Application of Derivatives PYQs - Last 10 Years

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Practice 112 TS EAMCET 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 10 Years Application of Derivatives Questions

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1Application Of Derivatives

If the expression $7+6 x-3 x^2$ attains its extreme value $\beta$ at $x=\alpha$, then the sum of the squares of the roots of the equation $x^2+\alpha x-\beta=0$ is

2Application Of Derivatives

If the function $f(x)=x^3+a x^2+b x+40$ satisfies the conditions of Rolle's theorem on the interval $[-5,4]$ and $-5,4$ are two roots of the equation $f(x)=0$, then one of the values of $c$ as stated in that theorem is

3Application Of Derivatives

$y=f(x)$ and $x=g(y)$ are two curves and $P(x, y)$ is a common point of the two curves. If at $P$ on the curve $y=f(x), \frac{d y}{d x}=Q(x)$ and at the same point $P$ on the curve $x=g(y), \frac{d x}{d y}=-Q(x)$, then

4Application Of Derivatives

If $A=\{P(\alpha, \beta) /$ the tangent drawn at $P$ to the curve $y^{3}-3 x y+2=0$ is horizontal line $\}$ and $B=\{Q(a, b) /$ the tangent drawn at $Q$ to the curve $y^{3}-3 x y+2=0$ is a vertical line $\}$, then $n(A)+n(B)=$

5Application Of Derivatives

The maximum interval in which the slopes of the tangents drawn to the curve $y=x^{4}+5 x^{3}+9 x^{2}+6 x+2$ increase is

6Application Of Derivatives

Let $m$ be the slope of the normal $L$ drawn at $(1,2)$ to the curve $x=t^2-7 t+7, y=t^2-4 t-10$ and $a x+b y+c=0$ be the equation of the normal $L$. If GCD of $(a, b, c)$ is 1 , then $m(a+b+c)=$

7Application Of Derivatives

If a line having slope 2 is a tangent to the curve $y=x^4-6 x^3+13 x^2-12 x+5$ at points $P\left(x_1, y_1\right)$ and $Q\left(x_2, y_2\right), x_1, x_2 \in N$, then $x_1 x_2-y_1 y_2=$

8Application Of Derivatives

If the function $f(x)=x e^{-x}, x \in R$ attains its maximum value $\beta$ at $x=\alpha$, then $(\alpha, \beta)=$

9Application Of Derivatives

The maximum volume (in cu. units) of the cylinder which can be inscribed in a sphere of radius 12 units is

10Application Of Derivatives

A ladder of length 13 m has one end resting against a vertical wall and the other on the ground. If the lower end moves away from the wall at a speed of $2 \mathrm{~m} / \mathrm{min}$ then the speed (in $\mathrm{m} / \mathrm{min}$ ) at whic...

11Application Of Derivatives

An angle between the curves $x^2-y^2=4$ and $x^2+y^2=4 \sqrt{2}$ is

12Application Of Derivatives

The slope of the normal drawn at a point $P$ to the curve $y=x^3-10 x^2+31 x-30$ is $-\frac{1}{14}$. If the co-ordinates of $P$ are integers, then the $X$-intercept of the tangent drawn at $P$ to the given curve is

13Application Of Derivatives

The nearest approximate value of $\sqrt{2023}$ is (let $\Delta x=87$ ).

14Application Of Derivatives

$x$ and $y$ are two positive integers such that $2 x+3 y=50$. If $x^2 y^3$ is maximum for $x=\alpha$ and $y=\beta$, then $\frac{\alpha}{2}+\frac{\beta}{5}=$

15Application Of Derivatives

For $h, k \in N$, let $P(h, k)$ be the point of intersection of the curves $x^2 y-x^3=8$ and $y^3-x y^2=32$. If $\theta$ is the acute angle between these two curves at $P$, then $\tan \theta=$

16Application Of Derivatives

The diameter of a sphere is measured as 42 cm . If there is an error of $1 / 77 \mathrm{~cm}$ in measuring it, then the error involved in the volume of that sphere (in cubic centimeters) is

17Application Of Derivatives

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

18Application Of Derivatives

If the slope of the tangent drawn at any point $(x, y)$ to the curve $y=f(x)$ is $3 x^2-5$ and $f(1)=2$, then the tangent at $(1,2)$ to the curve $y=f(x)$ intersects the curve at the point

19Application Of Derivatives

If the expression $x^3+3 x^2-9 x+\lambda$ is of the form $(x-\alpha)^2(x-\beta)$, then the values of $\lambda$ are

20Application Of Derivatives

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

21Application Of Derivatives

If the function $f(x)=\frac{x}{5}+\frac{5}{x},(x \neq 0)$ attains its relative maximum value at $x=\alpha$, then $\sqrt{\alpha^2+2 \alpha-6}=$

22Application Of Derivatives

For all real values of $x$, the minimum value of $\frac{1-x+\lambda^2}{1+x+x^2}$ is

23Application Of Derivatives

If the equation of a tangent drawn to the curve $y=\cos (x+y),-1 \leq x \leq 1+\pi$ is $x+2 y=k$, then $k=$

24Application Of Derivatives

Electric current $(I)$ is measured by galvanometer, the current being proportional to the tangent of the angle ( $\theta$ ) of deflection. If the deflection is read as $45^{\circ}$ and an error of $1 \%$ is made in reading it, the percentag...

25Application Of Derivatives

$f: R \rightarrow R$ is a function defined by $f(x)=\frac{1}{e^x+2 e^{-x}}$
Assertion (A) : $f(c)=\frac{1}{3}$ for some values of $c \in R$
Reason (R) : $0 < f(x) \leq \frac{1}{2 \sqrt{2}}$ for all $x \in R$
Then, which of the following opt...

