Application of Derivatives · Mathematics · AP EAPCET

The difference between the absolute maximum and absolute minimum values of the function $f(x)=2 x^3-15 x^2+36 x-30$ on $[-1,4]$ is

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

The angle between the curves $y^2=x$ and $x^2=y$ at the point $(1,1)$ is

AP EAPCET 2025 - 26th May Evening Shift

If the tangent of the curve $x y+a x+b y=0$ at $(1,1)$ makes an angle $\tan ^{-1} 2$ with $X$-axis, then $\frac{a b}{a+b}=$

If the displacement $S$ of a particle travelling along a straight line in $t$ seconds is given by $S=2 t^3+2 t^2-2 t-3$, then the time taken (in second) by the particle to change its direction is

AP EAPCET 2025 - 26th May Evening Shift

If the function $f(x)=x^3+b x^2+c x-6$ satisfies all the conditions of Rolle's theorem in $[1,3]$ and $f^{\prime}\left(\frac{2 \sqrt{3}+1}{\sqrt{3}}\right)=0$, then $b c=$

AP EAPCET 2025 - 26th May Evening Shift

If the surface area of a spherical bubble is increasing at the rate of $4 \mathrm{sq} . \mathrm{cm} / \mathrm{sec}$, then the rate of change in its volume (in cubic $\mathrm{cm} / \mathrm{sec}$ ) when its radius is 8 cms is

AP EAPCET 2025 - 24th May Morning Shift

The number of turning points of the curve $f(x)=2 \cos x-\sin 2 x$ in the interval $[-\pi, \pi]$ is

AP EAPCET 2025 - 24th May Morning Shift

The radius and the height of a right circular solid cone are measured as 7 feet each. If there is an error of 0.002 ft for every feet in measuring them, then the error in the total surface area of the cone (in sq. ft ) is

AP EAPCET 2025 - 24th May Morning Shift

The slope of a tangent drawn at the point $P(\alpha, \beta)$ lying on the curve $y=\frac{1}{2 x-5}$ is -2 . If $P$ lies in the fourth quadrant, then $\alpha-\beta=$

AP EAPCET 2025 - 23rd May Evening Shift

The function $f(x)=x e^{-x} \forall x \in R$ attains a maximum value at $x=k$, then $k=$

AP EAPCET 2025 - 23rd May Evening Shift

If $m$ and $M$ are the absolute minimum and absolute maximum values of the function $f(x)=2 \sqrt{2} \sin x-\tan x$ in the interval $[0, \pi / 3]$, then $m+M=$

AP EAPCET 2025 - 23rd May Evening Shift

If $\frac{1}{2} \leq \frac{x^2+x+a}{x^2-x+a} \leq 2 \forall x \in R$, then $a=$

$P$ and $Q$ are the ends of a diameter of the circle $x^2+y^2=a^2\left(a>\frac{1}{\sqrt{2}}\right) . s$ and $t$ are the lengths of the perpendiculars drawn from $P$ and $Q$ onto the line $x+y=1$ respectively. When the product st is maximum, the greater value among $s, t$ is

Let $P(x)=x^4+a x^3+b x^2+c x+d$ be such that $x=0$ is the only real root of $P^1(x)=0$. If $P(-1)

If the volume of a sphere is increasing at the rate of 12 c.c. $/ \mathrm{sec}$, then the rate (in $\mathrm{sq} . \mathrm{cm} / \mathrm{sec}$ ) at which its surface area is increasing, when the diameter of the sphere is 12 cm is

If the lengths of the tangent, subtangent, normal and subnormal for the curve $y=x^2+x-1$ at the point $(1,1)$ are $a, b, c$ and $d$ respectively, then their increasing order is

If the tangent drawn at the point $\left(x_1, y_1\right), x_1, y_1 \in N$ on the curve $y=x^4-2 x^3+x^2+5 x$ passes through origin, then $x_1+y_1=$

Which one of the following functions is monotonically increasing in its domain?

