Gamma Question Bank for Engineering Mathematics>Chapter 25 - Application of Derivatives>Assignment>Q 7

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Consider the curve represented parametrically by the equation $x = t^{3} - 4 t^{2} - 3 t$ and $y = 2 t^{2} + 3 t - 5$ where $t \in R$ If $H$ denotes the number of points on the curve where the tangent is horizontal and $V$ the number of points where the tangent is vertical then-

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Important Questions on Application of Derivatives

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Gamma Question Bank for Engineering Mathematics>Chapter 25 - Application of Derivatives>Assignment>Q 8

The normal of the curve

x = a (\cos θ + θ \sin θ)

and

y = a (\sin θ - θ \cos θ)

at any point

θ

, is such that

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Gamma Question Bank for Engineering Mathematics>Chapter 25 - Application of Derivatives>Assignment>Q 9

The equation of tangent to the curve

x = a \cos^{3} t

,

y = a \sin^{3} t

at '

t

' is

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Gamma Question Bank for Engineering Mathematics>Chapter 25 - Application of Derivatives>Assignment>Q 10

If

m

be the slope of a tangent to the curve

e^{y} = 1 + x^{2}

, then

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Gamma Question Bank for Engineering Mathematics>Chapter 25 - Application of Derivatives>Assignment>Q 11

The point(s) on the curve,

y^{3} + 3 x^{2} = 12 y

where the tangent is vertical, is (are):

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Gamma Question Bank for Engineering Mathematics>Chapter 25 - Application of Derivatives>Assignment>Q 12

The curve

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

touches the line

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

at the point

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Gamma Question Bank for Engineering Mathematics>Chapter 25 - Application of Derivatives>Assignment>Q 13

If the tangent to the curve

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

at the point

(a, a)

cuts off intercepts

α

and

β

on the co-ordinate axes, (where

α^{2} + β^{2} = 61

) then

a^{2}

equals

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Gamma Question Bank for Engineering Mathematics>Chapter 25 - Application of Derivatives>Assignment>Q 14

Suppose

x_{1}

and

x_{2}

are the point of maximum and the point of minimum respectively of the function

f (x) = 2 x^{3} - 9 a x^{2} + 12 a^{2} x + 1

respectively, then for the equality

x_{1}^{2} = x_{2}

to be true the value of '

a

' must be

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Gamma Question Bank for Engineering Mathematics>Chapter 25 - Application of Derivatives>Assignment>Q 15

Point '

A

' lies on the curve

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

and has the coordinate

(x, e^{- x^{2}})

where

x > 0

. Point

B

has the coordinates

(x, 0)

. If '

O

' is the origin, then the maximum area of the triangle

A O B

is