Question 1: Palette 1/107

AP-EAPCET 2025 MCQ

If the tangent of the curve $4 y^3=3 a x^2+x^3$ drawn at the point $(a, a)$ forms a triangle of area $\frac{25}{24}$ sq. units with the coordinates axes, then $a=$

A

$\pm 10$

B

$\pm 5$

C

$\pm 6$

D

$\pm 3$

Question 2: Palette 2/107

AP-EAPCET 2025 MCQ

If the function $f(x)=\sin x-\cos ^2 x$ is defined on the interval $[-\pi, \pi]$, then $f$ is strictly increasing in the interval

A

$\left(\frac{-5 \pi}{6}, \frac{-\pi}{6}\right) \cup\left(\frac{-\pi}{6}, \frac{\pi}{2}\right)$

B

$\left(\frac{-\pi}{2}, \frac{-\pi}{6}\right)$

C

$\left(\frac{-5 \pi}{6}, \frac{\pi}{2}\right)$

D

$\left(\frac{-5 \pi}{6}, \frac{-\pi}{2}\right) \cup\left(\frac{-\pi}{6}, \frac{\pi}{2}\right)$

Question 3: Palette 3/107

AP-EAPCET 2025 MCQ

If the Lagrange' mean value theorem is applied to the function $f(x)=e^x$ defined on the interval $[1,2]$ and the value of $c \in(1,2)$ is $k$, then $e^{k-1}=$

A

2

B

$e-1$

C

$e+1$

D

1

Question 4: Palette 4/107

AP-EAPCET 2025 MCQ

Consider the quadratic equation $a x^2+b x+c=0$, where $2 a+3 b+6 c=0$ and let $g(x)=\frac{a x^3}{3}+\frac{b x^2}{2}+c x$

Statement I The given quadratic equation $a x^2+b x+c=0$ has atleast one root in $(0,1)$.

Statement II Rolle's theorem is applicable to $g(x){\text {on }}$ [0, 1].

Then

A

Statement I is false, Statement II is true

B

Statement I is true, Statement II is false

C

Statement I is true, Statement II is true but Statement IIs not a correct explanation of Statement I

D

Statement I is true, Statement II is true and Statement I is a correct explanation of Statement I

