RBSE Solutions for Class 12 Maths Chapter 8 Application of Derivatives Miscellaneous Exercise

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 = 2πr² + 2πrh

Question 2.
Find the value of x and y for function, y = x³ + 21 where, the rate of change ofy is three times as the rate of change of x.
Solution:
Given, function y = x³ + 21
Diff w.r.t. t,
$\frac { dy }{ dt }$ = 3x² $\frac { dx }{ dt }$ …..(i)
According to question
$\frac { dy }{ dt }$ = 3 $\frac { dx }{ dt }$ …..(ii)
From equation (i) and (ii)
3x² = 3
⇒ x = ± 1
In equation y = x³ + 21
Putting x = 1
y = 1³ + 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 – y₁ = m(x – x₁) 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 V² be also maximum or minimum

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