26Application Of Derivatives

If $x$ and $y$ are two positive integers such that $x+2 y=10$ and $x^2 y^3$ is maximum, then $x^2+2 y^3=$

27Application Of Derivatives

The approximate value of $\sqrt[3]{28}$ rounded up to 3 decimal places is

28Application Of Derivatives

$y=x^2$ is the given curve. Imagine that this curve is dragged along the positive $X$-axis to a distance of ' $a$ ' units. If the acute angle between the curves at two positions is $\theta$, then

29Application Of Derivatives

If $\theta$ is the angle between the curves $x^2-y^2=4$ and $y^2=3 x$, then $\tan \theta=$

30Application Of Derivatives

The equation of the tangent to the curve $x^2+y-7=4 x$ at the point $(1,10)$ is

31Application Of Derivatives

The absolute maximum value of the function $f(x)=2 x^3-3 x^2-36 x+9$ defined on $[-3,3]$ is

32Application Of Derivatives

Let $\sqrt{3}$ be the radius and $\frac{\pi}{3}$ be the semi-vertical angle of the given cone. Then, the height of the right circular cylinder of maximum volume that can be inscribed in the given cone is

33Application Of Derivatives

If $\theta$ is the acute angle between the curves $x^2+y^2=4$ and $y^2=3 x$, then $\tan \theta=$

34Application Of Derivatives

Assertion (A) The curves $y^2=4 x$ and $x^2=-2 y$ intersect at $(1,2)$ orthogonally.
Reason (R) If the product of the slopes of the tangents drawn to two curves at their point of intersection is -1 , then the curves are said to cut each oth...

35Application Of Derivatives

The equation of the normal to the curve $\sin y=\sqrt{3} x \sin \left(\frac{\pi}{6}+y\right)$ at $x=0$, is

36Application Of Derivatives

Let $f(x)=\left\{\begin{array}{cc}1+6 x-3 x^2 & x \leq 1 \\ x+\log _2\left(b^2+7\right) & x>1\end{array}\right.$. Then, the set of all possible values of $b$ such that $f(1)$ is the maximum value of $f(x)$ is

37Application Of Derivatives

Let a function $f(x)$ be continuous in an interval $[a, b]$. Let $\delta>0$ be a very small real number. Let $c \in(a, b)$ be such that $f(c-\delta)0$. Let $(f(\alpha-\delta)-f(\alpha))(f(\alpha+\delta))<0 \forall \alpha \in(a, b)$ and $\al...

38Application Of Derivatives

The area of the triangle formed by the tangent and the normal drawn to the curve $y^2=4 x$ at $(1,2)$ with $Y$-axis is (in square units)

39Application Of Derivatives

Consider two families of curves $y^2=4 a x$ ( $a$ is a parameter) and $x^2+\frac{y^2}{2}=c^2(c$ is parameter). If one curve from each family is chosen, then the angle between those two curves is

40Application Of Derivatives

The local maximum value of the function $f(x)=-(x-2)^3(x+2)^2$ is

41Application Of Derivatives

If $\theta$ is the angle between the curves $y^2=4 x$ and $x^2+y^2=5$, then $|\tan \theta|=$

42Application Of Derivatives

If an error of $0.02 \mathrm{sq} . \mathrm{cm}$ is found in the surface area of a sphere when its radius is measured as 10 cm , then the approximate error that occurs in the volume of the sphere, in cubic centimeters, is

43Application Of Derivatives

The area (in sq. units) of the triangle formed by the tangent and normal drawn to the curve $\left(\frac{x}{3}\right)^n+\left(\frac{y}{4}\right)^n=2$ at $(3,4)$ and $x$-axis is

44Application Of Derivatives

A vessel in the shape of an inverted cone of height 10 ft and semi vertical angle $30^{\circ}$ is full of water. Due to a hole at the vertex, the slant height of the water in the vessel is decreasing at a constant rate of $\frac{1}{\sqrt{3}...

45Application Of Derivatives

If the curves $a x^2+b y^2=1$ and $c x^2+d y^2=1$ intersect orthogonally, then $\frac{b-a}{d-c}=$

46Application Of Derivatives

The angle $A$ of $\triangle A B C$ is found by measurement to be $67 \frac{1^{\circ}}{2}$ and the area of $\triangle A B C$ is calculated from the measurements of $b, c, A$. In measuring $A$, an error of 9 min is made then the percentage er...

47Application Of Derivatives

In $\triangle A B C, \angle B=90^{\circ}$ and $(b+a)$ is always a constant. In order that $\triangle A B C$ encloses the maximum area, $\angle C=$

48Application Of Derivatives

The $x$-coordinate changes on the curve $y=3 x^5+15 x-8$ at the rate of $\frac{1}{5}$ units/sec. $A\left(x_1, y_1\right), B\left(x_2, y_2\right)$ are the points on the curve at which the $y$-coordinate changes at the rate of 6 units/sec, th...

49Application Of Derivatives

Let $f: R \rightarrow R$ be a bijection. A curve represented by $y=f(x)$ is such that $f^{\prime}(x)>0 \forall x \in \mathbf{R}$. The tangent and normal drawn at $P(\alpha, 1)$ on the curve cuts the $X$-axis at $A, B$ respectively and $C$ i...

50Application Of Derivatives

If $\alpha$ is a root of multiplicity 3 of the equation $x^5-8 x^4+25 x^3-38 x^2+28 x-8=0$, then $\alpha^2-5 \alpha+6=$