Practice Book for KVPY Aptitude Test - Stream SX Mathematics>Chapter 6 - Application of Derivatives>EXERCISE>Q 12

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A curve is represented by the equations $x = \sec^{2} t$ and $y = \cot t,$ where $t$ is a parameter. If the tangent at the point $P$ on the curve, where $t = \frac{π}{4}$ , meets the curve again at the point $Q,$ then $| P Q |$ is equal to

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Important Questions on Application of Derivatives

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Practice Book for KVPY Aptitude Test - Stream SX Mathematics>Chapter 6 - Application of Derivatives>EXERCISE>Q 13

The

x

-intercept of the tangent at any arbitrary point of the curve

\frac{a}{x^{2}} + \frac{b}{y^{2}} = 1

is proportional to

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Practice Book for KVPY Aptitude Test - Stream SX Mathematics>Chapter 6 - Application of Derivatives>EXERCISE>Q 14

The number of points in the rectangle

{(x, y) - 12 \leq x \leq 12 & - 3 \leq y \leq 3}

, which lie on the curve

y = x + \sin x

and at which the tangent to the curve is parallel to the

x

-axis, is

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Practice Book for KVPY Aptitude Test - Stream SX Mathematics>Chapter 6 - Application of Derivatives>EXERCISE>Q 15

The lines tangent to the curves

y^{3} - x^{2} y + 5 y - 2 x = 0

and

x^{4} - x^{3} y^{2} + 5 x + 2 y = 0

at the origin intersect at an angle

θ

equal to

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Practice Book for KVPY Aptitude Test - Stream SX Mathematics>Chapter 6 - Application of Derivatives>EXERCISE>Q 16

The curves

4 x^{2} + 9 y^{2} = 72

and

x^{2} - y^{2} = 5

at

(3, 2)

EASY

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Practice Book for KVPY Aptitude Test - Stream SX Mathematics>Chapter 6 - Application of Derivatives>EXERCISE>Q 17

If

f (x) = x^{3} + 7 x - 1

, then

f (x)

has a zero between

x = 0

and

x = 1 .

The theorem that best describes this is

EASY

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Practice Book for KVPY Aptitude Test - Stream SX Mathematics>Chapter 6 - Application of Derivatives>EXERCISE>Q 18

Let

f (x)

and

g (x)

be differentiable for

0 \leq x \leq 1,

such that

f (0) = 0, g (0) = 0, f (1) = 6 .

Let there exists a real number

c

in

(0, 1)

such that

f^{'} (c) = 2 g^{'} (c),

then the value of

g (1)

must be

EASY

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Practice Book for KVPY Aptitude Test - Stream SX Mathematics>Chapter 6 - Application of Derivatives>EXERCISE>Q 19

A value of

c

for which the conclusion of mean value theorem holds for the function

f (x) = \log_{e} x

on the interval

[1, 3]

is

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Practice Book for KVPY Aptitude Test - Stream SX Mathematics>Chapter 6 - Application of Derivatives>EXERCISE>Q 20

Let

f (x)

be a twice differentiable function for all real values of

x

and satisfies

f (1) = 1, f (2) = 4, f (3) = 9 .