Question 5: Palette 5/107

AP-EAPCET 2025 MCQ

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

A

80

B

1

C

85

D

4

Question 6: Palette 6/107

AP-EAPCET 2025 MCQ

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

A

increasing on $\left[-\frac{1}{2}, 1\right]$

B

decreasing on $R$

C

increasing on $R$

D

decreasing on $\left[-\frac{1}{2}, 1\right]$

Question 7: Palette 7/107

AP-EAPCET 2025 MCQ

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

A

$\tan ^{-1}\left(\frac{4}{3}\right)$

B

$\tan ^{-1}\left(\frac{3}{4}\right)$

C

$90^{\circ}$

D

$45^{\circ}$

Question 8: Palette 8/107

AP-EAPCET 2025 MCQ

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 $\frac{a b}{a+b}=$

A

1

B

2

C

3

D

4

Question 9: Palette 9/107

AP-EAPCET 2025 MCQ

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

A

$\frac{1}{3}$

B

2

C

3

D

$\frac{1}{2}$

Question 10: Palette 10/107

AP-EAPCET 2025 MCQ

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 $b c=$

A

18

B

-66

C

38

D

-46

Question 11: Palette 11/107

AP-EAPCET 2025 MCQ

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

A

8

B

12

C

15

D

16

Question 12: Palette 12/107

AP-EAPCET 2025 MCQ

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

A

4

B

3

C

1

D

2

Question 13: Palette 13/107

AP-EAPCET 2025 MCQ

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

A

$(0.088)(\sqrt{2}+1)$

B

$(0.616)(\sqrt{2}+1)$

C

$(0.616)(\sqrt{2})$

D

$(0.088)(\sqrt{2})$

Question 14: Palette 14/107

AP-EAPCET 2025 MCQ

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 $\alpha-\beta=$

A

4

B

3

C

2

D

1

Question 15: Palette 15/107

AP-EAPCET 2025 MCQ

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

A

1

B

2

C

$\frac{1}{e}$

D

3

Question 16: Palette 16/107

AP-EAPCET 2025 MCQ

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 $m+M=$

A

-1

B

0

C

1

D

2

Question 17: Palette 17/107

AP-EAPCET 2025 MCQ

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

A

$\frac{3}{4}$

B

$\frac{-3}{4}$

C

$\frac{9}{4}$

D

$\frac{-9}{4}$

Question 18: Palette 18/107

AP-EAPCET 2025 MCQ

$P$ 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

A

$a+\sqrt{2}$

B

$a+\frac{1}{\sqrt{2}}$

C

$a-\frac{1}{\sqrt{2}}$

D

$a-\sqrt{2}$

Question 19: Palette 19/107

AP-EAPCET 2025 MCQ

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)

