NCERT Exemplar Class 12 Maths Solutions Chapter 6 Applications Of Derivatives 2021

NCERT Exemplar Class 12 Maths Solutions Chapter 6 Applications Of Derivatives

Edited By Komal Miglani | Updated on Mar 31, 2025 03:34 AM IST | #CBSE Class 12th

Imagine, for the annual day of the school, the principal wants to decorate the school’s ballroom to its maximum capacity with fabric, ribbons, and flowers, but because of a tight budget, he needs to use the least amount of materials. Here comes the role of this chapter, Application of Derivatives, to minimize the material. This chapter comprises topics like the rate of change of quantities, increasing & decreasing functions, maxima & minima, tangents & normals, and approximation using differentials.

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NCERT exemplar solutions for Class 12 Maths chapter 6 Applications of Derivatives provides complete package for those who are willing to get a good score in exam. Students need to solve the NCERT exemplar Class 12 Maths chapter 6. Solving questions will not only help in understanding the concepts but will also help in scoring well.

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Class 12 Maths chapter 6 solutions Exercise: 6.1
Page number: 135-142
Total questions: 64

Question:1

A spherical ball of salt is dissolving in water in such a manner that the rate of decrease of the volume at any instant is proportional to the surface. Prove that the radius is decreasing at a constant rate.

Answer:

Given: When a spherical ball salt is dissolved, the rate of decrease of the volume at any instant is proportional to the surface
To prove: The rate of decrease of radius is constant at any given time
Explanation: Take the radius of the spherical ball at any time t be ‘r’
Assume S as the surface area of the spherical ball
Then, $S = 4 π r^{2}$ ……….(i)
Take the volume of the spherical ball be V
Then, $V = \frac{4}{3} π r^{3}$
According to the given criteria,
$- \frac{dV}{dt} \propto S$
The rate of decrease of volume is indicated by the negative sign It can also be written as
$- \frac{d V}{dt} = kS$
Here K is the proportional constant
After substitution of values from equation (i) and (ii), we get
$\Rightarrow - \frac{d (\frac{4}{3} π r^{3})}{dt} = k 4 π r^{2}$
When the constant term is taken outside the LHS, we get
$\Rightarrow - \frac{4}{3} π \frac{d (r^{3})}{dt} = k 4 π r^{2}$
After the derivatives are applied with respect to t, we get
$\Rightarrow - \frac{4}{3} π \times 3 r^{2} \times \frac{dr}{dt} = k 4 π r^{2}$
After cancelling of the like terms, we get
$\Rightarrow - \frac{dr}{dt} = k \Rightarrow \frac{dr}{dt} = - k$
Hence the rate of decrease of the radius of the spherical ball is constant.
Hence Proved

Question:2

If the area of a circle increases at a uniform rate, then prove that the perimeter varies inversely as the radius.

Answer:

Given: A circle with uniformly increasing area rate
To prove: relation between perimeter and radius is inversely proportional
Explanation: Take the radius of circle ‘r’
Let the area of the circle be A
Then $A = π r^{2}$ ……..(i)
After considering the give criteria of area increasing at a uniform rate,
$\frac{d A}{dt} = k$
Substitute the value of equation (i) into above equation,
$\frac{d (π r^{2})}{d t} = k$
Differentiating with respect to t results in
$\Rightarrow π \times 2 r \times \frac{d (r)}{dt} = k$

$\Rightarrow \frac{d (r)}{dt} = \frac{k}{2 π r} \dots (i i)$
Let P as the perimeter of the circle,
P = 2πr
Now differentiate the perimeter with respect to t,
$\frac{dP}{dt} = \frac{d (2 π r)}{dt}$
Apply all the derivatives,
$\Rightarrow \frac{dP}{dt} = 2 π \frac{d (r)}{dt}$
Substituting of equation (ii) in the above equation,
$\Rightarrow \frac{dP}{dt} = 2 π (\frac{k}{2 π r})$
Cancelling out of the like terms results into
$\Rightarrow \frac{dP}{dt} = (\frac{k}{r})$
Now covert it into proportionality,
$\Rightarrow \frac{dP}{dt} \propto \frac{1}{r}$
Hence in the given conditions, the relation between perimeter of circle and radius is inversely proportional.
Hence Proved

Question:3

A kite is moving horizontally at a height of 151.5 meters. If the speed of kite is 10 m/s, how fast is the string being let out; when the kite is 250 m away from the boy who is flying the kite? The height of boy is 1.5 m.

Answer:

Given: a boy of n height 1.5 m is flying a kite at a height of 151.5 m. The kite is moving with a speed of 10m/s. And the kite is 250 m away from the boy.
To find: the letting out speed of the string
Explanation: the above situation is explained by the figure,

Referring to the above figure,
Height of the kite, H = AD = 151.5 m
Height of the boy, b = BC = 1.5 m
x = CD = BE
Distance between kite and boy = AB = y =250
So, we need to calculate the increasing rate of the string
From figure, h = AE
= AD-ED
= 151.5-1.5
= 150m
The figure implies that the ΔABE is a right-angled triangle
Applying of the Pythagoras theorem results into,
$A B^{2} = B E^{2} + A E^{2}$

$\Rightarrow y^{2} = x^{2} + h^{2} \dots \dots \dots . . (i)$

$Substitute the corresponding values$

$y^{2} = (x)^{2} + (150)^{2}$
Let’s differentiate the equation (i) with respect to time,
$\frac{d (y^{2})}{dt} = \frac{d (x^{2} + h^{2})}{dt}$
After using the differentiation sum rule, we get
$\Rightarrow \frac{d (y^{2})}{dt} = \frac{d (x^{2})}{dt} + \frac{d (h^{2})}{dt}$
since the height is not increasing, it indicates that it is constant, thus
$\Rightarrow \frac{d (y^{2})}{dt} = \frac{d (x^{2})}{dt} + 0$
Let's apply the derivative with respect to t
$\Rightarrow 2 y \cdot \frac{d (y)}{dt} = 2 x \cdot \frac{d (x)}{dt}$
$\Rightarrow y \cdot \frac{d (y)}{dt} = x \cdot \frac{d (x)}{dt} \dots$ (iii)
since the speed of the kite is 10 m/s so
$\frac{d x}{d t} = 10 m / s$
When y = 250
$250^{2} = (x)^{2} + (150)^{2}$
$x = 200$
After substituting the corresponding values in equation (iii), we get
$\Rightarrow (250) \cdot \frac{d (y)}{dt} = (200) . (10)$
$\frac{d (y)}{dt} = \frac{(200) . (10)}{250} = 8$
Therefore, the letting out speed of the string is 8 m/s

Question:4

Two men A and B start with velocities v at the same time from the junction of two roads inclined at 45° to each other. If they travel by different roads, find the rate at which they are being separated.

Answer:

Given: two men A and B start with velocities v at the same time from the junction of the two roads inclined at 45° to each other
To find: The rate of separation of the two men
Explanation:

The distance x travelled by A and B on any given time t will be same as they have velocity.
Hence
$\begin{aligned} \frac{dx}{dt} = v \dots \dots (1) \\ In the △ A O B, after applying the cosine rule, we get \\ y^{2} = x^{2} + x^{2} - 2 x \cdot x \cdot \cos 45^{\circ} \end{aligned}$
$\begin{aligned} \Rightarrow y^{2} = 2 x^{2} - 2 x^{2} \cdot \frac{1}{\sqrt{2}} \\ \Rightarrow y^{2} = 2 x^{2} (1 - \frac{1}{\sqrt{2}}) \\ \Rightarrow y^{2} = 2 x^{2} (\frac{\sqrt{2} - 1}{\sqrt{2}}) \\ Lets multiple and divide by \sqrt{2}, \\ \Rightarrow y^{2} = 2 x^{2} (\frac{\sqrt{2} - 1}{\sqrt{2}}) \times \frac{\sqrt{2}}{\sqrt{2}} \\ \Rightarrow y^{2} = \sqrt{2} x^{2} (\sqrt{2} - 1) \\ \Rightarrow y = \sqrt{\sqrt{2} x^{2} (\sqrt{2} - 1)} \\ \Rightarrow y = x \cdot \sqrt{(2 - \sqrt{2})} \end{aligned}$
Apply the derivatives with respect to t,
$\Rightarrow \frac{d y}{d t} = \frac{d (x \cdot \sqrt{(2 - \sqrt{2}})}{d t}$
Now take out the constant terms, we get
$\Rightarrow \frac{d y}{d t} = \sqrt{(2 - \sqrt{2})} \frac{d (x)}{d t}$
After value substitution for equation (i) we get
$\Rightarrow \frac{d y}{d t} = \sqrt{(2 - \sqrt{2})} (v) m / s$
Thus, the above given amount is the rate of separation of the two roads.

Question:5

Find an angle $θ, 0 < θ < \frac{π}{2}$ which increases twice as fast as its sine.

Answer:

Given: a condition $θ, 0 < θ < \frac{π}{2}$
To find: the angle θ such that it increases twice as fast as its sine.
Explanation: Let x = sin θ
Let’s differentiate with respect to t,
$\frac{d x}{d t} = \frac{d (\sin θ)}{d t}$
Now applying the derivative results into,
$\frac{d x}{d t} = \cos θ \cdot \frac{d (θ)}{d t} \dots . . (i)$
As this is given in the question
$\frac{d (θ)}{dt} = 2 \cdot \frac{d x}{dt}$
Let's substitute this value in equation (i)
$\frac{d x}{d t} = \cos θ \cdot 2 \cdot \frac{d x}{d t}$
After cancelling the like terms, we get $1 = 2 \cos θ$
$\Rightarrow \cos θ = \frac{1}{2}$
But given $0 < θ < \frac{π}{2},$ this possibility occurs only when $\cos θ = \frac{π}{3}$
Hence the angle $θ$ is $\frac{π}{3}$ .

Question:6

Find the approximate value of $(1.999)^{5}$ .

Answer:

Given: $(1.999)^{5}$
And as the nearest integer to 1.999 is 2, $So, 1.999 = 2 - 0.001$
$∴ a = 2$ and $h = - 0.001$
Hence, $(1.999)^{5} = (2 + (- 0.001))^{5}$
Therefore, the function becomes, $f (x) = x^{5} \dots \dots \dots (i)$
After applying of first derivative, we get $f^{'} (x) = 5 x^{4} \dots \dots \dots .$ (ii)
Now let $f (a + h) = (1.999)^{5}$
Now we know,
$f (a + h) = f (a) + h f^{'} (a)$
From equations (i) and (ii), substituting of functions results in,
$f (a + h) = a^{5} + h (5 a^{4})$
Substitution of values of a and h, we get
$f (2 + (- 0.001)) = 2^{5} + (- 0.001) (5 (2^{4}))$
$\Rightarrow f (1.999) = 32 + (- 0.001) (5 (16))$ $\Rightarrow (1.999)^{5} = 32 + (- 0.001) (80)$
$\Rightarrow (1.999)^{5} = 32 - 0.08 \Rightarrow (1.999)^{5} = 31.92$
So, the approximate value of $(1.999)^{5}$ = 31.92.

Question:7

Find the approximate volume of metal in a hollow spherical shell whose internal and external radii are 3 cm and 3.0005 cm, respectively.

Answer:

Given: A hollow spherical shell having an internal radius of 3cm and external radii of 3.0005 cm
To find: The amount of metal used in the formation of the spherical shell.
Explanation: Let the r and R be the internal and external radii respectively.
So, it is given,
R = 3.0005 and r = 3
Let V be the volume of the hollow shell.
So according to question,
$\begin{aligned} V = \frac{4}{3} π (R^{3} - r^{3}) \\ R and r values are substituted, we get \\ V = \frac{4}{3} π ((3.0005)^{3} - 3^{3}) \dots (i) \end{aligned}$
To get the approximate value of $(3.0005)^{3}$ , differentiation is carried out
But the integer nearest to 3.0005 is 3,
So 3.0005 = 3+0.0005
So let a = 3 and h = 0.0005
Hence, $(3.0005)^{3} = (3 + 0.0005)^{3}$
Let the function becomes,
$f (x) = x^{3} \dots \dots \dots (i i)$
Now applying first derivative, we get
$f^{'} (x) = 2 x^{2} \dots \dots \dots . (i i i)$
Now let $f (a + h) = (3.0005)^{3}$
Now we know,
$f (a + h) = f (a) + h f^{'} (a)$
Now substituting the function from (ii) and (iii), we get
$f (a + h) = a^{3} + h (3 a^{2})$
Substituting the values of a and h, we get
$\begin{aligned} f (3 + 0.0005) = 3^{3} + (0.0005) (3 (3^{2})) \\ \Rightarrow f (3.0005) = 27 + (0.0005) (3 (9)) \\ \Rightarrow (3.0005)^{3} = 27 + (0.0005) (27) \\ \Rightarrow (3.0005)^{3} = 27 + 0.0135 \\ \Rightarrow (3.0005)^{3} = 27.0135 \\ Thus, the approximate value of (3.0005)^{3} = 27.0135 . \end{aligned}$
So, substituting of the value in equation (i),
$\begin{aligned} V & = \frac{4}{3} π (27.0135 - 27) \\ V & = \frac{4}{3} π (0.0135) \\ V & = 4 π (0.0045) \\ V & = 0.018 π {cm}^{3} \end{aligned}$
Hence, the approximate volume of the metal in the hollow spherical shell is $0.018 π c m^{3}$ .

