Chapterwise/Topicwise Daily Practice Problems (DPP) Calculus JEE Main & Advanced>Chapter 5 - Applications of Derivatives>DPP>Q 9

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JEE Main

IMPORTANT

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From the point $(1, 1)$ tangents are drawn to the curve represented parametrically as $x = 2 t - t^{2}$ and $y = t + t^{2} .$ The distance between the points of contact is equal to

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Important Questions on Applications of Derivatives

MEDIUM

JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Calculus JEE Main & Advanced>Chapter 5 - Applications of Derivatives>DPP>Q 10

The value of parameter

t

so that the line

(4 - t) x + t y + (a^{3} - 1) = 0

is normal to the curve

x y = 1

may lie in the interval:

MEDIUM

JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Calculus JEE Main & Advanced>Chapter 5 - Applications of Derivatives>DPP>Q 11

The portion of the tangent at any point on the curve

x = a t^{3}

,

y = a t^{4}

between the axes is divided by the abscissa of the point of contact externally in the ratio:

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JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Calculus JEE Main & Advanced>Chapter 5 - Applications of Derivatives>DPP>Q 12

Equation of a line which is tangent to both the curves

y = x^{2} + 1

and

y = - x^{2}

is:

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JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Calculus JEE Main & Advanced>Chapter 5 - Applications of Derivatives>DPP>Q 13

For the functions defined parametrically by the equations

$f (t) = x = \{\begin{matrix} 2 t + t^{2} \sin \frac{1}{t} & ; t \neq 0 \\ 0 & ; t = 0 \end{matrix}$ and

$g (t) = y = \{\begin{matrix} \frac{1}{t} \sin t^{2} & ; t \neq 0 \\ 0 & ; t = 0 \end{matrix}$ .

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JEE Main

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Chapterwise/Topicwise Daily Practice Problems (DPP) Calculus JEE Main & Advanced>Chapter 5 - Applications of Derivatives>DPP>Q 14

For the curve

x = 2 a \sin t + a \sin t \cos^{2} t

;

y = - a \cos^{3} t

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JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Calculus JEE Main & Advanced>Chapter 5 - Applications of Derivatives>DPP>Q 15

The curve

y = a x^{3} + b x^{2} + c x

is inclined at

45 °

to

x

-axis at (0,0) but it touches

x

-axis at

(1, 0)

, then:

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JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Calculus JEE Main & Advanced>Chapter 5 - Applications of Derivatives>DPP>Q 1

The length of the sub-tangent to the hyperbola

x^{2} - 4 y^{2} = 4

corresponding to the normal having slope unity is

\frac{1}{\sqrt{k}},

then the value of

k

is

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JEE Main

IMPORTANT

Chapterwise/Topicwise Daily Practice Problems (DPP) Calculus JEE Main & Advanced>Chapter 5 - Applications of Derivatives>DPP>Q 2

Cosine of the acute angle between the curves

y = 3^{x - 1} \log_{e} x

and

y = x^{x} - 1

is