A

$P(-1)$ is not minimum of $P(x)$, but $P(1)$ is the maximum of $P(x)$

B

$P(-1)$ is minimum of $P(x)$, but $P(1)$ is not the maximum of $P(x)$

C

Neither $P(-1)$ is the minimum nor $P(1)$ is the maximum of $P(x)$

D

$P(-1)$ is the minimum and $P(1)$ is the maximum of $P(x)$

Question 20: Palette 20/107

AP-EAPCET 2025 MCQ

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

A

2

B

3

C

4

D

6

Question 21: Palette 21/107

AP-EAPCET 2025 MCQ

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

A

$b, d, a, c$

B

b, a, c, d

C

$a, b, c, d$

D

$b, a, d, c$

Question 22: Palette 22/107

AP-EAPCET 2025 MCQ

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 $x_1+y_1=$

A

5

B

4

C

7

D

6

Question 23: Palette 23/107

AP-EAPCET 2025 MCQ

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

A

$f(x)=\log (1+x)-x+\frac{x^2}{2}$

B

$g(x)=2 \tan ^{-1} x-x-1$

C

$h(x)=4 \cos x+x$

D

$u(x)=\log (1+x)-\frac{x}{x+1}$

Question 24: Palette 24/107

AP-EAPCET 2025 MCQ

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

A

$\tan \beta=\frac{1}{\sqrt{3}} \sec 2 \theta$

B

$\cot \beta=\sqrt{3} \operatorname{cosec} 2 \theta$

C

$\sqrt{3} \cot \beta=\sin 2 \theta$

D

$\cot \beta=\frac{1}{\sqrt{2}} \sec 2 \theta$

Question 25: Palette 25/107

AP-EAPCET 2025 MCQ

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

A

12

B

$2 \sqrt{3}+\sqrt{13}+5$

C

$7+\sqrt{21}$

D

$5(\sqrt{2}+1)$

Question 26: Palette 26/107

AP-EAPCET 2025 MCQ

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}}= $

A

1

B

-1

C

2

D

0

Question 27: Palette 27/107

AP-EAPCET 2025 MCQ

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 $\alpha^2+\beta^2=$

A

10

B

9

C

28

D

19

Question 28: Palette 28/107

AP-EAPCET 2025 MCQ

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

A

8

B

4

C

$\frac{8}{3}$

D

$\frac{4}{3}$

Question 29: Palette 29/107

AP-EAPCET 2025 MCQ

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 $\cos k=$

A

$\frac{\sqrt{m}}{4}$

B

$\frac{\sqrt{m+1}}{\sqrt{2}}$

C

$\frac{\sqrt{5}}{\sqrt{m}}$

D

$\frac{1}{m}$

Question 30: Palette 30/107

AP-EAPCET 2025 MCQ

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 $P=$

A

$(e, e)$

B

$\left(\frac{1}{e}, \frac{-1}{e}\right)$

C

$\left(\frac{1}{e^2}, \frac{-2}{e^2}\right)$

D

$\left(e^3, 3 e^3\right)$

Question 31: Palette 31/107

AP-EAPCET 2025 MCQ

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

A

6

B

9

C

$\frac{9}{2}$

D

3

Question 32: Palette 32/107

AP-EAPCET 2025 MCQ

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

A

$\left(-\pi,-\frac{\pi}{3}\right) \cup\left(\frac{\pi}{3}, \pi\right)$

B

$\left(\frac{\pi}{6}, \frac{\pi}{2}\right)$

C

$\left(-\frac{\pi}{3}, \frac{\pi}{3}\right)$

D

$\left(-\pi,-\frac{\pi}{6}\right) \cup\left(\frac{\pi}{6}, \pi\right)$

Question 33: Palette 33/107

AP-EAPCET 2025 MCQ

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

A

constant

B

inversely proportional to its velocity

C

proportional to its velocity

D

proportional to its displacement

Question 34: Palette 34/107

AP-EAPCET 2025 MCQ

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 $2 \alpha-\beta=$

A

-1

B

3

C

5

D

-7

Question 35: Palette 35/107

AP-EAPCET 2025 MCQ

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

A

$1 / 2$

B

1

C

2

D

$3 / 2$

Question 36: Palette 36/107

AP-EAPCET 2025 MCQ

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

A

$\frac{\pi}{2}$

B

$\frac{\pi}{6}$

C

$\frac{\pi}{4}$

D

$\frac{\pi}{3}$

Question 37: Palette 37/107

AP-EAPCET 2025 MCQ

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

$\left[-\frac{1}{2}, \frac{4}{3}\right]$

B

$\left(\frac{-1}{2}, \frac{4}{3}\right)$

C

$R-\left(\frac{-1}{2}, \frac{3}{4}\right)$

D

$R-\left[\frac{-1}{2}, \frac{4}{3}\right]$

Question 38: Palette 38/107

AP-EAPCET 2024 MCQ

$A$ 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)

A

$24 \sqrt{3}$

B

24

C

$15 \sqrt{3}$

D

$48 \sqrt{3}$

Question 39: Palette 39/107

AP-EAPCET 2024 MCQ

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)

A

$y^k+x^k=C$

B

$x^{\frac{1}{k}} C=y^k$

C

$(x+y)^k=C$

D

$y=x^{\frac{1}{k}} C$

Question 40: Palette 40/107

AP-EAPCET 2024 MCQ

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)

A

14

B

8

C

5

D

3

Question 41: Palette 41/107

AP-EAPCET 2024 MCQ

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 )

A

7

B

5

C

3

D

2

Question 42: Palette 42/107

AP-EAPCET 2024 MCQ

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

A

5

B

3

C

2

D

1

Question 43: Palette 43/107

AP-EAPCET 2024 MCQ

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

A

$(1, \infty)$

B

$(0,1)$

C

$(-\infty, 0) \cup(1, \infty)$

D

$(-\infty, 0)$

Question 44: Palette 44/107

AP-EAPCET 2024 MCQ

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

A

$\log (e-1)$

B

$\log (e+1)$

C

$\log (e+24)$

D

$\log (e-24)$

Question 45: Palette 45/107

AP-EAPCET 2024 MCQ

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

A

$x-3 y+5=0$

B

$x+3 y+5=0$

C

$x+3 y+7=0$

D

$x+3 y-7=0$

Question 46: Palette 46/107

AP-EAPCET 2024 MCQ

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.

A

6

B

$6 \sqrt{3}$

C

$6 \sqrt{6}$

D

$3 \sqrt{6}$

Question 47: Palette 47/107

AP-EAPCET 2024 MCQ

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)