Question:8

A man, 2m tall, walks at the rate of $1 \frac{2}{3}$ m/s towards a street light which is $5 \frac{1}{3}$ m above the ground. At what rate is the tip of his shadow moving? At what rate is the length of the shadow changing when he is $3 \frac{1}{3}$ m from the base of the light?

Answer:

Given: a 2m tall man walks at the rate of $1 \frac{2}{3}$ m/s towards a $5 \frac{1}{3}$ m tall street light.
To find: calculate the rate of movement of the tip of the shadow and also the rate of change in the length of the shadow when he is $3 \frac{1}{3}$ m from the base of the light.
Explanation:

Here the street light is AB = $5 \frac{1}{3}$
And man is DC = 2m
Let BC = x m and CE = y m
The rate of the man’s walk towards the streetlight is $1 \frac{2}{3} m / s$ , and as the man is moving towards the street light, the entity carries a negative charge
Hence, $\frac{dx}{dt} = - 1 \frac{2}{3} m / s$ ....(i)
Now consider ΔABE and ΔDCE
∠DEC = ∠AEB (same angle)
∠DCE = ∠ABD = 90°
Hence by AA similarity,
ΔABE≅ΔDCE
Hence by CPCT,
$\frac{A B}{D C} = \frac{B E}{C E}$
$\begin{aligned} After substituting the values mentioned in the figure, we get \\ \Rightarrow \frac{(5 \frac{1}{3})}{2} = \frac{x + y}{y} \\ \Rightarrow (\frac{16}{3}) y = 2 (x + y) \\ \Rightarrow 16 y = 6 (x + y) \\ \Rightarrow 16 y = 6 x + 6 y \\ \Rightarrow 16 y - 6 y = 6 x \\ \Rightarrow 10 y = 6 x \\ \Rightarrow y = \frac{6}{10} x \\ \Rightarrow y = \frac{3}{5} x \end{aligned}$
Apply the first derivative with respect to t,
$\Rightarrow \frac{d y}{d t} = \frac{d (\frac{3}{5} x)}{d t}$ \ $\Rightarrow \frac{dy}{dt} = \frac{3 d (x)}{5}$
After substituting the value in the above equation from equation (i), we get
$\Rightarrow \frac{d y}{d t} = \frac{3}{5} (- 1 \frac{2}{3})$ \ $\Rightarrow \frac{d y}{d t} = \frac{3}{5} (- \frac{5}{3})$ \ $\Rightarrow \frac{dy}{dt} = - 1 m / s$
So, the rate of movement of tip of the shadow is -1m/s, i.e., the length of the shadow is decreasing at the rate of 1m/s.
Let BE = z
So from fig,
z = x+y
Let’s apply the first derivative with respect to t on the above equation.
$\begin{aligned} \frac{dz}{dt} = \frac{d (x)}{dt} + \frac{dy}{dt} \\ Substituting oft eh corresponding values result in, \\ \frac{d z}{d t} = - 1 \frac{2}{3} - 1 \\ \Rightarrow \frac{d z}{d t} = - \frac{5}{3} - 1 \\ \Rightarrow \frac{dz}{dt} = \frac{- 5 - 3}{3} \\ \Rightarrow \frac{dz}{dt} = \frac{- 8}{3} \\ \Rightarrow \frac{dz}{dt} = - 2 \frac{2}{3} m / s \end{aligned}$
So, the rate of tip of the shadow moving towards the light source is of $2 \frac{2}{3}$ m/s.

Question:9

A swimming pool is to be drained for cleaning. If L represents the number of litres of water in the pool t seconds after the pool has been plugged off to drain and $L = 200 (10 - t)^{2}$ . How fast is the water running out at the end of 5 seconds? What is the average rate at which the water flows out during the first 5 seconds?

Answer:

Given: L represents the number of litres of water in the pool t seconds after the pool has been plugged off to drain and
$L = 200 (10 - t)^{2}$
To find: the rate of water running out of the pool at the last 5s. and also the average rate of water flowing at during the first 5s.
Explanation: Take the rate of Water running out given by $- \frac{d L}{d t}$
Given $L = 200 (10 - t)^{2}$
Differentiation of the above given equation results in,
$\Rightarrow - \frac{d L}{dt} = - \frac{d (200 (10 - t)^{2})}{dt}$ Removing all the constant terms, we get

$\Rightarrow - \frac{d L}{dt} = - 200 \cdot \frac{d ((10 - t)^{2})}{dt}$

Using the power rule of differentiation, we get

$\Rightarrow - \frac{d L}{dt} = - 200.2 (10 - t) \cdot \frac{d (10 - t)}{dt}$

$\Rightarrow - \frac{dL}{dt} = - 400 (10 - t) . (- 1)$

$\Rightarrow - \frac{dL}{dt} = 400 (10 - t) \dots (i)$
Now, to find the speed of water running out at the end of 5s, we need to find the value of equation (i) with t=5
$\Rightarrow {(- \frac{d L}{d t})}_{t = 5} = (400 (10 - t))_{t = 5}$

$\Rightarrow {(- \frac{d L}{d t})}_{t = 5} = 400 (10 - 5)$

$\Rightarrow {(- \frac{d L}{d t})}_{t = 5} = 400 (5) \Rightarrow {(- \frac{d L}{d t})}_{t = 5} = 2000 \frac{L}{s} \dots . . (i i)$
Therefore, 2000 L/s is the rate of water running out at the end of 5s
To calculate the initial rate we need to take t=o in equation (i)
$\Rightarrow {(- \frac{dL}{dt})}_{t = 0} = (400 (10 - t)) t = 0 \Rightarrow {(- \frac{d L}{dt})}_{t = 0} = 400 (10 - 0) \Rightarrow {(- \frac{d L}{dt})}_{t = 0} = 4000 L / s$
Equation (ii) tells about the final rate of water flowing whereas the equation (iii) is the initial rate.
Thus, the average rate during 5s is
$= \frac{initial rate + final rate}{2}$ Substituting the corresponding values, we get $= \frac{4000 + 2000}{2} = 3000 L / s$
After calculation, the final rate of water flowing out of the pool in 5s is 3000L/s.

Question:10

The volume of a cube increases at a constant rate. Prove that the increase in its surface area varies inversely as the length of the side.

Answer:

Given: a cube with volume increasing at a constant rate
To prove: the relation between the increase int eh surface area with the length of the side is inversely proportional.
Explanation: Let ‘a’ the length of the side of the cube.
Take the volume of the cube ‘V’
Then $V = a^{3} . . . . . . . . (i)$
As mentioned in the question, the rate of volume increase is constant, then
$\frac{d V}{dt} = k$
After substituting in the above equation, the values from equation (i) we get
$\frac{d (a^{3})}{dt} = k$
Differentiating the equation with respect to t,
$\Rightarrow 3 a^{2} \times \frac{d (a)}{d t} = k$

$\Rightarrow \frac{d (a)}{dt} = \frac{k}{3 a^{2}} \dots \dots (ii)$
Take S as the surface area of the cube, then $S = 6 a^{2}$
After differentiating the surface area with respect to t, we get
$\frac{dS}{dt} = \frac{d (6 a^{2})}{dt}$
Applying the derivatives, we get
$\Rightarrow \frac{dS}{dt} = 6 \times 2 a \times \frac{d (a)}{dt}$
After substituting value from equation (ii) in the given equation we get
$\Rightarrow \frac{dS}{dt} = 12 a (\frac{k}{3 a^{2}})$
After taking out the like terms,
$\Rightarrow \frac{dS}{dt} = 4 (\frac{k}{a})$
Now converting it to proportional, we get
$\Rightarrow \frac{dS}{dt} \propto \frac{1}{a}$
Therefore, the relation between the length and the side of the cube is inversely proportional in the given condition.
Hence Proved.

Question:11

x and y are the sides of two squares such that $y = x - x^{2}$ . Find the rate of change of the area of the second square with respect to the area of the first square.

Answer:

Given: two squares of sides x and y, such that $y = x - x^{2}$
To find: The rate of change of area of both the squares with respect to each other
Explanation: Take $A_{1}$ and $A_{2}$ as the area of first and 2nd square respectively
Thus, the area of the 1st square will be
$A_{1} = x^{2}$
Differentiating the equation with respect to time, we get
$\frac{d A_{1}}{dt} = \frac{d (x^{2})}{dt} \Rightarrow \frac{d A_{1}}{dt} = 2 x \frac{d (x)}{dt} \dots \dots (i)$
And the area of the second square is
$A^{2} = y^{2} \dots \dots . (i i)$
But given, $y = x - x^{2}$
Now substituting the known value in equation (ii),
$A^{2} = (x - x^{2})^{2} \dots \dots . (i i i)$
After differentiating equation (iii) with respect to time it results into,
$\frac{d A_{2}}{dt} = \frac{d ({(x - x^{2})}^{2})}{dt}$
Apply the power rule of differentiation to get,
$\frac{d A_{2}}{dt} = 2 (x - x^{2}) \times \frac{d (x - x^{2})}{dt}$
Applying the sum rule of differentiation, we get
$\begin{aligned} \frac{d A_{2}}{dt} = 2 (x - x^{2}) (\frac{d (x)}{dt} - \frac{d (x^{2})}{dt}) \\ Applying the derivative, we get \\ \frac{d A_{2}}{dt} = 2 (x - x^{2}) (\frac{d (x)}{dt} - 2 x \frac{d (x)}{dt}) \\ \Rightarrow \frac{d A_{2}}{dt} = 2 x (1 - x) (\frac{d (x)}{dt} (1 - 2 x)) \\ \Rightarrow \frac{d A_{2}}{dt} = 2 x (1 - x) (1 - 2 x) \frac{d (x)}{dt} \dots \dots (iv) \end{aligned}$
Since we need to find the rate of change of area of both the squares with respect to each other, which is
$\frac{d A_{2}}{d A_{1}} = \frac{\frac{d A_{2}}{dt}}{\frac{d A_{1}}{dt}}$
Substituting the known values from equation (i) and (iv), we get
$\frac{d A_{2}}{d A_{1}} = \frac{2 x (1 - x) (1 - 2 x) \frac{d (x)}{dt}}{2 x \frac{d (x)}{dt}}$
By canceling the like terms, we get
$\begin{aligned} \frac{d A_{2}}{d A_{1}} = (1 - x) (1 - 2 x) \\ \Rightarrow \frac{d A_{2}}{d A_{1}} = 1 (1 - 2 x) - x (1 - 2 x) \\ \Rightarrow \frac{d A_{2}}{d A_{1}} = 1 - 2 x - x + 2 x^{2} \\ \Rightarrow \frac{d A_{2}}{d A_{1}} = 2 x^{2} - 3 x + 1 \\ Therefore, the rate of change of area of second square with respect to area of first square is \\ 2 x^{2} - 3 x + 1 \end{aligned}$

Question:12

Find the condition that the curves $2 x = y^{2}$ and 2xy = k intersect orthogonally.