If $\beta$ is an angle between the normals drawn to the curve $x^2+3 y^2=9$ at the points $(3 \cos \theta, \sqrt{3} \sin \theta)$ and $(-3 \sin \theta, \sqrt{3} \cos \theta), \theta \in\left(0, \frac{\pi}{2}\right)$, then

If the area of a right-angle triangle with hypotenuse 5 is maximum, then its perimeter is

If $y=|\cos x-\sin x|+|\tan x-\cot x|$, then

$$ \left(\frac{d y}{d x}\right)_{x=\frac{\pi}{3}}+\left(\frac{d y}{d x}\right)_{x=\frac{\pi}{6}}= $$

If the tangent drawn at the point $(\alpha, \beta)$ on the curve $x^{\frac{2}{3}}+y^{\frac{2}{3}}=4$ is parallel to the line $\sqrt{3 x}+y=1$, then $\alpha^2+\beta^2=$

The displacement $S$ of a particle measured from a fixed point $O$ on a line is given by $S=t^3-16 t^2+64 t-16$. Then, the time at which displacement of the particle is maximum is

If the extreme value of the function $f(x)=\frac{4}{\sin x}+\frac{1}{1-\sin x}$ in $\left[0, \frac{\pi}{2}\right]$ is $m$ and it exists at $x=k$, then $\cos k=$

If the normal drawn at the point $P$ on the curve $y=x \log x$ is parallel to the line $2 x-2 y=3$, then $P=$

If the curves $y^2=16 x$ and $9 x^2+\alpha y^2=25$ intersect at right angles, then $\alpha=$

If the function $y=\sin x(1+\cos x)$ is defined in the interval $[-\pi, \pi]$, then $y$ is strictly increasing in the interval

If the velocity of a particle moving on a straight line is proportional to the cube root of its displacement, then its acceleration is

If $\alpha$ and $\beta(\alpha>\beta)$ are the multiple roots of the equation $4 x^4+4 x^3-23 x^2-12 x+36=0$, then $2 \alpha-\beta=$

The area (in square units) of the triangle formed by the $X$-axis, the tangent and the normal drawn at $(1,1)$ to the curve $x^3+y^3=2 x y$ is

The value of the Rolle's theorem for the function $f(x)=2 \sin x+\sin 2 x$ in the interval $[0, \pi]$ is

If the function $y=g(x)$ representing the slopes of the tangents drawn to the curve $y=3 x^4-5 x^3-12 x^2+18 x+3$ is strictly increasing, then the domain of $g(x)$ is

$A$ is a point on the circle with radius 8 and centre at $O$. A particle $P$ is moving on the circumference of the circle starting from $A . M$ is the foot of the perpendicular from $P$ on $O A$ and $\angle P O M=\theta$. When $O M$ $=4$ and $\frac{d \theta}{d t}=6$ radians $/ \mathrm{sec}$, then the rate of change of $P M$ is (in units/sec)

AP EAPCET 2024 - 23th May Morning Shift

If the length of the sub-tangent at any $P$ on a curve is proportional to the abscissa of the point $P$, then the equation of that curve is ( $C$ is an arbitrary constant)

AP EAPCET 2024 - 23th May Morning Shift

The semi-vertical angle of a right circular cone is $45^{\circ} \%$ If the radius of the base of the cone is measured as 14 cm with an error of $\left(\frac{\sqrt{2}-1}{11}\right) \mathrm{cm}$, then the approximate error in measuring its total surface area is (in sq cm)

If a man of height 1.8 mt , is walking away from the foot of a light pole of height 6 mt , with a speed of 7 km per hour on a straight horizontal road opposite to the pole, then the rate of change of the length of his shadow is (in kmph )

If the curves $2 x^2+k y^2=30$ and $3 y^2=28 x$ cut each other orthogonally, then $k$ is equal to

The interval containing all the real values of $x$ such that the real valued function $f(x)=\sqrt{x}+\frac{1}{\sqrt{x}}$ is strictly increasing is

The value of Lagrange's mean value theorem for $f(x)=e^x+24$ in $[0,1]$ is

Equation of the normal to the curve $y=x^2+x$ at the point $(1,2)$ is

Displacement $s$ of a particle at time $t$ is expressed as $s=2 t^3-9 t$. Find the acceleration at the time when $b^{t 5}$ velocity vanishes.

If a running track of 500 ft is to be laid out enclosing a playground the shape of which is a rectangle with a semi-circle at each end, then the length of the rectangular portion such that the area of the rectangular portion is to be maximum is (in feet)

If $x$ is real and $\alpha, \beta$ are maximum and minimum values of $\frac{x^2-x+1}{x^2+x+1}$ respectively, then $\alpha+\beta=$

The value of $c$ such that the straight line joining the points $(0,3)$ and $(5,-2)$ is tangent to the curve $y=\frac{c}{x+1}$ is

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

The equation of the tangent to the curve $y=x^3-2 x+7$ at the point $(1,6)$ is

The distance ( s ) travelled by a particle in time $t$ is given by $S=4 t^2+2 t+3$. The velocity of the particle, when $t=3 \mathrm{sec}$ is

