## Rajasthan Board RBSE Class 12 Maths Chapter 8 Application of Derivatives Miscellaneous Exercise

Question 1.

If radius and height of a cylinder is r and A, then find the rate of change of surface area with respect to its radius.

Solution:

Radius of cylinder = r

and Height of cylinder = h

Rate of change of surface area w.r.t. r, = \(\frac { ds }{ dr } \)

Surface area of cylinder S = 2πr^{2} + 2πrh

Question 2.

Find the value of x and y for function, y = x^{3} + 21 where, the rate of change ofy is three times as the rate of change of x.

Solution:

Given, function y = x^{3} + 21

Diff w.r.t. t,

\(\frac { dy }{ dt } \) = 3x^{2} \(\frac { dx }{ dt } \) …..(i)

According to question

\(\frac { dy }{ dt } \) = 3 \(\frac { dx }{ dt } \) …..(ii)

From equation (i) and (ii)

3x^{2} = 3

⇒ x = ± 1

In equation y = x^{3} + 21

Putting x = 1

y = 1^{3} + 21

= 1 + 21 =22

Putting x = -1

y = (- 1)^{3} + 21 = -1 + 21 = 20

So, x = ± 1 and y = 22, 20

Question 3.

Prove that exponential function (ex) is an increasing function,

Solution:

y = e^{x}

Then, \(\frac { dy }{ dx } \) = e^{x}

= + ve ∀ x ∈ R

So, for x ∈ R, e^{x} is an increasing function.

Question 4.

Prove that f(x) = log (sinx) is increasing in (0,\(\frac { \pi }{ 2 } \)) and decreasing in (\(\frac { \pi }{ 2 } \),π)

Solution:

f(x) = log (sin x)

Diff w.r.t. t,

Question 5.

If tangent OX and OY of curve \(\sqrt { x }\) – \(\sqrt { y }\) = \(\sqrt { a }\) at some point cut the axis at point P and Q, then show that OP + OQ – a, where O is origin.

Solution:

Given,

Equation \(\sqrt { x }\) – \(\sqrt { y }\) = \(\sqrt { a }\) …..(i)

Let, tangent cut the axis OX and OY at point P and Q, point on curve contact with tangent is (h, k).

Since, (h, k) situated on curve.

So, \(\sqrt { x }\) – \(\sqrt { y }\) = \(\sqrt { a }\) …..(ii)

DifF. (i) w.r.t. x,

So, from y – y_{1} = m(x – x_{1}) equation of tangent at the point (h, k).

Question 6.

Find equation of tangent of curve y = cos (x + y), x ∈ [- 2π, 2π] which is parallel to the line x + 2y = 0

Solution:

Hence, equation (i) and (ii) are the required equations of tangents.

Question 7.

Find the percentage error in calculating the volume of a cubical box if an error of 5% is made in measuring the length of edge of the cube.

Solution:

Let side of cube is x and volume is V.

Percentage error in volume = 3 × 5% = 15%

Hence, required percentage error is 15%

Question 8.

A circular metal plate expands under heating so that its radius increased by 2%. Find the approximate increase in the area of the plate, if the radius of the plate before heating is 10 cm.

Solution:

Let radius of circular plate is r and surface area S.

Hence, approximate increase in the area of plate is 4π cm2.

Question 9.

Prove that the volume of the largest cone inside a sphere is \(\frac { 8 }{ 27 } \) of the volume of sphere.

Solution:

Let r and h be radius and height of the cone respectively inscribed in a sphere of radius R.

Question 10.

Prove that semivertical angle of a cylinderical cone of given surface area and maximum volume is sin^{-1} (\(\frac { 1 }{ 3 } \)).

Solution:

Let, Slant height of cone = l

Height = h

radius = r

T.S.A. = S

Volume = V

and Semi vertical angle = α

It is clear that, for S when V is maximum or minimum, then V^{2} be also maximum or minimum

Hence, when semi-vertical angle of cone is sin-1 (\(\frac { 1 }{ 3 } \)) then its volume will be maximum.

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