Answer:

Given: two curves $2 x = y^{2}$ and 2xy = k
To find: to track the condition where both the curves intersect orthogonally
Explanation: Given 2xy = k
$\Rightarrow y = \frac{k}{2 x} \dots \dots (i)$
Put in the value of y in another curve equation, i.e., $2 x = y^{2},$ we get
$2 x = {(\frac{k}{2 x})}^{2}$
$\Rightarrow 2 x = \frac{k^{2}}{4 x^{2}}$ $\Rightarrow x^{3} = \frac{k^{2}}{8}$
Putting both the sides under cube root, we get
$\Rightarrow x = \frac{k^{\frac{2}{3}}}{2} \dots \dots (i i)$
Substituting equation (ii) in equation (i), we get
$\Rightarrow y = \frac{k}{2 (\frac{k^{\frac{2}{3}}}{2})}$
$\begin{aligned} \Rightarrow y = \frac{k}{k^{\frac{2}{3}}} \\ \Rightarrow y = k^{1 - \frac{2}{3}} \\ \Rightarrow y = k^{\frac{1}{3}} \end{aligned}$
$Thus, (\frac{k^{\frac{2}{3}}}{2}, k^{\frac{1}{3}})$ is the point of intersection of two curves

Now given $2 x = y^{2}$
After differentiating the equation with respect to x, we get
$\frac{d (2 x)}{dx} = \frac{d (y^{2})}{dx}$ $\Rightarrow 2 = 2 y \frac{d (y)}{dx}$

$\Rightarrow \frac{dy}{dx} = \frac{1}{y}$
After tracking the value of differentiation at the point of intersection i.e., at $(\frac{k^{\frac{2}{3}}}{2}, k^{\frac{1}{3}})$ we
get
$(\frac{d y}{d x})$ $(\frac{k^{\frac{2}{3}}}{2}, k^{\frac{1}{3}})$ $= \frac{1}{k^{\frac{1}{3}}} = m_{1} . . . (i i)$
Also given 2xy = k
Differentiating this with respect to x, we get
$\begin{aligned} \frac{d (2 xy)}{dx} = \frac{d (k)}{dx} \\ \Rightarrow 2 \frac{d (xy)}{dx} = 0 \\ When applying the product rule of differentiation, it results into \\ \Rightarrow 2 (x \frac{d y}{d x} + y \frac{d x}{d x}) = 0 \\ \Rightarrow x \frac{dy}{dx} + y = 0 \\ \Rightarrow x \frac{d y}{d x} = - y \\ \Rightarrow \frac{d y}{d x} = - \frac{y}{x} \end{aligned}$
Then again finding the about given differentiation value at the point of intersection i.e., at $(\frac{k^{\frac{2}{3}}}{2}, k^{\frac{1}{3}})$ , we get
$(\frac{d y}{d x})$ $(\frac{k^{\frac{2}{3}}}{2}, k^{\frac{1}{3}})$
$= - \frac{k^{\frac{1}{3}}}{\frac{k^{\frac{2}{3}}}{2}} = - 2 k^{\frac{1}{3} - \frac{2}{3}} = - 2 k^{- \frac{1}{3}} = m_{2} \dots (iv)$
But the orthogonal interaction of two curves occurs if
m1.m2 = -1
Then Substituting the values from equation (iii) and equation (iv), we get
$\begin{aligned} \frac{1}{k^{\frac{1}{3}}} \cdot (- 2 k^{- \frac{1}{3}}) = - 1 \\ \Rightarrow k^{- \frac{1}{3}} \cdot (- 2 k^{- \frac{1}{3}}) = - 1 \\ \Rightarrow - 2 k^{- \frac{1}{3} - \frac{1}{3}} = - 1 \\ \Rightarrow 2 k^{- \frac{2}{3}} = 1 \\ \Rightarrow \frac{2}{k^{\frac{2}{3}}} = 1 \\ \Rightarrow k^{\frac{2}{3}} = 2 \\ Now taking cube on both sides, we get \\ \Rightarrow k^{2} = 2^{3} = 8 \\ \Rightarrow k = \pm 2 \sqrt{2} \end{aligned}$
This condition proves to fulfill the orthogonal interaction point for the two curves.

Question:13

Prove that the curves $x y = 4$ and $x^{2} + y^{2} = 8$ touch each other.
Answer:

Given: two curves $x y = 4$ and $x^{2} + y^{2} = 8$
To prove: two curves meet each other at a point
Explanation:
Now given $x^{2} + y^{2} = 8$
Differentiating this with respect to x, we get
$\begin{aligned} \frac{d (x^{2} + y^{2})}{d x} = \frac{d (8)}{dx} \\ By using the sum rule of differentiation, we get \\ \Rightarrow \frac{d (x^{2})}{dx} + \frac{d (y^{2})}{dx} = 0 \\ \Rightarrow 2 x + 2 y \frac{d y}{d x} = 0 \\ \Rightarrow 2 x = - 2 y \frac{d y}{d x} \\ \Rightarrow \frac{d y}{d x} = - \frac{x}{y} = m_{1} \dots \dots (i) \end{aligned}$

Also given xy = 4
Differentiating this with respect to x, we get
$\frac{d (xy)}{dx} = \frac{d (4)}{dx}$
The using the product rule of differentiation, we get
$\Rightarrow (x \frac{d y}{d x} + y \frac{d x}{d x}) = 0$ $\Rightarrow x \frac{d y}{d x} + y = 0$
$\Rightarrow x \frac{dy}{dx} = - y$

$\Rightarrow \frac{dy}{dx} = - \frac{y}{x} = m_{2} \dots \dots$ (ii)
But the touch of 2 curves is possible if
$m_{1} = m_{2}$
Now substituting the values from equation (ii) and equation (ii), we get
$- \frac{y}{x} = - \frac{x}{y}$ \ $\Rightarrow y^{2} = x^{2}$ \ $\Rightarrow x = y$
Now substituting $x = y$ in $x^{2} + y^{2} = 8,$ we get $y^{2} + y^{2} = 8$
$\Rightarrow 2 y^{2} = 8$

$\Rightarrow y^{2} = 4$

$\Rightarrow y = \pm 2$
When $y = 2$ $x y = 4$ becomes $x (2) = 4 \Rightarrow x = 2$
when $y = - 2$ $x y = 4$ becomes $x (- 2) = 4 \Rightarrow x = - 2$
Therefore, (2,2) and (-2, -2) is the intersection point of the two curve
Substituting these points of intersection equation (i) and equation (ii), we get
For (2,2),
$m_{1} = - \frac{x}{y} = - \frac{2}{2} = - 1$

$m_{2} = - \frac{y}{x} = - \frac{2}{2} = - 1$

$∴ m_{1} = m_{2}$

$For (- 2, - 2) m_{1} = - \frac{x}{y} = - \frac{- 2}{- 2} = - 1$

$m_{2} = - \frac{y}{x} = - \frac{- 2}{- 2} = - 1 ∴ m_{1} = m_{2}$
Thus, the condition for both the curves to touch is possible if that they have same slope
Hence the two given curves touch each other.
Hence proved

Question:14

Find the co-ordinates of the point on the curve $\sqrt{x} + \sqrt{y} = 4$ at which the tangent is equally inclined to the axes.

Answer:

Given: curve $\sqrt{x} + \sqrt{y} = 4$
To find: point coordinates on which tangent is equally inclined to the axis on the curve
Explanation: given $\sqrt{x} + \sqrt{y} = 4$
After differentiating with respect to,
$\frac{d (\sqrt{x} + \sqrt{y})}{d x} = \frac{d (4)}{dx}$
Now using the sum rule of differentiation
$\frac{d (\sqrt{x})}{d x} + \frac{d (\sqrt{y})}{d x} = 0$

$\Rightarrow \frac{d (x^{\frac{1}{2}})}{dx} + \frac{d (y^{\frac{1}{2}})}{d x} = 0$
Then differentiating the equation, we get
$\Rightarrow \frac{1}{2} x^{\frac{1}{2} - 1} + \frac{1}{2} y^{\frac{1}{2} - 1} \frac{d (y)}{d x} = 0$

$\Rightarrow \frac{1}{2} x^{\frac{- 1}{2}} + \frac{1}{2} y^{\frac{- 1}{2}} \frac{d (y)}{d x} = 0$

$\Rightarrow \frac{1}{2} x^{\frac{- 1}{2}} = - \frac{1}{2} y^{\frac{- 1}{2}} \frac{d (y)}{d x}$

$\Rightarrow \frac{d y}{d x} = - \frac{x^{\frac{- 1}{2}}}{y^{\frac{- 1}{2}}}$

$\Rightarrow \frac{d y}{d x} = - \sqrt{\frac{y}{x}} \dots . . (i)$
The given curve has this tangent
As mentioned in the question tangent is equally inclined to the axis,
$\frac{d y}{d x} = \pm 1$

$\Rightarrow - \sqrt{\frac{y}{x}} = \pm 1$

$\Rightarrow \frac{y}{x} = 1$

$\Rightarrow y = x$
Substituting values in the curve equation from equation (ii)
$\sqrt{y} + \sqrt{y} = 4$

$\Rightarrow 2 \sqrt{y} = 4$

$\Rightarrow \sqrt{y} = 2$

$\Rightarrow y = 4$
When y = 4, then x = 4 from equation (ii)
Show the points on the curve $\sqrt{x} + \sqrt{y} = 4$ at which the tangent equally inclined to the axis has the coordinates (4,4).

Question:15

Find the angle of intersection of the curves $y = 4 - x^{2}$ and $y = x^{2} .$

Answer:

Given: the curves $y = 4 - x^{2}$ and $y = x^{2} .$

To find: the interaction angle between two curves

Explanation: acknowledging first curve

$y = 4 - x^{2}$

when the above curve is differentiated with respect to X

$\begin{aligned} \frac{d y}{d x} = \frac{d (4 - x^{2})}{d x} \\ \frac{d y}{d x} = - 2 x = m_{1} \dots \\ Considering the second curve \\ y = x^{2} \end{aligned}$

second curve differentiated with respect to X

$\frac{d y}{d x} = \frac{d (x^{2})}{d x}$

$\frac{dy}{dx} = 2 x = m_{2} \dots (i i)$

Given $y = x^{2}$ Substituting the other curve equation with this

$x^{2} = 4 - x^{2}$

$\Rightarrow 2 x^{2} = 4$

$\Rightarrow x^{2} = 2$

$\Rightarrow x = \pm \sqrt{2}$

When $x = \sqrt{2}$ , we get $y = (\sqrt{2})^{2} \Rightarrow y = 2$

When $x = - \sqrt{2}$ we get $y = (- \sqrt{2})^{2} \Rightarrow y = 2$

Hence the intersection points are $(\sqrt{2}, 2)$ and $(- \sqrt{2}, 2)$ since angle of intersection can be found using the formula

i.e., $\tan θ = | \frac{m_{1} - m_{2}}{1 + m_{1} m_{2}} |$

Substituting the values from equation (i) and equation (ii), we get

$\Rightarrow \tan θ = | \frac{- 2 x - 2 x}{1 + (- 2 x) (2 x)} |$

$\Rightarrow \tan θ = | \frac{- 4 x}{1 - 4 x^{2}} |$

For $(\sqrt{2}, 2)$ , the equation gets converted into,

$\Rightarrow \tan θ = | \frac{- 4 (\sqrt{2})}{1 - 4 (\sqrt{2})^{2}} |$

$\Rightarrow \tan θ = | \frac{- 4 (\sqrt{2})}{- 7} |$

$\Rightarrow θ = \tan^{- 1} | \frac{4 \sqrt{2}}{7} |$

Hence, the angle at which the curve intersect at $y = 4 - x^{2}$ and $y = x^{2}$ is $\tan^{- 1} | \frac{4 \sqrt{2}}{7} |$

Question:16

Prove that the curves $y^{2} = 4 x$ and $x^{2} + y^{2} - 6 x + 1 = 0$ touch each other at the point (1, 2).

Answer:

Given: two curves $y^{2} = 4 x$ and $x^{2} + y^{2} - 6 x + 1 = 0$

To prove: Two curves have the possibilities of meeting at a point(1,2)

Explanation:

Now given $x^{2} + y^{2} - 6 x + 1 = 0$

Differentiating the above equation with respect to x

$\frac{d (x^{2} + y^{2} - 6 x + 1)}{d x} = \frac{d (0)}{dx}$

Now using the sum rule of differentiation

$\begin{aligned} \Rightarrow \frac{d (x^{2})}{dx} + \frac{d (y^{2})}{dx} - \frac{d (6 x)}{dx} + \frac{d (1)}{dx} = 0 \\ \Rightarrow 2 x + 2 y \frac{d y}{d x} - 6 + 0 = 0 \\ \Rightarrow 2 x - 6 = - 2 y \frac{dy}{dx} \\ \Rightarrow 6 - 2 x = 2 y \frac{dy}{dx} \\ \Rightarrow \frac{d y}{d x} = \frac{6 - 2 x}{2 y} \\ \Rightarrow \frac{d y}{d x} = \frac{3 - x}{y} \\ Finding the solution of above equation at point (1, 2), we get \\ \Rightarrow {(\frac{dy}{dx})}_{(1, 2)} = \frac{3 - 1}{2} = 1 = m_{1} \dots \dots (i) \end{aligned}$

Also given $y^{2} = 4 x$

Differentiating the value with respect to x, we get

$\frac{d (y^{2})}{d x} = \frac{d (4 x)}{dx}$

Using a product rule of differentiation, we get

$\Rightarrow 2 y \frac{dy}{dx} = 4$

$\Rightarrow \frac{dy}{dx} = \frac{4}{2 y}$

$\Rightarrow \frac{dy}{dx} = \frac{2}{y} = m_{2}$

Finding the solution of the above equation at point (1,2), we get

$\Rightarrow {(\frac{d y}{d x})}_{(1, 2)} = \frac{2}{2} = 1 = m_{2} \dots \dots (i i)$

From equation (i) and (ii),

∴ $m_{1} = m_{2}$

Therefore it is possible for both the curves to touch each other at point(1,2).