If $a^2 x^4+b^2 y^4=c^6$, then maximum value of $x y$ is equal to

If a number is drawn at random from the set $\{1,3,5,7, \ldots . .59\}$, then the probability that it lies in the interval in which the function $f(x)=x^3-16 x^2+20 x-5$ is stricly decreasing is

The equation of the normal drawn to the parabola $y^2=6 x$ at the point $(24,12)$ is

The point which lies on the tangent drawn to the curve $x^4 e^y+2 \sqrt{y+1}=3$ at the point $(1,0)$ is

If $f(x)=x^x$, then the interval in which $f(x)$ decrease is

If the Rolle's theorem is applicable for the function $f(x)$ defined by $f(x)=x^3+P x-12$ on $[0,1]$ then the value of $C$ of the Rolle's theorem is

The number of all the value of $x$ for which the function $f(x)=\sin x+\frac{1-\tan ^2 x}{1+\tan ^2 x}$ attains it maximum value on [ $0.2 \pi$ ] is

Equation of a tagent line of the parabola $y^2=8 x$, which passes through the point $(1,3)$ is

$p_1$ and $p_2$ are the perpendicular distances from the origin to the tangent and normal drawn at any point on the curve $x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}$ respectively. If $k_1 p_1^2+k_2 p_2^2=a^2$, then $k_1+k_2=$

The length of the subnormal at any point on the curve $y=\left(\frac{x}{2024}\right)^k$ is constant, if the value of $k$ is

The acute angle between the curves $x^2+y^2=x+y$ and $x^2+y^2=2 y$ is

A' value of $C$ according to the Lagrange's mean value theorem for $f(x)=(x-1)(x-2)(x-3)$ in $[0,4]$ is

If $T=2 \pi \sqrt{\frac{L}{g}}, \mathrm{~g}$ is a constant and the relative error in $T$ is $k$ times to the percentage error in $l$, then $\frac{1}{K}=$

The angle between the curves $y^2=2 x$ and $x^2+y^2=8$ is

If the function $f(x)=\sqrt{x^2-4}$ satisfies the Lagrange's mean value theorem on $[2,4]$, then the value of $C$ is

If $x, y$ are two positive integers such that $x+y=20$ and the maximum value of $x^3 y$ is $k$ at $x=\alpha$ and $y=\beta$, then $\frac{k}{\alpha^2 \beta^2}=$

If $y=\left(1+\alpha+\alpha^2+\ldots\right) e^{\eta x}$, where $\alpha$ and $n$ are constants, then the relative error in $y$ is

If the equation of tangent at $(2,3)$ on $y^2=a x^3+b$ is $y=4 x-5$, then the value of $a^2+b^2=$

If Rolle's theorem is applicable for the function $f(x)=x(x+3) e^{-x / 2}$ on $[3,0]$, then the value of $c$ is

For all $x \in[0,2024]$ assume that $f(x)$ is differentiable, $f(0)=-2$ and $f^{\prime}(x) \geq 5$. Then, the least possible value of $f(2024)$ is

A point is moving on the curve $y=x^3-3 x^2+2 x-1$ and the $y$-coordinate of the point is increasing at the rate d 6 units per second. When the point is at $(2,-1)$, the rate of change of $x$-coordinate of the point is

AP EAPCET 2024 - 18th May Morning Shift

The set of all real values of a such that the real valued function $f(x)=x^3+2 a x^2+3(a+1) x+5$ is strictly increasing in its entire domain is

AP EAPCET 2024 - 18th May Morning Shift

If $$3 f(\cos x)+2 f(\sin x)=5 x$$, then $$f^{\prime}(\cos x)+f^{\prime}(\sin x)=$$

If the normal drawn at a point $$P$$ on the curve $$3 y=6 x-5 x^3$$ passes through $$(0,0)$$, then the positive integral value of the abscissa of the point $$P$$ is

The line joining the points $$(0,3)$$ and $$(5,-2)$$ is a tangent to the curve $$y=\frac{c}{x+1}$$, then $$c=$$

If $$a, b>0$$, then minimum value of $$y=\frac{b^2}{a-x}+\frac{a^2}{x}, 0< x< a$$ is

The point on the curve $$y=x^2+4 x+3$$ which is closest to the line $$y=3 x+2$$ is

The number of those tangents to the curve $$y^2-2 x^3-4 y+8=0$$ which pass through the point $$(1,2)$$ is