A

100

B

150

C

125

D

200

Question 48: Palette 48/107

AP-EAPCET 2024 MCQ

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

A

$\frac{10}{3}$

B

$\frac{8}{3}$

C

$\frac{4}{3}$

D

$\frac{2}{3}$

Question 49: Palette 49/107

AP-EAPCET 2024 MCQ

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

A

3

B

4

C

5

D

2

Question 50: Palette 50/107

AP-EAPCET 2024 MCQ

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

A

6

B

$3 / 2$

C

2

D

4

Question 51: Palette 51/107

AP-EAPCET 2024 MCQ

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

A

$y=x+5$

B

$x+y=7$

C

$2 x+y=8$

D

$x+2 y=13$

Question 52: Palette 52/107

AP-EAPCET 2024 MCQ

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

A

26 unit/sec

B

20 unit/sec

C

24 unit $/ \mathrm{sec}$

D

30 unit/sec

Question 53: Palette 53/107

AP-EAPCET 2024 MCQ

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

A

$\frac{c^3}{2 a b}$

B

$\frac{c^3}{\sqrt{2 a b}}$

C

$\frac{c^3}{a b}$

D

$\frac{c^3}{\sqrt{a b}}$

Question 54: Palette 54/107

AP-EAPCET 2024 MCQ

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

A

$\frac{1}{5}$

B

$\frac{1}{3}$

C

$\frac{1}{2}$

D

$\frac{1}{6}$

Question 55: Palette 55/107

AP-EAPCET 2024 MCQ

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

A

$3 x-y=60$

B

$4 x+y=108$

C

$2 x+y=60$

D

$x-2 y=0$

Question 56: Palette 56/107

AP-EAPCET 2024 MCQ

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

A

$(2,6)$

B

$(2,-6)$

C

$(-2,-6)$

D

$(-2,6)$

Question 57: Palette 57/107

AP-EAPCET 2024 MCQ

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

A

$\left[0, \frac{1}{e}\right]$

B

$[0, \mathrm{e}]$

C

$\left[\frac{1}{\theta}, \infty\right]$

D

$\left[0, e^e\right]$

Question 58: Palette 58/107

AP-EAPCET 2024 MCQ

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

A

$\pm \frac{1}{\sqrt{3}}$

B

$-\frac{1}{\sqrt{3}}$

C

$\frac{1}{\sqrt{3}}$

D

$\frac{1}{3}$

Question 59: Palette 59/107

AP-EAPCET 2024 MCQ

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

A

4

B

1

C

2

D

infinite

Question 60: Palette 60/107

AP-EAPCET 2024 MCQ

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

A

$y=2 x+1$

B

$2 y=x+5$

C

$y=-2 x+5$

D

$2 y=3 x+3$

Question 61: Palette 61/107

AP-EAPCET 2024 MCQ

$p_1$ 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 $k_1+k_2=$

A

7

B

6

C

5

D

4

Question 62: Palette 62/107

AP-EAPCET 2024 MCQ

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

A

1

B

$\frac{1}{3}$

C

$\frac{1}{2}$

D

0

Question 63: Palette 63/107

AP-EAPCET 2024 MCQ

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

A

$\frac{2 \pi}{3}$

B

$\frac{\pi}{2}$

C

$\frac{\pi}{3}$

D

$\frac{\pi}{4}$

Question 64: Palette 64/107

AP-EAPCET 2024 MCQ

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

A

$2+\frac{2}{\sqrt{3}}$

B

$2-\sqrt{\frac{16}{3}}$

C

$1+\sqrt{\frac{5}{4}}$

D

$2+\sqrt{\frac{8}{3}}$

Question 65: Palette 65/107

AP-EAPCET 2024 MCQ

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 $\frac{1}{K}=$

A

2

B

$\frac{1}{200}$

C

200

D

$\frac{1}{2}$

Question 66: Palette 66/107

AP-EAPCET 2024 MCQ

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

A

$\tan ^{-1}(1)$

B

$\tan ^{-1}(2)$

C

$\tan ^{-1}(3)$

D

$\tan ^{-1}\left(-\frac{1}{2}\right)$(d) $\tan ^{-1}\left(-\frac{1}{2}\right)$

Question 67: Palette 67/107

AP-EAPCET 2024 MCQ

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

A

$2 \sqrt{3}$

B

$-2 \sqrt{3}$

C

$\sqrt{6}$

D

$-\sqrt{6}$

Question 68: Palette 68/107

AP-EAPCET 2024 MCQ

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 $\frac{k}{\alpha^2 \beta^2}=$

A

$\frac{\alpha}{\beta}+\frac{\beta}{\alpha}$

B

$\frac{\alpha}{\beta}-\frac{\beta}{\alpha}$

C

$\frac{\alpha}{\beta}$

D

$\frac{\alpha+\beta}{\alpha \beta}$

Question 69: Palette 69/107

AP-EAPCET 2024 MCQ

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

A

error in $x$

B

percentage error in $x$

C

$n$, (error in $x$ )