Hence proved

Question:17

Find the equation of the normal lines to the curve $3 x^{2} - y^{2} = 8$ which are parallel to the line $x + 3 y = 4$ .

Answer:

Given: equation of the curve $3 x^{2} - y^{2} = 8$ equation of line $x + 3 y = 4$

To find: the equation of the normal lines to the given curve which are parallel to the given line

Explanation:

Now given equation of curve as $3 x^{2} - y^{2} = 8$

Differentiating the equation with respect to X

$\frac{d (3 x^{2} - y^{2})}{dx} = \frac{d (8)}{dx}$ $\Rightarrow \frac{d (3 x^{2})}{dx} - \frac{d (y^{2})}{d x} = 0$

$\Rightarrow 3 \times 2 x - 2 y \frac{dy}{dx} = 0$ $\Rightarrow 6 x = 2 y \frac{dy}{dx}$

$\Rightarrow \frac{dy}{dx} = \frac{6 x}{2 y} = \frac{3 x}{y} = m_{1} \dots \dots (i)$

Assume the slope of the normal curve to be m₂ is given by

$\Rightarrow m_{2} = \frac{- 1}{m_{1}}$

Substituting value from equation (i), we get

$\Rightarrow m_{2} = \frac{- 1}{\frac{3 x}{y}}$

$\Rightarrow m_{2} = \frac{- y}{3 x}$

The known equation of the line is

$x + 3 y = 4$

$\Rightarrow 3 y = 4 - x$

$\Rightarrow y = \frac{4}{3} - \frac{x}{3}$

$\Rightarrow y = - \frac{x}{3} + \frac{4}{3}$

$∴$ the slope of this line will be

$m_{3} = - \frac{1}{3} \dots \dots (i i i)$

since slope of normal to the curve should be equal to the slope of the line which is parallel to the curve,

$∴ m_{2} = m_{3}$

Substituting values from equation (ii) and (iii), we get

$\Rightarrow - \frac{y}{3 x} = - \frac{1}{3}$

$\Rightarrow 3 y = 3 x$

$\Rightarrow y = x_{\dots \dots} (i v)$

After putting y=x in the equation of the curve, we get

$3 x^{2} - y^{2} = 8$

$\Rightarrow 3 x^{2} - x^{2} = 8$

$\Rightarrow 2 x^{2} = 8$

$\Rightarrow x^{2} = 4$

$\Rightarrow x = \pm 2$

But from equation (iv)

y=x

$\Rightarrow y = \pm 2$

Therefore, the points at which normal to the given curve is parallel to the given line are (2, 2) and (-2, -2)

Thus, the equations of the normal can be calculated by

$\begin{aligned} y - 2 = m_{2} (x - 2) and y + 2 = m_{2} (x + 2) \\ \Rightarrow y - 2 = - \frac{1}{3} (x - 2) And y + 2 = - \frac{1}{3} (x + 2) \\ 3 y - 6 = - x + 2 and 3 y + 6 = - x - 2 \\ 3 y + x = 8 and 3 y + x = - 8 \\ Hence the equation of the normal lines to the curve 3 x^{2} - y^{2} = 8 which are parallel to the line \\ x + 3 y = 8 and 3 y + y = - 8 \end{aligned}$

Question:18

At what points on the curve $x^{2} + y^{2} - 2 x - 4 y + 1 = 0$ , the tangents are parallel to the y-axis?

Answer:

Given: equation of a curve $x^{2} + y^{2} - 2 x - 4 y + 1 = 0$

To find: the points on the curve $x^{2} + y^{2} - 2 x - 4 y + 1 = 0$ , the tangents are parallel to the y-axis

Explanation: the given equation of curve as $x^{2} + y^{2} - 2 x - 4 y + 1 = 0$

Differentiating the equation with respect to X

$\begin{aligned} \frac{d (x^{2} + y^{2} - 2 x - 4 y + 1)}{dx} = \frac{d (0)}{dx} \\ Now applying the sum rule of differentiation \\ \Rightarrow \frac{d (x^{2})}{dx} + \frac{d (y^{2})}{dx} - \frac{d (2 x)}{dx} - \frac{d (4 y)}{dx} + \frac{d (1)}{dx} = 0 \\ \Rightarrow 2 x + 2 y \frac{d (y)}{d x} - 2 - 4 \frac{d y}{d x} + 0 = 0 \\ \Rightarrow (2 y - 4) \frac{d (y)}{d x} = 2 - 2 x \\ \Rightarrow \frac{d y}{d x} = \frac{2 (1 - x)}{2 (y - 2)} \end{aligned}$

Sense the tangents are parallel to the axis as mentioned in the question

Thus,

$\begin{aligned} \tan θ = \tan 90^{\circ} = \frac{dy}{dx} \\ Value from equation (i) can be substituted equation \\ \frac{1}{0} = \frac{(1 - x)}{(y - 2)} \end{aligned}$

$\Rightarrow y - 2 = 0$

$\Rightarrow y = 2$

Putting y = 2 in curve equation, we get

$x^{2} + y^{2} - 2 x - 4 y + 1 = 0$

$\Rightarrow x^{2} + 2^{2} - 2 x - 4 (2) + 1 = 0$

$\Rightarrow x^{2} + 4 - 2 x - 8 + 1 = 0$

$\Rightarrow x^{2} - 2 x - 3 = 0$

After the splitting of the middle term,

$\Rightarrow x^{2} - 3 x + x - 3 = 0$

$\Rightarrow x (x - 3) + 1 (x - 3) = 0$

$\Rightarrow (x + 1) (x - 3) = 0$

$\Rightarrow x + 1 = 0 o r x - 3 = 0$

$\Rightarrow x = - 1 o r x = 3$

Thus, the needed points are(-1, 2) and (3, 2).

Hence the points on the curve $x^{2} + y^{2} - 2 x - 4 y + 1 = 0$ , the tangents are parallel to the y-axis are (-1, 2) and (3, 2).

Question:19

Show that the line $\frac{x}{a} + \frac{y}{b} = 1$ touches the curve $y = {b . e}^{- \frac{x}{a}}$ at the point where the curve intersects the axis of y.

Answer:

Given: equation of line $\frac{x}{a} + \frac{y}{b} = 1$

the curve $y = {b . e}^{- \frac{x}{a}}$ intersects the y-axis

To show: the line touches the curve at the point where the curve intersects the axis of y

Explanation: given the curve $y = {b . e}^{- \frac{x}{a}}$ intersects the y-axis, i.e., at x = 0

Now differentiate the given curve equation with respect to x, i.e.,

$\frac{d y}{d x} = \frac{d (b \cdot e^{- \frac{x}{a}})}{d x}$

Removing all the constant term,

$\Rightarrow \frac{d y}{d x} = b \frac{d (e^{- \frac{x}{a}})}{d x}$

After differentiate in the equation we get

$\Rightarrow \frac{d y}{d x} = b \cdot e^{- \frac{x}{a}} \frac{d (- \frac{x}{a})}{d x}$ $\Rightarrow \frac{d y}{d x} = b \cdot e^{- \frac{x}{a}} \cdot (- \frac{1}{a})$

Now put the value of $x = 0$ ,

$\Rightarrow {(\frac{d y}{d x})}_{x = 0} = b \cdot e^{- \frac{0}{a}} \cdot (- \frac{1}{a})$

$\Rightarrow {(\frac{d y}{d x})}_{x = 0} =$ b. $1 \cdot (- \frac{1}{a})$ $\Rightarrow {(\frac{d y}{d x})}_{x = 0} = (- \frac{b}{a}) = m_{1} \dots \dots (i)$

Then considering the line equation,

$\frac{x}{a} + \frac{y}{b} = 1$ Now differentiate it with respect to X $\frac{d (\frac{X}{a})}{dx} + \frac{d (\frac{y}{b})}{dx} = \frac{d (1)}{dx}$

Removing all the constant terms

$\Rightarrow \frac{1}{a} + \frac{1}{b} \frac{d y}{d x} = 0$

$\Rightarrow \frac{1}{b} \frac{dy}{dx} = - \frac{1}{a}$ \ $\Rightarrow \frac{d y}{d x} = - \frac{b}{a} = m_{2} \dots .$

Line touches the curve only if their slopes are equal

From equation (i) and (ii), we see that

$m_{1} = m_{2}$

Hence, the line touches the curve at the point where the curve intersects the axis of y.

Question:20

Show that $f (x) = 2 x + \cot^{- 1} x + \log (\sqrt{1 + x^{2}} - x)$ is increasing in R.

Answer:

Given: $f (x) = 2 x + \cot^{- 1} x + \log (\sqrt{1 + x^{2}} - x)$

To show: the mentioned function increases in R

Explanation: Given $f (x) = 2 x + \cot^{- 1} x + \log (\sqrt{1 + x^{2}} - x)$

Substituting the first derivative in respect to x

$f^{'} (x) = \frac{d (2 x + \cot^{- 1} x + \log (\sqrt{1 + x^{2}} - x))}{dx}$

$\Rightarrow f^{'} (x) = \frac{d (2 x)}{dx} + \frac{d (\cot^{- 1} x)}{dx} + \frac{d (\log (\sqrt{1 + x^{2}} - x))}{dx}$

But the derivative of 2x is 2,so

$f^{'} (x) = 2 + \frac{d (\cot^{- 1} x)}{dx} + \frac{d (\log (\sqrt{1 + x^{2}} - x))}{dx}$

But the derivative of $\cot^{- 1} x = - \frac{1}{x^{2} + 1}$ ,so

$f^{'} (x) = 2 + (- \frac{1}{x^{2} + 1}) + \frac{d (\log (\sqrt{1 + x^{2}} - x))}{dx}$

$\Rightarrow f^{'} (x) = 2 + (- \frac{1}{x^{2} + 1}) + \frac{1}{\sqrt{1 + x^{2}} - x} \cdot \frac{d (\sqrt{1 + x^{2}} - x)}{d x}$

When the sum rule is applied to the last part we get,

$f^{'} (x) = 2 + (- \frac{1}{x^{2} + 1}) + \frac{1}{\sqrt{1 + x^{2}} - x} (\frac{d (\sqrt{1 + x^{2}})}{dx} - \frac{d (x)}{dx})$

$\Rightarrow f^{'} (x) = 2 + (- \frac{1}{x^{2} + 1}) + \frac{1}{\sqrt{1 + x^{2}} - x} (\frac{d ({(1 + x^{2})}^{\frac{1}{2}})}{dx} - 1)$

$\Rightarrow f^{'} (x) = 2 + (- \frac{1}{x^{2} + 1}) + \frac{1}{\sqrt{1 + x^{2}} - x} (\frac{1}{2} \cdot {(1 + x^{2})}^{\frac{1}{2} - 1} \cdot \frac{d (1 + x^{2})}{d x} - 1)$

$\Rightarrow f^{'} (x) = 2 + (- \frac{1}{x^{2} + 1}) + \frac{1}{\sqrt{1 + x^{2}} - x} (\frac{1}{2} \cdot {(1 + x^{2})}^{- \frac{1}{2}} \cdot [\frac{d (1)}{d x} + \frac{d (x^{2})}{d x}] - 1)$

$\Rightarrow f^{'} (x) = 2 + (- \frac{1}{x^{2} + 1}) + \frac{1}{\sqrt{1 + x^{2}} - x} (\frac{1}{2 \sqrt{1 + x^{2}}} \cdot [0 + 2 x] - 1)$

$\Rightarrow f^{'} (x) = 2 + (- \frac{1}{x^{2} + 1}) + \frac{1}{\sqrt{1 + x^{2}} - x} (\frac{2 x}{2 \sqrt{1 + x^{2}}} - 1)$

$\Rightarrow f^{'} (x) = 2 - \frac{1}{x^{2} + 1} + \frac{1}{\sqrt{1 + x^{2}} - x} (\frac{x}{\sqrt{1 + x^{2}}} - 1)$ $\Rightarrow f^{'} (x) = 2 - \frac{1}{x^{2} + 1} + \frac{1}{\sqrt{1 + x^{2}} - x} (\frac{x - \sqrt{1 + x^{2}}}{\sqrt{1 + x^{2}}})$

Cutting out all the like terms, $f^{'} (x) = 2 - \frac{1}{x^{2} + 1} + \frac{- 1}{\sqrt{1 + x^{2}}}$

Then, adding by taking the LCM, we get

$f^{'} (x) = \frac{2 + 2 x^{2} - 1 - \sqrt{1 + x^{2}}}{x^{2} + 1}$

$\Rightarrow f^{'} (x) = \frac{2 x^{2} + 1 - \sqrt{1 + x^{2}}}{x^{2} + 1}$

Then to calculate any real value x, the above value of f(x) is larger than or equal to zero

Hence $\frac{2 x^{2} + 1 - \sqrt{1 + x^{2}}}{x^{2} + 1} \forall x \in R$

And we know, if $f^{'} (x) \geq 0$ , then f(x) is increasing function.

Hence, the given function is an increasing function in R.

Question:21

Show that for $a \geq 1, f (x) = \sqrt{3} \sin x - \cos x - 2 a x + b$ is decreasing in R.

Answer:

Given: $a \geq 1, f (x) = \sqrt{3} \sin x - \cos x - 2 a x + b$

To show: the above function is decreasing in R.

Explanation: Given $f (x) = \sqrt{3} \sin x - c o s x - 2 a x + b$

The first derivative is applied with respect to x,

$f^{'} (x) = \frac{d (\sqrt{3} \sin x - \cos x - 2 ax + b)}{dx}$

By using the sum rule of differentiation, we get

$f^{'} (x) = \frac{d (\sqrt{3} \sin x)}{dx} - \frac{d (\cos x)}{dx} - \frac{d (2 ax)}{dx} + 0$

Removing all the constant terms, we get

$f^{'} (x) = \sqrt{3} \frac{d (\sin x)}{dx} - \frac{d (\cos x)}{dx} - 2 a \frac{d (x)}{dx}$

But the derivative of sin X = cos x and that of cos x = -sin x, so

$f^{'} (x) = \sqrt{3} c o s x - (- s i n x) - 2 a f^{'} (x) = \sqrt{3} \cos x + \sin x - 2 a$

Multiplying and dividing RHS by 2,

$f^{'} (x) = 2 [\frac{\sqrt{3}}{2} \cos x + \frac{1}{2} \sin x] - 2 a$
$\sin \cos \frac{π}{6} = \frac{\sqrt{3}}{2}$ and $\sin \frac{π}{6} = \frac{1}{2}$ , putting these values in the above equation,
$f^{'} (x) = 2 [\cos \frac{π}{6} \cos x + \sin \frac{π}{6} \cdot \sin x] - 2 a$

But $\cos (A - B) = \cos A \cos B + \sin A \cdot \sin B$ , placing the values in the above given equation,
$f^{'} (x) = 2 [\cos (\frac{π}{6} - x)] - 2 a$

Now, we know $\cos x$ always belong to $[- 1, 1]$ for $a \geq 1$
$2 [\cos (\frac{π}{6} - x)] - 2 a \leq 0$

And we know, if $f^{'} (x) \leq 0$ , then $f (x)$ is decreasing function.
Therefore, the given function is decreasing function in $R$ .

Question:22

Show that $f (x) = t a n^{- 1} (\sin x + \cos x)$ is an increasing function in $(0, \frac{π}{4})$

Answer:

Given: $f (x) = t a n^{- 1} (\sin x + \cos x)$

To show: the given function is increasing in $(0, \frac{π}{4})$

Explanation: Given

$f (x) = t a n^{- 1} (\sin x + \cos x)$

First derivative is applied with respect to x,

$f^{'} (x) = \frac{d (\tan^{- 1} (\sin x + \cos x))}{dx}$

Using the differentiation rule for $t a n$ ^{-1}$, results into

$f^{'} (x) = \frac{1}{(\sin x + \cos x)^{2} + 1} \cdot \frac{d (\sin x + \cos x)}{dx}$

Now use the sum rule of differentiation,

$\begin{aligned} f^{'} (x) = \frac{1}{(\sin x + \cos x)^{2} + 1} [\frac{d (\sin x)}{d x} + \frac{d (\cos x)}{d x}] \\ But the derivative of \sin X = \cos x and that of \cos x = - \sin x, s \circ \\ f^{'} (x) = \frac{1}{(\sin x + \cos x)^{2} + 1} [\cos x + (- \sin x)] \\ Expanding (\sin x + \cos x)^{2}, we get \\ f^{'} (x) = \frac{\cos x - \sin x}{\sin^{2} x + \cos^{2} x + 2 \sin x \cos x + 1} \\ But \sin^{2} x + \cos^{2} x = 1 and 2 \sin x \cos x = \sin 2 x, thus the equation given above gets converted \\ into, \\ f^{'} (x) = \frac{\cos x - \sin x}{1 + \sin 2 x + 1} \\ f^{'} (x) = \frac{\cos x - \sin x}{\sin 2 x + 2} \end{aligned}$

To make f(x) to be increasing function,

$f^{'} (x) \geq 0$

$\Rightarrow \frac{\cos x - \sin x}{\sin 2 x + 2} \geq 0$ $\Rightarrow \cos x - \sin x \geq 0$

$\Rightarrow \cos x \geq \sin x$

But this is possible only when $x \in (0, \frac{π}{4})$

Hence, the given function is increasing function in $(0, \frac{π}{4})$

Question:23

At what point, the slope of the curve $y = - x^{3} + 3 x^{2} + 9 x - 27$ is maximum? Also find the maximum slope.

Answer:

Given: $y = - x^{3} + 3 x^{2} + 9 x - 27$

To find: the point in the curve where the slope is maximum and the maximum value of the slope.

Explanation: given $y = - x^{3} + 3 x^{2} + 9 x - 27$

The slope of the curve can be found by calculating the first derivative of the known curve equation,

Thus, slope of the curve is

$\frac{d y}{d x} = \frac{d (- x^{3})}{d x} + \frac{d (3 x^{2})}{d x} + \frac{d (9 x)}{d x} - \frac{d (27)}{d x}$

Then using the derivative,

$\frac{d y}{d x} = - 3 x^{2} + 3.2 x + 9 - 0$

$\Rightarrow \frac{d y}{d x} = - 3 x^{2} + 6 x + 9$

To know the critical point, it is necessary to have the value of the slope,

$\frac{d^{2} y}{{dx}^{2}} = \frac{d (- 3 x^{2})}{dx} + \frac{d (6 x)}{dx} + \frac{d (9)}{dx}$

Using the derivative leads to,

$\frac{d^{2} y}{d x^{2}} = - 3 \cdot 2 x + 6 + 0$

$\Rightarrow \frac{d^{2} y}{d x^{2}} = - 6 x + 6$

When the second derivative is equated to 0 it gives the critical point, i.e.,

$\frac{d^{2} y}{d x^{2}} = 0$

$\Rightarrow - 6 x + 6 = 0$

$\Rightarrow 6 x = 6$

$\Rightarrow x = 1 \dots \dots . (i)$

Then finding the third derivative of the curve,

i.e.,

$\frac{d^{3} y}{d x^{3}} = \frac{d (- 6 x)}{d x} + \frac{d (6)}{d x}$

Putting the values of the derivative results in

$\frac{d^{3} y}{d x^{3}} = - 6 < 0$

As the third derivative is less than 0, so the maximum slope of the given curve is at x=1.

Equating the first derivative with x=1 leads to the maximum value of the slope, i.e.,

${(\frac{d y}{d x})}_{x = 1} = - 3 x^{2} + 6 x + 9$

$\Rightarrow {(\frac{d y}{d x})}_{x = 1} = - 3 (1)^{2} + 6 (1) + 9$

$\Rightarrow {(\frac{d y}{d x})}_{x = 1} = - 3 + 6 + 9$ $\Rightarrow {(\frac{dy}{dx})}_{x = 1} = 12$

Therefore, the slope of the curve is maximum at x=1, and the maximum value of the slope is 12

Question:24

Prove that $f (x) = \sin x + \sqrt{3} \cos x$ has maximum value at $x = \frac{π}{6}$ .

Answer:

Given: $f (x) = \sin x + \sqrt{3} \cos x$

To prove: the given function has maximum value at $x = \frac{π}{6}$

Explanation: given $f (x) = \sin x + \sqrt{3} \cos x$

While calculating the first derivative we get,

$f^{'} (x) = \frac{d (\sin x)}{dx} + \frac{d (\sqrt{3} \cos x)}{d x}$

Then putting the derivative, we get

$f^{'} (x) = c o s x - \sqrt{3} s i n x$

Critical point ca be calculated by equating the derivative with 0,

$f^{'} (x) = 0$

$\Rightarrow c o s x - \sqrt{3} s i n x = 0 \Rightarrow \sqrt{3} s i n x = c o s x$

$\Rightarrow \frac{\sin x}{\cos x} = \frac{1}{\sqrt{3}}$ \ $\Rightarrow \tan x = \frac{1}{\sqrt{3}}$ \ $x = \frac{π}{6}$

This is possible only when

The second derivative if the c=function can be calculated by,

$f^{''} (x) = \frac{d (\cos x)}{dx} - \sqrt{3} \frac{d (\sin x)}{dx}$

Putting the value of derivative, we get

$f^{''} (x) = - \sin x - \sqrt{3} \cos x$

Then, we will substitute $x = \frac{π}{6}$ in the above equation, we get

$f^{''} (x) = - \sin x - \sqrt{3} \cos x$ \ $f^{'} (\frac{π}{6}) = - \sin \frac{π}{6} - \sqrt{3} \cos \frac{π}{6}$

Putting the corresponding value, we get

$f^{'} (\frac{π}{6}) = - \frac{1}{2} - \sqrt{3} \times \frac{\sqrt{3}}{2} f^{'} (\frac{π}{6}) = - \frac{1}{2} - \frac{3}{2} f^{'} (\frac{π}{6}) = - 2 < 0$

Hence f(x) has a maximum value at $x = \frac{π}{6}$ .

Hence proved.

Question:25

If the sum of the lengths of the hypotenuse and a side of a right angled triangle is given, show that the area of the triangle is maximum when the angle between them is $\frac{π}{3}$ .

Answer:

Given: a right-angled triangle with the sum of the lengths of its hypotenuse and side.
To show: at this angle $\frac{π}{3}$ the area of the triangle is maximum
Explanation:

Let ΔABC be the right-angled triangle,

Let hypotenuse, AC = y,

side, BC = x, AB = h

then the calculation of sum of the side and hypotenuse is done using,

⇒ x+y = k, where k is any constant value

⇒ y = k-x………..(i)

Take A as the area of the triangle, as we know

$A = \frac{1}{2} h \cdot x \dots (i i)$

Then using the Pythagoras theorem, we get

$y^{2} = x^{2} + h^{2}$

$\Rightarrow h = \sqrt{y^{2} - x^{2}}$

Putting the value from equation (i) in above equation, we get

$\Rightarrow h = \sqrt{(k - x)^{2} - x^{2}}$

$\Rightarrow h = \sqrt{k^{2} + x^{2} - 2 kx - x^{2}}$

$\Rightarrow h = \sqrt{k^{2} - 2 kx} \dots \dots \dots (i i i)$

Applying the values from equation (iii) into equation (ii), we get

$A = \frac{1}{2} (\sqrt{k^{2} - 2 k x}) \cdot x$

The above equation is differentiated with respect to x,

$\frac{d A}{dx} = \frac{d (\frac{1}{2} (\sqrt{k^{2} - 2 kx}) \cdot x)}{dx}$

Then the constant terms are taken out,

$\frac{d A}{d x} = \frac{1}{2} \frac{d ((\sqrt{k^{2} - 2 k x}) \cdot x)}{d x}$

$\begin{aligned} By using the product rule, \\ \frac{d A}{dx} = \frac{1}{2} [(\sqrt{k^{2} - 2 kx}) \frac{d (x)}{dx} + (x) \frac{d ((\sqrt{k^{2} - 2 kx}))}{dx}] \\ \Rightarrow \frac{d A}{dx} = \frac{1}{2} [(\sqrt{k^{2} - 2 kx}) (1) + (x) \frac{d ({(k^{2} - 2 kx)}^{\frac{1}{2}})}{dx}] \end{aligned}$

Power rule pf differentiation is applied on the second part of the above equation,

$\frac{d A}{dx} = \frac{1}{2} [(\sqrt{k^{2} - 2 kx}) + (x) [\frac{1}{2} \cdot {(k^{2} - 2 kx)}^{\frac{1}{2} - 1} \cdot \frac{d (k^{2} - 2 kx)}{dx}]] \Rightarrow \frac{d A}{dx} = \frac{1}{2} [(\sqrt{k^{2} - 2 kx}) + \frac{1}{2} (x) {(k^{2} - 2 kx)}^{\frac{- 1}{2}} [\frac{d (k^{2})}{d x} - \frac{d (2 kx)}{d x}]] \Rightarrow \frac{d A}{dx} = \frac{1}{2} [(\sqrt{k^{2} - 2 kx}) + \frac{(x)}{2 \sqrt{k^{2} - 2 kx}} [0 - 2 k]]$

$\Rightarrow \frac{d A}{dx} = \frac{1}{2} [(\sqrt{k^{2} - 2 kx}) - \frac{2 k (x)}{2 \sqrt{k^{2} - 2 kx}}]$

$\begin{aligned} \Rightarrow \frac{d A}{dx} = \frac{1}{2} [(\sqrt{k^{2} - 2 kx}) - \frac{kx}{\sqrt{k^{2} - 2 kx}}] \dots . (iv) \\ When the first derivative is equated with 0, it gives critical point, \\ \frac{d A}{dx} = 0 \end{aligned}$

$\Rightarrow \frac{1}{2} [(\sqrt{k^{2} - 2 kx}) - \frac{kx}{\sqrt{k^{2} - 2 kx}}] = 0 \Rightarrow (\sqrt{k^{2} - 2 kx}) - \frac{kx}{\sqrt{k^{2} - 2 kx}} = 0 \Rightarrow (\sqrt{k^{2} - 2 kx}) = \frac{kx}{\sqrt{k^{2} - 2 kx}} \Rightarrow (\sqrt{k^{2} - 2 kx}) (\sqrt{k^{2} - 2 kx}) = kx \Rightarrow k^{2} - 2 kx = kx \Rightarrow k^{2} = 2 kx + kx \Rightarrow k^{2} = 3 kx \Rightarrow k = 3 x \Rightarrow x = \frac{k}{3}$

Once again, differentiating equation (iv) with respect to x, we get

$\begin{aligned} \Rightarrow \frac{d^{2} A}{{dx}^{2}} = \frac{d (\frac{1}{2} [(\sqrt{k^{2} - 2 kx}) - \frac{kx}{\sqrt{k^{2} - 2 kx}}])}{dx} \\ Taking out the constant term and taking the LCM, we get \\ \Rightarrow \frac{d^{2} A}{{dx}^{2}} = \frac{1}{2} \frac{d ([\frac{(k^{2} - 2 kx) - kx}{\sqrt{k^{2} - 2 kx}}])}{dx} \\ \Rightarrow \frac{d^{2} A}{d x^{2}} = \frac{1}{2} \frac{d ([\frac{(k^{2} - 3 k x)}{\sqrt{k^{2} - 2 k x}}])}{d x} \\ \Rightarrow \frac{d^{2} A}{{dx}^{2}} = \frac{1}{2} \frac{d ((k^{2} - 3 kx) {(k^{2} - 2 kx)}^{\frac{- 1}{2}})}{dx} \end{aligned}$

Using the product rule of differentiation,

$\Rightarrow \frac{d^{2} A}{{dx}^{2}} = \frac{1}{2} [({(k^{2} - 2 kx)}^{\frac{- 1}{2}} \frac{d (k^{2} - 3 kx)}{dx}) - (k^{2} - 3 kx) \frac{d ([{(k^{2} - 2 kx)}^{\frac{- 1}{2}}])}{dx}]$

Then using the power rule of differentiation,

$\begin{aligned} \Rightarrow \frac{d^{2} A}{{dx}^{2}} = \frac{1}{2} & [({(k^{2} - 2 kx)}^{\frac{- 1}{2}} (\frac{d (k^{2})}{d x} - \frac{d (3 kx)}{d x})) \\ - (k^{2} - 3 kx) (- \frac{1}{2} \cdot (k^{2} - 2 kx) \frac{- 1}{2} - 1 \cdot \frac{d ([k^{2} - 2 kx])}{dx})] \\ \Rightarrow \frac{d^{2} A}{{dx}^{2}} = \frac{1}{2} & [{(k^{2} - 2 kx)}^{\frac{- 1}{2}} (- 3 k)) \\ - (k^{2} - 3 kx) (- \frac{1}{2} \cdot {(k^{2} - 2 kx)}^{\frac{- 1}{2}} \cdot {(k^{2} - 2 kx)}^{- 1} \cdot (- 2 k))] \end{aligned}$

$\Rightarrow \frac{d^{2} A}{{dx}^{2}} = \frac{1}{2} [- \frac{3 k}{\sqrt{k^{2} - 2 kx}} - (k^{2} - 3 kx) (\frac{k}{(k^{2} - 2 kx) \sqrt{k^{2} - 2 kx}})] \Rightarrow \frac{d^{2} A}{{dx}^{2}} = \frac{1}{2} [- \frac{3 k}{\sqrt{k^{2} - 2 kx}} - \frac{k (k^{2} - 3 kx)}{(k^{2} - 2 kx) \sqrt{k^{2} - 2 kx}}]$

Putting $x = \frac{k}{3}$ , in above equation, we get

$\Rightarrow {(\frac{d^{2} A}{{dx}^{2}})}_{x = \frac{k}{3}} = \frac{1}{2} [- \frac{3 k}{\sqrt{k^{2} - 2 k (\frac{k}{3})}} - \frac{k (k^{2} - 3 k (\frac{k}{3}))}{(k^{2} - 2 k (\frac{k}{3})) \sqrt{k^{2} - 2 k (\frac{k}{3})}}]$ \ $\Rightarrow {(\frac{d^{2} A}{{dx}^{2}})}_{x = \frac{k}{3}} = \frac{1}{2} [- \frac{3 k}{\sqrt{k^{2} (1 - \frac{2}{3})}} - \frac{k (k^{2} - k^{2})}{(k^{2} (1 - \frac{2}{3})) \sqrt{k^{2} (1 - \frac{2}{3})}}]$

$\Rightarrow {(\frac{d^{2} A}{{dx}^{2}})}_{x = \frac{k}{3}} = \frac{1}{2} [- \frac{3 k}{k \sqrt{(\frac{1}{3})}} - 0] \Rightarrow {(\frac{d^{2} A}{{dx}^{2}})}_{x = \frac{k}{3}} = \frac{1}{2} [- \frac{3}{\sqrt{(\frac{1}{3})}]} \Rightarrow {(\frac{d^{2} A}{{dx}^{2}})}_{x = \frac{k}{3}} = \frac{1}{2} [- 3 \sqrt{3}] \Rightarrow {(\frac{d^{2} A}{{dx}^{2}})}_{x = \frac{k}{3}} = \frac{- 3 \sqrt{3}}{2} < 0$

Hence the maximum value of A is at $X = \frac{k}{3}$

We know,

$\cos θ = \frac{adjacent side}{hypotenuse}$

Then from figure,

$\Rightarrow \cos θ = \frac{x}{y}$

Applying the value of y=k -x from equation (i), we get

$\Rightarrow \cos θ = \frac{x}{k - x}$

Putting the value of $x = \frac{k}{3},$ we get

$\Rightarrow \cos θ = \frac{\frac{k}{3}}{k - \frac{k}{3}}$ \ $\Rightarrow \cos θ = \frac{1}{2}$

This possibility is present when $θ = \frac{π}{3}$

Therefore, the area of the triangle is maximum only when the angle between them is $\frac{π}{3}$

Question:26

Find the points of local maxima, local minima and the points of inflection of the function $f (x) = x^{5} - 5 x^{4} + 5 x^{3} - 1$ . Also find the corresponding local maximum and local minimum values.

Answer:

Given: function $f (x) = x^{5} - 5 x^{4} + 5 x^{3} - 1$

To find: the points of local maxima, local minima and the points of inflection of f(x) and also to find the corresponding local maximum and local minimum values.

Explanation: given $f (x) = x^{5} - 5 x^{4} + 5 x^{3} - 1$

Calculating the first derivative of f(x), i.e.,

$f^{'} (x) = \frac{d (x^{5} - 5 x^{4} + 5 x^{3} - 1)}{d x}$

When the sum rule of differentiation is applied followed by taking out all the constant term, we get $f^{'} (x) = \frac{d (x^{5})}{dx} - \frac{d (5 x^{4})}{dx} + \frac{d (5 x^{3})}{dx} - \frac{d (1)}{dx}$

$f^{'} (x) = 5 x^{4} - 5.4 x^{3} + 5.3 x^{2} - 0$

$\Rightarrow f^{'} (x) = 5 x^{4} - 20 x^{3} + 15 x^{2}$

Equating the first derivative with 0 to find out the critical point,

$f^{'} (x) = 0 \Rightarrow 5 x^{4} - 20 x^{3} + 15 x^{2} = 0 \Rightarrow 5 x^{2} (x^{2} - 4 x + 3) = 0 \Rightarrow 5 x^{2} (x^{2} - 4 x + 3) = 0$

Then splitting the middle term, we get $\Rightarrow 5 x^{2} (x^{2} - 3 x - x + 3) = 0$

$\Rightarrow 5 x^{2} [x (x - 3) - 1 (x - 3)] = 0$

$\Rightarrow 5 x^{2} (x - 1) (x - 3) = 0$

$\Rightarrow 5 x^{2} = 0 o r (x - 1) = 0 o r (x - 3) = 0$

$\Rightarrow x = 0 o r x = 1 o r x = 3$

Now we will find the corresponding y value by putting the numerous values of x in given function

$f (x) = x^{5} - 5 x^{4} + 5 x^{3} - 1$

$W h e n x = 0, f (0) = 0^{5} - 5 (0)^{4} + 5 (0)^{3} - 1 = - 1$

Hence the point is (0,-1)

When x = 1, $f (1) = 1^{5} - 5 (1)^{4} + 5 (1)^{3} - 1 = 1 - 5 + 5 - 1 = 0$

Hence the point is (1,0)

When x = 3, $f (3) = 3^{5} - 5 (3)^{4} + 5 (3)^{3} - 1 = 243 - 405 + 135 - 1 = - 28$

Hence the point is (3,-28)

Therefore, we see that

At x = 3, y has minimum value = -28. Hence x = 3 is point of local minima.

At x = 1, y has maximum value = 0. Hence x = 1 is point of local maxima.

And at x = 0, y has neither maximum nor minimum value, hence this is point of inflection.

Question:27

A telephone company in a town has 500 subscribers on its list and collects fixed charges of Rs 300/- per subscriber per year. The company proposes to increase the annual subscription and it is believed that for every increase of Re 1/- one subscriber will discontinue the service. Find what increase will bring maximum profit?

Answer:

Given: a telephone company in a town has 500 subscribers and collects fixed charges of Rs 300/- per subscriber per year, company increase the annual subscription and for every increase of Re 1/- one subscriber will discontinue the service
To find: the best increase amount for the company to earn maximum profit
Explanation: company has 500 subscribers, and collects 300 per subscriber per year.
Let x as the increase in annual subscription by the company
As per the question, the number of subscribers to discontinue the service will be x
The total revenue earned after the increment would be calculated by,
$R (x) = (500 - x) (300 + x) \Rightarrow R (x) = 500 (300 + x) - x (300 + x) \Rightarrow R (x) = 150000 + 500 x - 300 x - x \textsuperscript 2 \Rightarrow R (x) = 150000 + 200 x - x \textsuperscript 2$
We need to calculate the first derivative of the above equation,
$R^{'} (x) = \frac{d (150000 + 200 x - x^{2})}{d x} R^{'} (x) = \frac{d (150000)}{dx} + \frac{d (200 x)}{d x} - \frac{d (x^{2})}{d x}$
$R^{'} (x) = 0 + 200 - 2 x \dots . . (i)$
The critical point is calculated by equating the first derivative with 0,
$R^{'} (x) = 0 \Rightarrow 200 - 2 x = 0 \Rightarrow 2 x = 200 \Rightarrow x = 100 \dots \dots (i i)$
Then we calculate the second derivative of the total revenue function, i.e., again differentiate equation (i), i.e.,
$R^{''} (x) = \frac{d (200 - 2 x)}{dx} R^{''} (x) = \frac{d (200)}{dx} - \frac{d (2 x)}{d x} R^{''} (x) = 0 - 2 < 0$
Hence R’’(100) is also less than 0,
Therefore, R(x) is maximum at x = 0, i.e.,
Thus, the required increase on the subscription fee for the company to make profit is by Rs 100.

Question:28

If the straight-line $x \cos α + y \sin α = p$ touches the curve $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ then prove that $a^{2} \cos 2 α + b^{2} \sin 2 α = p^{2} .$

Answer:

Given: equation of straight-line $x \cos α + y \sin α = p$ , equation of curve $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ and the straight line touches the curve

To prove: $a^{2} c o s 2 α + b^{2} s i n 2 α = p^{2} .$

Explanation: the know line equation is,

$x \cos α + y \sin α = p$

$\Rightarrow y s i n α = p - x c o s α$

$\Rightarrow y = \frac{p - x \cos α}{\sin α}$

$\Rightarrow y = \frac{p}{\sin α} - \frac{x \cos α}{\sin α}$

$\Rightarrow y = \frac{p}{\sin α} - x (\frac{\cos α}{\sin α})$

$\Rightarrow y = - x (\frac{\cos α}{\sin α}) + \frac{p}{\sin α}$

Comparing this with the equation $y = mx + c$

the slope and intercept of the given line can be seen as $m = - \frac{\cos α}{\sin α}, C = \frac{p}{\sin α}$

We know that if a line y = mx+c touches the eclipse

$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ , then required condition is

$c^{2} = a^{2} m^{2} + b^{2}$

Then putting the corresponding values, we get

${(\frac{p}{\sin α})}^{2} = a^{2} {(- \frac{\cos α}{\sin α})}^{2} + b^{2}$ $\frac{p^{2}}{\sin^{2} α} = \frac{\cos^{2} α}{\sin^{2} α} (a^{2}) + b^{2}$ $\frac{p^{2}}{\sin^{2} α} = \frac{a^{2} \cos^{2} α + b^{2} \sin^{2} α}{\sin^{2} α}$

Removing the like terms we get,

$p^{2} = b^{2} \sin^{2} α + a^{2} \cos^{2} α$

Hence, proved.

Question:29

An open box with square base is to be made of a given quantity of card board of area c2. Show that the maximum volume of the box is $\frac{c^{3}}{6 \sqrt{3}}$ cubic units.

Answer:

Given: a cardboard box that is open and square in shape has $c^{2}$ area
To show: $\frac{c^{3}}{6 \sqrt{3}}$ cubic units is the maximum volume of the box.
Explanation:

Take the side of the square be x cm and

Take the height the box be y cm.

So, the total area of the cardboard used is

A = area of square base + 4x area of rectangle

$\Rightarrow A = x^{2} + 4 x y$

But it is given this is equal to $c^{2}$ , hence

$c^{2} = x^{2} + 4 x y \Rightarrow 4 x y = c^{2} - x^{2}$

$\Rightarrow y = \frac{c^{2} - x^{2}}{4 x} \dots \dots (i)$

According to the given condition the area of the square base will be

V = base × height

Since the base is square, the volume is

$V = x^{2} y \dots \dots (i i)$

Then putting the values of equation (i) in equation (ii), we get

$V = x^{2} (\frac{c^{2} - x^{2}}{4 x}) V = \frac{x}{4} (c^{2} - x^{2}) V = \frac{1}{4} ({xc}^{2} - x^{3})$

Calculation of the first derivative of the equation,

$V^{'} = \frac{d (\frac{1}{4} ({xc}^{2} - x^{3}))}{dx}$

Removing all the constant terms

$V^{'} = \frac{1}{4} \frac{d ({xc}^{2} - x^{3})}{dx}$

Using the sum rule of differentiation, we get

$V^{'} = \frac{1}{4} [\frac{d ({xc}^{2})}{dx} - \frac{d (x^{3})}{dx}]$

Removing all the constant terms, we get

$V^{'} = \frac{1}{4} [c^{2} \frac{d (x)}{dx} - \frac{d (x^{3})}{d x}]$

After differentiating the equation, we get

$V^{'} = \frac{1}{4} [c^{2} - 3 x^{2}] \dots \dots (i i i)$

We need to calculate the second derivative to find out the maximum value of x, so for that let $V^{'} = 0,$

So, equating the above equation with 0, we get

$\frac{1}{4} [c^{2} - 3 x^{2}] = 0$

$\Rightarrow c^{2} - 3 x^{2} = 0$

$\Rightarrow 3 x^{2} = c^{2}$

$\Rightarrow x^{2} = \frac{c^{2}}{3}$

$\Rightarrow x = \frac{c}{\sqrt{3}}$

Differentiating equation (iii) again with respect to x, we get

$\Rightarrow V^{''} = \frac{d (\frac{1}{4} [c^{2} - 3 x^{2}])}{dx}$

Removing all the constant terms results into

$\Rightarrow V^{''} = \frac{1}{4} \frac{d (c^{2} - 3 x^{2})}{dx}$

Using the differentiation rule of sum, we get

$\Rightarrow V^{''} = \frac{1}{4} [\frac{d (c^{2})}{d x} - \frac{d (3 x^{2})}{dx}]$ $\Rightarrow V^{''} = \frac{1}{4} [0 - 3 (2 x)]$

$\Rightarrow V^{''} = \frac{- 6 x}{4}$

$\Rightarrow V^{''} = \frac{- 3 x}{2}$

$x = \frac{c}{\sqrt{3}}$ , the above equation becomes,

$\Rightarrow {(V^{''})}_{x = \frac{c}{\sqrt{3}}} = \frac{- 3 (\frac{c}{\sqrt{3}})}{2}$

$\Rightarrow {(V^{''})}_{x = \frac{c}{\sqrt{3}}} = \frac{- 3 c}{2 \sqrt{3}} < 0$

Thus, the volume (V) is maximum at $x = \frac{c}{\sqrt{3}}$

∴ The box has a maximum value of

$V_{x = \frac{c}{\sqrt{3}}} = \frac{1}{4} ((\frac{c}{\sqrt{3}}) c^{2} - {(\frac{c}{\sqrt{3}})}^{3}) V_{x = \frac{c}{\sqrt{3}}} = \frac{1}{4} ((\frac{c^{3}}{\sqrt{3}}) - \frac{c^{3}}{3 \sqrt{3}}) V_{x = \frac{c}{\sqrt{3}}} = \frac{1}{4} (\frac{c^{3}}{\sqrt{3}} (1 - \frac{1}{3})) V_{x = \frac{c}{\sqrt{3}}} = \frac{1}{4} (\frac{c^{3}}{\sqrt{3}} (\frac{2}{3})) V_{x = \frac{c}{\sqrt{3}}} = \frac{c^{3}}{6 \sqrt{3}} units$

So, the box has a maximum value of is $\frac{c^{3}}{6 \sqrt{3}}$ cubic units.

Hence, proved.

Question:30

Find the dimensions of the rectangle of perimeter 36 cm which will sweep out a volume as large as possible, when revolved about one of its sides. Also, find the maximum volume.

Answer:

Given: rectangle of perimeter 36cm

To find: to estimate the dimensions of a rectangle in a way that it can sweep out maximum amount of volume when resolved to about one of its sides. Also, to find the maximum volume

Explanation: x and y can be the length and the breadth of the rectangle

The known perimeter of the rectangle is 36cm

$\Rightarrow 2 x + 2 y = 36$

$\Rightarrow x + y = 18$

$\Rightarrow y = 18 - x \dots \dots \dots (i)$

Now when the rectangle revolve about side y it will form a cylinder with y as the height and x as the radius, then if the volume of the cylinder is V, then we know

$V = π x^{2} y$

Applying the value from equation (i) in above equation we get

$V = π x^{2} (18 - x)$

$\Rightarrow V = π (18 x^{2} - x^{3})$

Then find out the first derivative of the given equation,

$V^{'} = \frac{d (π (18 x^{2} - x^{3}))}{dx}$

Taking out the constant terms from equation followed by using the sum rule of differentiation,

$V^{'} = π [\frac{d (18 x^{2})}{d x} - \frac{d (x^{3})}{d x}]$

$V^{'} = π [18 (2 x) - 3 x^{2}]$

$V^{'} = π [36 x - 3 x^{2}] \dots \dots$ (ii)

To calculate critical point, we will equate the first derivative to 0, i.e.,

$V^{'} = 0$

$\Rightarrow π (36 x - 3 x^{2}) = 0$

$\Rightarrow 36 x - 3 x^{2} = 0$

$\Rightarrow 36 x = 3 x^{2}$

$\Rightarrow 3 x = 36$

$\Rightarrow x = 12 \dots \dots . . (i i i)$

By differentiating the second equation, second derivative of the volume equations can be easily calculated,

$V^{''} = \frac{d (π (36 x - 3 x^{2}))}{dx}$

Taking out the constant terms from equation followed by using the sum rule of differentiation,

$V^{''} = π [\frac{d (36 x)}{dx} - \frac{d (3 x^{2})}{dx}]$

$\Rightarrow V^{''} = π [36 - 3 (2 x)]$

$\Rightarrow V^{''} = π [36 - 6 x]$

Now substituting x = 12 (from equation (iii)), we get

$V_{x}^{″} = π [36 - 6 (12)]$

$\Rightarrow V_{x}^{″} = π [36 - 72]$

$\Rightarrow V_{x}^{″} = 12 = - 36 π < 0$

Hence at x = 12, V will have maximum value.

The maximum value of V can be found by substituting x = 12 in

$V = π (18 x^{2} - x^{3}),$ i. e $V_{x} = 12 = π (18 (12)^{2} - (12)^{3})$

$\Rightarrow V_{x} = π (18 (144) - (1728))$

$\Rightarrow V_{x} = π (2592 - (1728))$

$\Rightarrow V_{x} = 864 π c m^{3}$

Therefore, the dimensions of the rectangle which will sweep out a volume as large as possible, when revolved about one of its sides equal to 12cm.

And the maximum volume is $864 π c m^{3}$ .

Question:31

If the sum of the surface areas of cube and a sphere is constant, what is the ratio of an edge of the cube to the diameter of the sphere, when the sums of their volumes are minimum?

Answer:

Given: The combined surface area of a cube and sphere are constant
To find: the ratio of an edge of the cube to the diameter of the sphere, when the sums of their volumes are minimum
Explanation: Let ‘a’ be the side of the cube
Then surface area of the cube = $6 a^{2}$ ….(i)
Take ‘r’ as the radius of the sphere
Then the surface area of the sphere = $4 π r^{2}$ …(ii)
According to the question, the surface area of both the figures is added, thus adding the equation (i) and (ii), we get
$6 a^{2} + 4 π r^{2} = k$ (where k is the constant)

