Application of Derivatives · Mathematics · AP EAPCET

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MCQ (Single Correct Answer)

1
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
2
Equation of the normal to the curve $y=x^2+x$ at the point $(1,2)$ is
AP EAPCET 2024 - 22th May Morning Shift
3
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
4
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
5
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
6
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
7
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
8
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
9
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
10
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
11
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
12
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
13
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
14
If $f(x)=x^x$, then the interval in which $f(x)$ decrease is
AP EAPCET 2024 - 21th May Morning Shift
15
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
16
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
17
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
18
$p_1$ 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
19
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
20
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
21
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
22
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
23
The angle between the curves $y^2=2 x$ and $x^2+y^2=8$ is
AP EAPCET 2024 - 20th May Morning Shift
24
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
25
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
26
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
27
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
28
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
29
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
30
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
31
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
32

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
33

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
34

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
35

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
36

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
37

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
38

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
39

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

AP EAPCET 2022 - 4th July Evening Shift
40

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
41

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
42

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
43

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
44

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

AP EAPCET 2022 - 4th July Morning Shift
45

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

AP EAPCET 2022 - 4th July Morning Shift
46

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
47

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
48

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
49

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
50

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
51

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
52

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
53

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
54

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
55

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
56

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
57

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
58

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

AP EAPCET 2021 - 19th August Morning Shift
59

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
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