D

$n$, (relative error in $x$ )

Question 70: Palette 70/107

AP-EAPCET 2024 MCQ

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

A

51

B

53

C

58

D

25

Question 71: Palette 71/107

AP-EAPCET 2024 MCQ

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

A

3

B

3 and -2

C

-2

D

-1

Question 72: Palette 72/107

AP-EAPCET 2024 MCQ

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

10120

B

10118

C

10122

D

2024

Question 73: Palette 73/107

AP-EAPCET 2024 MCQ

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

A

3

B

$\frac{1}{2}$

C

$-\frac{1}{2}$

D

-3

Question 74: Palette 74/107

AP-EAPCET 2024 MCQ

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

A

$\left(-\infty-\frac{3}{4}\right) \cup(3, \infty)$

B

$\left(-\frac{3}{4}, 3\right)$

C

( 13

D

$(-\infty, 1) \cup(3-=)$

Question 75: Palette 75/107

AP-EAPCET 2022 MCQ

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

A

$-5(\sin x+\cos x)$

B

$-5 \sin x \cos x$

C

$\frac{-5}{\sin x}-\frac{5}{\cos x}$

D

$\frac{5}{\sin x}+\frac{5}{\cos x}$

Question 76: Palette 76/107

AP-EAPCET 2022 MCQ

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

A

1

B

$\frac{2}{3}$

C

$\frac{1}{3}$

D

$-\frac{2}{3}$

Question 77: Palette 77/107

AP-EAPCET 2022 MCQ

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

A

1

B

$-$2

C

4

D

5

Question 78: Palette 78/107

AP-EAPCET 2022 MCQ

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

A

4a

B

4b

C

2a

D

2b

Question 79: Palette 79/107

AP-EAPCET 2022 MCQ

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

A

$\left(\frac{1}{2}, \frac{5}{4}\right)$

B

$\left(\frac{-1}{2}, \frac{5}{4}\right)$

C

$\left(2, \frac{-5}{3}\right)$

D

$\left(2, \frac{5}{3}\right)$

Question 80: Palette 80/107

AP-EAPCET 2022 MCQ

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

A

0

B

2

C

1

D

3

Question 81: Palette 81/107

AP-EAPCET 2022 MCQ

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 $k=$

A

4

B

5

C

6

D

7

Question 82: Palette 82/107

AP-EAPCET 2022 MCQ

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

A

$c^2=16 a^2$

B

$c^2=32 a^2$

C

$c^4=16 a^4$

D

$c^4=32 a^4$

Question 83: Palette 83/107

AP-EAPCET 2022 MCQ

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

A

$2: 1$

B

$1: 2$

C

$2: 3$

D

$3: 2$

Question 84: Palette 84/107

AP-EAPCET 2022 MCQ

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

A

1

B

5

C

169

D

49

Question 85: Palette 85/107

AP-EAPCET 2022 MCQ

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

A

$\frac{-197}{27}$

B

$\frac{125}{31}$

C

12

D

$\frac{-238}{27}$

Question 86: Palette 86/107

AP-EAPCET 2022 MCQ

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

A

$x-y+1=0$

B

$2 x-y+1=0$

C

$x+y+1=0$

D

$2 x+y+1=0$

Question 87: Palette 87/107

AP-EAPCET 2022 MCQ

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

A

$1 / 4$

B

$1 / 5$

C

$1 / 6$

D

$1 / 7$

Question 88: Palette 88/107

AP-EAPCET 2022 MCQ

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

A

$-$1

B

$-$2

C

1

D

2

Question 89: Palette 89/107

AP-EAPCET 2022 MCQ

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

A

$b^2-4 a c$

B

$b^2=3 a c$

C

$b^2<3 a c$

D

$b^2>3 a c$

Question 90: Palette 90/107

AP-EAPCET 2022 MCQ

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

$(1,0)$

B

$(0,1)$

C

$(\sqrt{5}, 0)$

D

$(0, \sqrt{5})$

Question 91: Palette 91/107

AP-EAPCET 2021 MCQ

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.

