Mathematics Crash Course JEE Main>Chapter 1 - Application of Derivatives>Level 2>Q 33

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IMPORTANT

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If a rectangle is inscribed in an equilateral triangle of side length $2 \sqrt{2}$ as shown in the figure, then the square of the largest area of such a rectangle is _____.

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Important Questions on Application of Derivatives

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IMPORTANT

Mathematics Crash Course JEE Main>Chapter 1 - Application of Derivatives>Level 2>Q 34

Let

f (x)

be a polynomial of degree

6

in

x,

in which the coefficient of

x^{6}

is unity and it has extrema at

x = - 1

and

x = 1

. If

\lim_{x \to 0} \frac{f (x)}{x^{3}} = 1,

then

5 \cdot f (2)

is equal to

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IMPORTANT

Mathematics Crash Course JEE Main>Chapter 1 - Application of Derivatives>Level 3>Q 1

Water runs into an inverted conical tent at the rate of

20 f t^{3} / m i n

and leaks out at the rate of

5 f t^{3} / m i n

. The height of the water is three times the radius of the water's surface. The rate at which the radius of the water surface is increasing when the radius is

5 f t

, is

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IMPORTANT

Mathematics Crash Course JEE Main>Chapter 1 - Application of Derivatives>Level 3>Q 2

A curve is represented parametrically by the equations

x = t + e^{a t}

and

y = - t + e^{a t}

,

t \in R

and

a > 0

. If the curve touches the axis of

x

at the point

A

, then the coordinates of the point

A

are

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IMPORTANT

Mathematics Crash Course JEE Main>Chapter 1 - Application of Derivatives>Level 3>Q 3

The equation of the straight lines which are both tangent and normal to the curve

27 x^{2} = 4 y^{3}

are:

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IMPORTANT

Mathematics Crash Course JEE Main>Chapter 1 - Application of Derivatives>Level 3>Q 4

Cosine of the acute angle between the curves

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

and

y = x^{x} - 1

is

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IMPORTANT

Mathematics Crash Course JEE Main>Chapter 1 - Application of Derivatives>Level 3>Q 5

The minimum distance between a point on the curve

y = e^{x}

and a point on the curve

y = \log_{e} x

is:

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IMPORTANT

Mathematics Crash Course JEE Main>Chapter 1 - Application of Derivatives>Level 3>Q 6

The number of the tangents that can be drawn to the curve

y^{2} - 2 x^{3} - 4 y + 8 = 0

from the point

(1, 2)

, is

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IMPORTANT

Mathematics Crash Course JEE Main>Chapter 1 - Application of Derivatives>Level 3>Q 7

If

f : [1, 10] \to [1, 10]

is a non-decreasing function and

g : [1, 10] \to [1, 10]

is a non-increasing function. Let

h (x) = f (g (x))

with

h (1) = 1

. Then find

h (2)

: