Application Of Derivatives - Calculus - Mathematics Previous Year Questions

1

2024 · Mathematics · Calculus · Application Of Derivatives

TG EAPCET 2024 ONLINE 10TH MAY EVENING SHIFT

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 $n(A)+n(B)=$

A

12

B

1

C

0

D

4

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2

2024 · Mathematics · Calculus · Application Of Derivatives

TG EAPCET 2024 ONLINE 10TH MAY EVENING SHIFT

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

A

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

B

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

C

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

D

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

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3

2024 · Mathematics · Calculus · Application Of Derivatives

TG EAPCET 2024 ONLINE 10TH MAY EVENING SHIFT

$y=f(x)$ 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

A

the two curves have common tangent

B

the angle between two curves is $45^{\circ}$

C

tangent drawn at $P$ to one curve is normal to the other curve at $P$

D

the two curves never intersect orthogonally

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4

2024 · Mathematics · Calculus · Application Of Derivatives

TG EAPCET 2024 ONLINE 10TH MAY MORNING SHIFT

A point $P$ is moving on the curve $x^3 y^4=2^7$. The $x$-coordinate of $P$ is decreasing at the rate of 8 units per second. When the point $P$ is at $(2,2)$, the $y$-coordinate of $P$

A

increases at the rate of 6 units per second

B

decreases at the rate of 6 units per second

C

increases at the rate of 4 units per second

D

decreases at the rate of 4 units per second

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5

2024 · Mathematics · Calculus · Application Of Derivatives

TG EAPCET 2024 ONLINE 10TH MAY MORNING SHIFT

If $x$ and $y$ are two positive integers such that $x+y=24$ and $x^3 y^5$ is maximum, then $x^2+y^2=$

A

288

B

296

C

306

D

320

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6

2024 · Mathematics · Calculus · Application Of Derivatives

TG EAPCET 2024 ONLINE 10TH MAY MORNING SHIFT

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

A

21

B

-19

C

19

D

-21

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7

2024 · Mathematics · Calculus · Application Of Derivatives

TG EAPCET 2024 ONLINE 10TH MAY MORNING SHIFT

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

A

3

B

$\frac{1+\sqrt{67}}{3}$

C

$\frac{1+\sqrt{65}}{3}$

D

-2

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8

2024 · Mathematics · Calculus · Application Of Derivatives

TG EAPCET 2024 ONLINE 10TH MAY MORNING SHIFT

The equation of the normal drawn to the curve $y^3=4 x^5$ at the point $(4,16)$ is

A

$20 x+3 y=128$

B

$20 x-3 y=32$

C

$3 x-20 y+308=0$

D

$3 x+20 y=332$

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9

2024 · Mathematics · Calculus · Application Of Derivatives

TG EAPCET 2024 ONLINE 11TH MAY MORNING SHIFT

The length of the normal drawn at $t=\frac{\pi}{4}$ on the curve $x=2(\cos 2 t+t \sin 2 t), y=4(\sin 2 t+t \cos 2 t)$ is

A

$\frac{4}{\pi} \sqrt{1+\pi^{2}}$

B

$4 \sqrt{1+\pi^{2}}$

C

$4 \pi$

D

$\frac{4}{\pi}$

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10

2024 · Mathematics · Calculus · Application Of Derivatives

TG EAPCET 2024 ONLINE 11TH MAY MORNING SHIFT

If $m$ and $M$ are respectively the absolute minimum and absolute maximum values of a function $f(x)=2 x^{3}+9 x^{2}+12 x+1$ defined on $[-3,0]$, then $m+M=$

A

-7

B

0

C

1

D

5

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11

2024 · Mathematics · Calculus · Application Of Derivatives

TG EAPCET 2024 ONLINE 11TH MAY MORNING SHIFT

For a given function $y=f(x), \delta y$ denote the actual error in $y$ corresponding to actual error $\delta x$ in $x$ and $d y$ denotes the approximately value of $\delta y$. If $y=f(x)=2 x^{2}-3 x+4$ and $\delta x=0.02$, then the value of $\delta y-d y$ when $x=5$ is

