Application of Derivatives · Mathematics · AP EAPCET

Start Practice

MCQ (Single Correct Answer)

1
$A$ 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
2
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
3

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)

AP EAPCET 2024 - 22th May Evening Shift
4

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 )

AP EAPCET 2024 - 22th May Evening Shift
5

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

AP EAPCET 2024 - 22th May Evening Shift
6
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
AP EAPCET 2024 - 22th May Evening Shift
7
The value of Lagrange's mean value theorem for $f(x)=e^x+24$ in $[0,1]$ is
AP EAPCET 2024 - 22th May Morning Shift
8
Equation of the normal to the curve $y=x^2+x$ at the point $(1,2)$ is
AP EAPCET 2024 - 22th May Morning Shift
9
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.
AP EAPCET 2024 - 22th May Morning Shift
10
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)
AP EAPCET 2024 - 22th May Morning Shift
11
If $x$ is real and $\alpha, \beta$ are maximum and minimum values of $\frac{x^2-x+1}{x^2+x+1}$ respectively, then $\alpha+\beta=$
AP EAPCET 2024 - 21th May Evening Shift
12
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
AP EAPCET 2024 - 21th May Evening Shift
13
If the percentage error in the radius of circle is 3 , then the percentage error in its area is
AP EAPCET 2024 - 21th May Evening Shift
14
The equation of the tangent to the curve $y=x^3-2 x+7$ at the point $(1,6)$ is
AP EAPCET 2024 - 21th May Evening Shift
15
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
AP EAPCET 2024 - 21th May Evening Shift
16
If $a^2 x^4+b^2 y^4=c^6$, then maximum value of $x y$ is equal to
AP EAPCET 2024 - 21th May Evening Shift
17
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
AP EAPCET 2024 - 21th May Morning Shift
18
The equation of the normal drawn to the parabola $y^2=6 x$ at the point $(24,12)$ is
AP EAPCET 2024 - 21th May Morning Shift
19
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
AP EAPCET 2024 - 21th May Morning Shift
20
If $f(x)=x^x$, then the interval in which $f(x)$ decrease is
AP EAPCET 2024 - 21th May Morning Shift
21
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
AP EAPCET 2024 - 21th May Morning Shift
22
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
AP EAPCET 2024 - 21th May Morning Shift
23
Equation of a tagent line of the parabola $y^2=8 x$, which passes through the point $(1,3)$ is
AP EAPCET 2024 - 20th May Evening Shift
24
$p_1$ 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 $k_1+k_2=$
AP EAPCET 2024 - 20th May Evening Shift
25
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
AP EAPCET 2024 - 20th May Evening Shift
26
The acute angle between the curves $x^2+y^2=x+y$ and $x^2+y^2=2 y$ is
AP EAPCET 2024 - 20th May Evening Shift
27
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
AP EAPCET 2024 - 20th May Evening Shift
28
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 $\frac{1}{K}=$
AP EAPCET 2024 - 20th May Morning Shift
29
The angle between the curves $y^2=2 x$ and $x^2+y^2=8$ is
AP EAPCET 2024 - 20th May Morning Shift
30
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
AP EAPCET 2024 - 20th May Morning Shift
31
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 $\frac{k}{\alpha^2 \beta^2}=$
AP EAPCET 2024 - 20th May Morning Shift
32
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
AP EAPCET 2024 - 19th May Evening Shift
33
If the equation of tangent at $(2,3)$ on $y^2=a x^3+b$ is $y=4 x-5$, then the value of $a^2+b^2=$
AP EAPCET 2024 - 19th May Evening Shift
34
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
AP EAPCET 2024 - 19th May Evening Shift
35
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
AP EAPCET 2024 - 19th May Evening Shift
36
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
37
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
38

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

AP EAPCET 2022 - 5th July Morning Shift
39

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

AP EAPCET 2022 - 5th July Morning Shift
40

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

AP EAPCET 2022 - 5th July Morning Shift
41

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

AP EAPCET 2022 - 5th July Morning Shift
42

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

AP EAPCET 2022 - 5th July Morning Shift
43

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

AP EAPCET 2022 - 4th July Evening Shift
44

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 $$k=$$

AP EAPCET 2022 - 4th July Evening Shift
45

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

AP EAPCET 2022 - 4th July Evening Shift
46

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

AP EAPCET 2022 - 4th July Evening Shift
47

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

AP EAPCET 2022 - 4th July Evening Shift
48

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

AP EAPCET 2022 - 4th July Morning Shift
49

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

AP EAPCET 2022 - 4th July Morning Shift
50

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

AP EAPCET 2022 - 4th July Morning Shift
51

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

AP EAPCET 2022 - 4th July Morning Shift
52

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

AP EAPCET 2022 - 4th July Morning Shift
53

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

AP EAPCET 2022 - 4th July Morning Shift
54

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

AP EAPCET 2021 - 20th August Morning Shift
55

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 $$f(x)=2 x^3-9 a x^2+12 a^2 x+1$$.

AP EAPCET 2021 - 20th August Morning Shift
56

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.

AP EAPCET 2021 - 20th August Morning Shift
57

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?

AP EAPCET 2021 - 20th August Morning Shift
58

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

AP EAPCET 2021 - 19th August Evening Shift
59

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

AP EAPCET 2021 - 19th August Evening Shift
60

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

AP EAPCET 2021 - 19th August Evening Shift
61

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

AP EAPCET 2021 - 19th August Evening Shift
62

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

AP EAPCET 2021 - 19th August Morning Shift
63

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

AP EAPCET 2021 - 19th August Morning Shift
64

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

AP EAPCET 2021 - 19th August Morning Shift
65

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

AP EAPCET 2021 - 19th August Morning Shift
EXAM MAP
Medical
NEETAIIMS
Graduate Aptitude Test in Engineering
GATE CSEGATE ECEGATE EEGATE MEGATE CEGATE PIGATE IN
Civil Services
UPSC Civil Service
Defence
NDA
Staff Selection Commission
SSC CGL Tier I
CBSE
Class 12