P$