A

0.0008

B

0.008

C

0.0004

D

0.004

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12

2024 · Mathematics · Calculus · Application Of Derivatives

TG EAPCET 2024 ONLINE 11TH MAY MORNING SHIFT

$y=2 x^{3}-8 x^{2}+10 x-4$ is a function defined on [1,2]. If the tangent drawn at a point $(a, b)$ on the graph of this function is parallel to X-axis $a \in(1,2)$, then $a=$

A

0

B

5

C

1

D

$\frac{5}{3}$

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13

2024 · Mathematics · Calculus · Application Of Derivatives

TG EAPCET 2024 ONLINE 11TH MAY MORNING SHIFT

If Water is poured into a cylindrical tank of radius 3.5 ft at the rate of $1 \mathrm{cu} \mathrm{ft} / \mathrm{min}$, then the rate at which the level of the water in the tank increases (in $\mathrm{ft} / \mathrm{min}$ ) is

A

$\frac{1}{154}$

B

$\frac{8}{77}$

C

$\frac{2}{77}$

D

$\frac{1}{11}$

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14

2024 · Mathematics · Calculus · Application Of Derivatives

TG EAPCET 2024 ONLINE 9TH MAY EVENING SHIFT

If $x=\cos 2 t+\log (\tan t)$ and $y=2 t+\cot 2 t$, then $\frac{d y}{d x}=$

A

$\tan 2 t$

B

$-\operatorname{cosec} 2 t$

C

$-\cot 2 t$

D

$\sec 2 t$

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15

2024 · Mathematics · Calculus · Application Of Derivatives

TG EAPCET 2024 ONLINE 9TH MAY EVENING SHIFT

The approximate value of $\sqrt[3]{730}$ obtained by the application of derivatives is

A

9.0041

B

9.01

C

9.006

D

9.05

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16

2024 · Mathematics · Calculus · Application Of Derivatives

TG EAPCET 2024 ONLINE 9TH MAY EVENING SHIFT

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

A

1

B

3

C

2

D

4

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17

2024 · Mathematics · Calculus · Application Of Derivatives

TG EAPCET 2024 ONLINE 9TH MAY EVENING SHIFT

If $4+3 x-7 x^2$ attains its maximum value $M$ at $x=\alpha$ and $5 x^2-2 x+1$ attains its minimum value $m$ at $x=\beta$, then $\frac{28(M-a)}{5(m+\beta)}=$

A

28

B

23

C

5

D

1

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18

2024 · Mathematics · Calculus · Application Of Derivatives

TG EAPCET 2024 ONLINE 9TH MAY EVENING SHIFT

If $f(x)=k x^3-3 x^2-12 x+8$ is strictly decreasing for all $x \in R$, then

A

$k<-\frac{1}{4}$

B

$k>-\frac{1}{4}$

C

$k>\frac{1}{4}$

D

$k<\frac{1}{4}$

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19

2024 · Mathematics · Calculus · Application Of Derivatives

TG EAPCET 2024 ONLINE 9TH MAY EVENING SHIFT

The vertical angle of a right circular cone is $60^{\circ}$. If water is being poured in to the cone at the rate of $\frac{1}{\sqrt{3}} \mathrm{~m}^3 / \mathrm{min}$, then the rate ( $\mathrm{m} / \mathrm{min}$ ) at which the radius of the water level is increasing when the height of the water level is 3 m is

A

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

B

$\frac{1}{9 \sqrt{3 \pi}}$

C

$\frac{1}{9 \pi}$

D

$\frac{1}{3 \pi}$

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20

2024 · Mathematics · Calculus · Application Of Derivatives

TG EAPCET 2024 ONLINE 9TH MAY EVENING SHIFT

A right circular cone is inscribed in a sphere of radius 3 units. If the volume of the cone is maximum, then semi-vertical angle of the cone is

A

$\frac{\pi}{4}$

B

$\frac{\pi}{6}$

C

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

D

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

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