A

5/6$\pi$

B

1/54$\pi$

C

1/18$\pi$

D

1/36$\pi$

Question 92: Palette 92/107

AP-EAPCET 2021 MCQ

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

A

9

B

12

C

8

D

6

Question 93: Palette 93/107

AP-EAPCET 2021 MCQ

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 $6 \mathrm{~cm}$.

A

$5 \mathrm{~cm}^2 / \mathrm{min}^{-1}$

B

$30 \mathrm{~cm}^2 / \mathrm{min}^{-1}$

C

$10 \mathrm{~cm}^2 / \mathrm{min}^{-1}$

D

$20 \mathrm{~cm}^2 / \mathrm{min}^{-1}$

Question 94: Palette 94/107

AP-EAPCET 2021 MCQ

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

A

$\left(\frac{-1}{\sqrt{2}}, 0\right) \cup\left(\frac{1}{\sqrt{2}}, \infty\right)$

B

$\left(\frac{-1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$

C

$(-1,0) \cup(1,2)$

D

$\left(-\infty, \frac{-1}{\sqrt{2}}\right) \cup\left(\frac{1}{\sqrt{2}}, \infty\right)$

Question 95: Palette 95/107

AP-EAPCET 2021 MCQ

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

A

0

B

1

C

2

D

$-$1

Question 96: Palette 96/107

AP-EAPCET 2021 MCQ

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

A

$a=\frac{4}{9}$ and $b=\frac{-4}{9}$

B

$a=0$ and $b=\frac{4}{9}$

C

$a=\frac{-4}{9}$ and $b=\frac{-4}{9}$

D

$a=\frac{4}{9}$ and $b=0$

Question 97: Palette 97/107

AP-EAPCET 2021 MCQ

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

A

0

B

2

C

1

D

$\frac{1}{4}$

Question 98: Palette 98/107

AP-EAPCET 2021 MCQ

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.

A

2.16 $\pi$ cm$^2$

B

21.6 $\pi$ cm$^2$

C

216 $\pi$ cm$^2$

D

0.216 $\pi$ cm$^2$

Question 99: Palette 99/107

AP-EAPCET 2021 MCQ

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?

A

4 cm/s

B

6 cm/s

C

$-$4 cm/s

D

$-$8 cm/s

Question 100: Palette 100/107

AP-EAPCET 2021 MCQ

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

A

0.08

B

$-$0.08

C

0.8

D

$-$0.8

Question 101: Palette 101/107

AP-EAPCET 2021 MCQ

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

A

$\frac{2}{3}$

B

$\frac{2}{\sqrt{3}}$

C

$\frac{4}{3}$

D

$\frac{3}{4}$

Question 102: Palette 102/107

AP-EAPCET 2021 MCQ

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

A

$1-3 x+2 x^2$

B

$1+3 x-2 x^2$

C

$2 x$

D

$x+2 x^3-3 x^2$

Question 103: Palette 103/107

AP-EAPCET 2021 MCQ

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

A

$\left(\frac{\pi}{6}, \frac{\pi}{3}\right)$

B

$\left(0, \frac{\pi}{4}\right)$

C

$\left(0, \frac{\pi}{3}\right)$

D

$\left(\frac{\pi}{4}, \frac{\pi}{2}\right)$

Question 104: Palette 104/107

AP-EAPCET 2021 MCQ

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

A

$y=\frac{1}{4}$

B

$y=\frac{1}{2}$

C

$y=1$

D

$y=4$

Question 105: Palette 105/107

AP-EAPCET 2021 MCQ

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

A

0.05%

B

0.0025%

C

0.25%

D

0.1%

Question 106: Palette 106/107

AP-EAPCET 2021 MCQ

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

A

$(0,-4),(2,12),(-2,12)$

B

$(0,4),(-2,12),(1,2)$

C

$(0,-4),(-1,2),(2,12)$

D

$(0,4),(-1,2),(1,2)$

Question 107: Palette 107/107

AP-EAPCET 2021 MCQ

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

A

2

B

$\frac{2}{\sqrt{3}}$

C

$\frac{2}{\sqrt{5}}$

D

$\frac{1}{2}$