If the straight line $$x \cos \alpha+y \sin \alpha=p$$ touches the curve $$\left(\frac{x}{a}\right)^n+\left(\frac{y}{b}\right)^n=2$$ at the point $$(a, b)$$ on it and $$\frac{1}{a^2}+\frac{1}{b^2}=\frac{k}{p^2}$$, then $$k=$$

Condition that 2 curves $$y^2=4 a x, x y=c^2$$ cut orthogonally is

A closed cylinder of given volume will have least surface area when the ratio of its height and base radius is

Two particles $$P$$ and $$Q$$ located at the points $$P\left(t, t^3-16 t-3\right), Q\left(t+1, t^3-6 t-6\right)$$ are moving in a plane, the minimum distance between the points in their motion is

If $$x^3-2 x^2 y^2+5 x+y-5=0$$, then at $$(\mathrm{l}, \mathrm{l}), y^{\prime \prime}(\mathrm{l})=$$

If the curves $$y=x^3-3 x^2-8 x-4$$ and $$y=3 x^2+7 x+4$$ touch each other at a point $$P$$, then the equation of common tangent at $$P$$ is

The maximum value of $$f(x)=\frac{x}{1+4 x+x^2}$$ is

The minimum value of $$f(x)=x+\frac{4}{x+2}$$ is

The condition that $$f(x)=a x^3+b x^2+c x+d$$ has no extreme value is

At any point $$(x, y)$$ on a curve if the length of the subnormal is $$(x-1)$$ and the curve passes through $$(1,2)$$, then the curve is a conic. A vertex of the curve is

A spherical iron ball 10 cm in radius is coated with a layer of ice of uniform thickness, which melts at a rate of 50 cm$$^3$$ /min. When the thickness of the ice is 15 cm, the rate at which the thickness of ice decreases is ........ cm/min.

Find the minimum value of $$2x+3y$$, when $$xy=6$$.

The volume of a spherical balloon is increasing at the rate of $$30 \mathrm{~cm}^3$$ per minute. Find the rate of change of surface area of the balloon, when its radius is $$6 \mathrm{~cm}$$.

If $$g(x)=\frac{1}{6} f\left(3 x^2-1\right)+\frac{1}{2} f\left(1-x^2\right), \forall x \in R$$, where $$f^{\prime \prime}(x) > 0, \forall x \in R$$. Then, $$g(x)$$ is increasing in the interval

If the function $$f(x)=2 x^3-9 a x^2+12 a^2 x+1$$ attains its maximum and minimum at $$p$$ and $$q$$ respectively, such that $$p^2=q$$, then $$a$$ equals

If $$y=4 x-6$$ is a tangent to the curve $$y^2=a x^4+b$$ at $$(3,6)$$, then the values of $$a$$ and $$b$$ are

Find the positive value of $$a$$ for which the equality $$2 \alpha+\beta=8$$ holds, where $$\alpha$$ and $$\beta$$ are the points of maximum and minimum, respectively, of the function $$f(x)=2 x^3-9 a x^2+12 a^2 x+1$$.

If the radius of a sphere is measured as 9 cm with an error of 0.03 cm, then find the approximate error in calculating its surface area.

The diameter and altitude of a right circular cone, at a certain instant, were found to be 10 cm and 20 cm respectively. If its diameter is increasing at a rate of 2 cm/s, then at what rate must its altitude change, in order to keep its volume constant?

100

Given, $$f(x)=x^3-4x$$, if x changes from 2 to 1.99, then the approximate change in the value of $$f(x)$$ is

101

If the curves $$\frac{x^2}{a^2}+\frac{y^2}{4}=1$$ and $$y^3=16 x$$ intersect at right angles, then $$a^2$$ is equal to

102

Let $$x$$ and $$y$$ be the sides of two squares such that, $$y=x-x^2$$. The rate of change of area of the second square with respect to area of the first square is

103

If $$f^{\prime \prime}(x)$$ is a positive function for all $$x \in R, f^{\prime}(3)=0$$ and $$g(x)=f\left(\tan ^2(x)-2 \tan (x)+4\right)$$ for $$0 < x <\frac{\pi}{2}$$, then the interval in which $$g(x)$$ is increasing is

104

The line which is parallel to X-axis and crosses the curve $$y=\sqrt x$$ at an angle of 45$$\Upsilon$$ is

105

If the error committed in measuring the radius of a circle is 0.05%, then the corresponding error in calculating its area would be

106

The stationary points of the curve $$y=8 x^2-x^4-4$$ are

107

The distance between the origin and the normal to the curve $$y=e^{2 x}+x^2$$ drawn at $$x=0$$ is units