$\Rightarrow 6 a^{2} = k - 4 π r^{2}$
$\Rightarrow a^{2} = \frac{k - 4 π r^{2}}{6}$ $\Rightarrow a = {(\frac{k - 4 π r^{2}}{6})}^{\frac{1}{2}} \dots$ (iii)
As the formula of volume of cube is
$v_{c} = a^{3}$
Plus the volume of a sphere is
$V_{s} = \frac{4}{3} π r^{3}$
Hence adding both the volumes will result into,
$V = a^{3} + \frac{4}{3} π r^{3}$
Then putting the values from equation (iii) in above equation,
$\begin{aligned} V & = {({(\frac{k - 4 π r^{2}}{6})}^{\frac{1}{2}})}^{3} + \frac{4}{3} π r^{3} \\ V & = {(\frac{k - 4 π r^{2}}{6})}^{\frac{3}{2}} + \frac{4}{3} π r^{3} \end{aligned}$
After finding the first derivative of the volume, we get
$V^{'} = \frac{d ({(\frac{k - 4 π r^{2}}{6})}^{\frac{3}{2}} + \frac{4}{3} π r^{3})}{dr}$
After taking out the constant terms along with using the sum rule of differentiating,
$V^{'} = \frac{d ({(\frac{k - 4 π r^{2}}{6})}^{\frac{3}{2}})}{dr} + \frac{4}{3} π \frac{d (r^{3})}{dr}$
Using the power rule of differentiation,
$\begin{aligned} V^{'} & = \frac{3}{2} \cdot {(\frac{k - 4 π r^{2}}{6})}^{\frac{3}{2} - 1} \cdot \frac{d (\frac{k - 4 π r^{2}}{6})}{dr} + \frac{4}{3} π (3 r^{2}) \\ V^{'} & = \frac{3}{2} \cdot {(\frac{k - 4 π r^{2}}{6})}^{\frac{1}{2}} [\frac{1}{6} \cdot \frac{d (k - 4 π r^{2})}{dr}] + (4 π r^{2}) \\ V^{'} & = \frac{3}{2} \cdot {(\frac{k - 4 π r^{2}}{6})}^{\frac{1}{2}} [\frac{1}{6} \cdot (- 4 π (2 r))] + (4 π r^{2}) \\ V^{'} & = \frac{3}{2} \cdot {(\frac{k - 4 π r^{2}}{6})}^{\frac{1}{2}} [- \frac{4}{3} π r] + (4 π r^{2}) \end{aligned}$
$V^{'} = - 2 π r \cdot {(\frac{k - 4 π r^{2}}{6})}^{\frac{1}{2}} + (4 π r^{2})$
$\begin{aligned} V^{'} = - 2 π r [{(\frac{k - 4 π r^{2}}{6})}^{\frac{1}{2}} - 2 r] \dots . (iv) \\ Critical point can be calculated by equating the first derivative with 0, \\ V^{'} = 0 \\ \Rightarrow - 2 π r [{(\frac{k - 4 π r^{2}}{6})}^{\frac{1}{2}} - 2 r] = 0 \\ \Rightarrow - 2 π r = 0 or [{(\frac{k - 4 π r^{2}}{6})}^{\frac{1}{2}} - 2 r] = 0 \\ \Rightarrow r = 0 or {(\frac{k - 4 π r^{2}}{6})}^{\frac{1}{2}} = 2 r \\ \Rightarrow r = 0 or \frac{k - 4 π r^{2}}{6} = 4 r^{2} \end{aligned}$
$\Rightarrow r = 0 o r (k - 4 π r^{2}) = 24 r^{2} \Rightarrow r = 0 o r k = 4 π r^{2} + 24 r^{2} \Rightarrow r = 0 o r (4 π + 24) r^{2} = k$
$\Rightarrow r = 0$ or $r^{2} = \frac{k}{(4 π + 24)}$ $\Rightarrow r = 0$ or $r = \pm \sqrt{\frac{k}{(4 π + 24)}}$
Now we know, $r \neq 0$
Hence
$r = \sqrt{\frac{k}{(4 π + 24)}}$
To find the second derivative of this volume equation, we cam simply differentiate the equation (ii),
$V^{''} = \frac{d (- 2 π r [{(\frac{k - 4 π r^{2}}{6})}^{\frac{1}{2}} - 2 r])}{dx}$
$V^{''} = \frac{d (- 2 π r \cdot {(\frac{k - 4 π r^{2}}{6})}^{\frac{1}{2}} + (4 π r^{2}))}{dx}$
After removing the constant terms, we apply the sum rule of differentiation,
$V^{''} = - 2 π \frac{d (r \cdot {(\frac{k - 4 π r^{2}}{6})}^{\frac{1}{2}})}{dx} + 4 π \frac{d (r^{2})}{dx}$
Using the product rule of differentiation,
$V^{''} = - 2 π [r \frac{d ({(\frac{k - 4 π r^{2}}{6})}^{\frac{1}{2}})}{dx} + {(\frac{k - 4 π r^{2}}{6})}^{\frac{1}{2}} \cdot \frac{dr}{dr}] + 4 π \frac{d (r^{2})}{dx}$
Again, the power rule of differentiation is used,
$V^{''} = - 2 π [r \cdot \frac{1}{2} \cdot {(\frac{k - 4 π r^{2}}{6})}^{\frac{1}{2} - 1} \frac{d (\frac{k - 4 π r^{2}}{6})}{dx} + {(\frac{k - 4 π r^{2}}{6})}^{\frac{1}{2}}] + 4 π (2 r)$
Differentiating the equation, we get
$V " = - 2 π [r . \frac{1}{2} {(\frac{k - 4 π r^{2}}{6})}^{\frac{- 1}{2}} (\frac{1}{6} (- 4 π (2 r))) + {(\frac{k - 4 π r^{2}}{6})}^{\frac{1}{2}}] + 8 π r$
$V^{''} = - 2 π [\frac{- 2 π r^{2}}{3 ({(\frac{k - 4 π r^{2}}{6})}^{\frac{1}{2}})} + {(\frac{k - 4 π r^{2}}{6})}^{\frac{1}{2}}] + 8 π r$
$V^{''} = - 2 π [\frac{- 2 π r^{2} + 3 (\frac{k - 4 π r^{2}}{6})}{3 ({(\frac{k - 4 π r^{2}}{6})}^{\frac{1}{2}})}] + 8 π V^{''} = - 2 π [\frac{- 2 π r^{2} + (\frac{k - 4 π r^{2}}{2})}{3 ({(\frac{k - 4 π r^{2}}{6})}^{\frac{1}{2}})}]] + 8 π r$
$V^{''} = - 2 π [\frac{\frac{k - 4 π r^{2} - 4 π r^{2}}{2}}{3 ({(\frac{k - 4 π r^{2}}{6})}^{\frac{1}{2}})}] + 8 π r$
$V^{''} = - 2 π [\frac{k - 8 π r^{2}}{6 ({(\frac{k - 4 {π r}^{2}}{6})}^{\frac{1}{2}})}] + 8 π r$
$V^{''} = - π [\frac{k - 8 {π r}^{2}}{6 ({(\frac{k - 4 {π r}^{2}}{6})}^{\frac{1}{2}})}] + 8 π r > 0$
$r = \sqrt{\frac{k}{(4 π + 24)}}$
Hence for The substituting $r = \sqrt{\frac{k}{(4 π + 24)}}$ , in equation (iii), we get
$\Rightarrow a = {(\frac{k - 4 π {(\sqrt{\frac{k}{(4 π + 24)}})}^{2}}{6})}^{\frac{1}{2}}$
$\Rightarrow a = {(\frac{k - 4 π (\frac{k}{4 (π + 6)})}{6})}^{\frac{1}{2}} \Rightarrow a = {(\frac{k - (\frac{k π}{(π + 6)})}{6})}^{\frac{1}{2}} \Rightarrow a = {(\frac{k (π + 6) - k π}{6 (π + 6)})}^{\frac{1}{2}} \Rightarrow a = {(\frac{k π + 6 k - k π}{6 (π + 6)})}^{\frac{1}{2}} \Rightarrow a = {(\frac{k}{(π + 6)})}^{\frac{1}{2}}$
Now we will find the ratio of an edge of the cube to the diameter of the sphere, when the sums of their volumes are minimum, i.e.,
a:2r
$\Rightarrow {(\frac{k}{(π + 6)})}^{\frac{1}{2}} : 2 ({(\frac{k}{(4 π + 24)})}^{\frac{1}{2}}) \Rightarrow {(\frac{k}{(π + 6)})}^{\frac{1}{2}} : 2 (\frac{1}{4^{\frac{1}{2}}} {(\frac{k}{(π + 6)})}^{\frac{1}{2}}) \Rightarrow {(\frac{k}{(π + 6)})}^{\frac{1}{2}} : 2 (\frac{1}{2} {(\frac{k}{(π + 6)})}^{\frac{1}{2}})$
$\Rightarrow {(\frac{k}{(π + 6)})}^{\frac{1}{2}} : ({(\frac{k}{(π + 6)})}^{\frac{1}{2}})$
Hence the required ratio is
a:2r = 1:1

Question:32

AB is a diameter of a circle and C is any point on the circle. Show that the area of Δ ABC is maximum when it is isosceles.

Answer:

Given: a circle with AB as diameter and C is any point on the circle
To show: area of Δ ABC is maximum, when it is isosceles
Explanation: If we take r as the radius of the circle, the diameter will become 2r= AB

This indicates that any angle in a semicircle is 90°.
Hence ∠ACB = 90°
Now let AC = x and BC = y
Using the Pythagoras theorem in this right-angled triangle ABC,
$(2 r)^{2} = x^{2} + y^{2}$

$\Rightarrow y^{2} = 4 r^{2} - x^{2}$
$\begin{aligned} \Rightarrow y = \sqrt{4 r^{2} - x^{2}} \dots \dots \\ Now we know area of △ A B C is \\ A = \frac{1}{2} x y \end{aligned}$
Then putting the values from the equation (i), we get
$A = \frac{1}{2} x \sqrt{4 r^{2} - x^{2}} \dots \dots (i i)$
By finding the first derivative of the area,
$A^{'} = \frac{d (\frac{1}{2} x \sqrt{4 r^{2} - x^{2}})}{dx}$
Simultaneously using the product rule of differentiation and also taking out the constant terms,
$A^{'} = \frac{1}{2} [x \frac{d ({(4 r^{2} - x^{2})}^{\frac{1}{2}})}{dx} + {(4 r^{2} - x^{2})}^{\frac{1}{2}} \cdot \frac{dx}{dx}]$
Using the power rule of differentiation,
$A^{'} = \frac{1}{2} [x \cdot \frac{1}{2} \cdot {(4 r^{2} - x^{2})}^{\frac{1}{2} - 1} \frac{d (4 r^{2} - x^{2})}{d x} + {(4 r^{2} - x^{2})}^{\frac{1}{2}}]$

$\Rightarrow A^{'} = \frac{1}{2} [x \cdot \frac{1}{2} \cdot {(4 r^{2} - x^{2})}^{\frac{- 1}{2}} (- 2 x) + {(4 r^{2} - x^{2})}^{\frac{1}{2}}]$

$\Rightarrow A^{'} = \frac{1}{2} [\frac{- x^{2}}{{(4 r^{2} - x^{2})}^{\frac{1}{2}}} + {(4 r^{2} - x^{2})}^{\frac{1}{2}}]$ $\Rightarrow A^{'} = \frac{1}{2} [\frac{- x^{2} + (4 r^{2} - x^{2})}{{(4 r^{2} - x^{2})}^{\frac{1}{2}}}]$

$\Rightarrow A^{'} = [\frac{(2 r^{2} - x^{2})}{\sqrt{4 r^{2} - x^{2}}}] \dots \dots (iii)$
Critical point can be calculated by putting the first derivative equal to 0
$\begin{aligned} A^{'} = 0 \\ \Rightarrow \frac{(2 r^{2} - x^{2})}{\sqrt{4 r^{2} - x^{2}}} = 0 \\ \Rightarrow 2 r^{2} - x^{2} = 0 \\ \Rightarrow 2 r^{2} = x^{2} \\ \Rightarrow r^{2} = \frac{x^{2}}{2} \\ \Rightarrow r = \pm \sqrt{\frac{x^{2}}{2}} \\ \Rightarrow r = \pm \frac{x}{\sqrt{2}} \\ \Rightarrow x = \pm r \sqrt{2} \\ Now we know, r \neq 0 \end{aligned}$
Hence $r = \frac{x}{\sqrt{2}}$ and $x = r \sqrt{2}$
Differentiating the equation (ii) will give us the second derivative of the equation
$A^{''} = \frac{d ([\frac{(2 r^{2} - x^{2})}{\sqrt{4 r^{2} - x^{2}}}])}{dx}$

$\Rightarrow A^{''} = \frac{d ((2 r^{2} - x^{2}) {(4 r^{2} - x^{2})}^{- \frac{1}{2}})}{dx}$
Removing all the constant terms and then using the product rule of differentiation,
$A^{''} = (2 r^{2} - x^{2}) \frac{d ({(4 r^{2} - x^{2})}^{- \frac{1}{2}})}{dx} + {(4 r^{2} - x^{2})}^{- \frac{1}{2}} \frac{d (2 r^{2} - x^{2})}{dx}$
After using the power rule of differentiation,
$A^{''} = (2 r^{2} - x^{2}) \cdot (- \frac{1}{2}) \cdot {(4 r^{2} - x^{2})}^{- \frac{1}{2} - 1} \frac{d (4 r^{2} - x^{2})}{d x} + {(4 r^{2} - x^{2})}^{- \frac{1}{2}} (- 2 x)$

$\Rightarrow A^{''} = (2 r^{2} - x^{2}) \cdot (- \frac{1}{2}) \cdot {(4 r^{2} - x^{2})}^{- \frac{3}{2}} (- 2 x) + {(4 r^{2} - x^{2})}^{- \frac{1}{2}} (- 2 x)$

$\Rightarrow A^{''} = - (2 r^{2} - x^{2}) \cdot {(4 r^{2} - x^{2})}^{- \frac{3}{2}} - 2 x {(4 r^{2} - x^{2})}^{- \frac{1}{2}}$

$\Rightarrow A^{''} = - \frac{(2 r^{2} - x^{2})}{(4 r^{2} - x^{2}) \sqrt{4 r^{2} - x^{2}}} - \frac{2 x}{\sqrt{4 r^{2} - x^{2}}}$

$\Rightarrow A^{''} = \frac{- (2 r^{2} - x^{2}) - 2 x ((4 r^{2} - x^{2}))}{(4 r^{2} - x^{2}) \sqrt{4 r^{2} - x^{2}}}$

$\Rightarrow A^{''} = \frac{- (2 r^{2} - x^{2}) - 8 x r^{2} + 2 x^{3}}{(4 r^{2} - x^{2}) \sqrt{4 r^{2} - x^{2}}}$
For $x = r \sqrt{2}$ in above equation, we get
$A_{x = r \sqrt{2}}^{''} = \frac{- (2 r^{2} - (r \sqrt{2})^{2}) - 8 (r \sqrt{2}) r^{2} + 2 (r \sqrt{2})^{3})}{(4 r^{2} - (r \sqrt{2})^{2}) \sqrt{4 r^{2} - (r \sqrt{2})^{2}}}$

$\Rightarrow A_{x = r \sqrt{2}}^{''} = \frac{- (2 r^{2} - 2 r^{2}) - 8 \sqrt{2} r^{3} + 4 \sqrt{2} r^{3}}{(4 r^{2} - 2 r^{2}) \sqrt{4 r^{2} - 2 r^{2}}}$

$\Rightarrow A_{x = r \sqrt{2}}^{''} = \frac{- (0) - 4 \sqrt{2} r^{3}}{(2 r^{2}) \sqrt{2 r^{2}}}$

$\Rightarrow A_{x = r \sqrt{2}}^{''} = \frac{- 4 \sqrt{2} r^{3}}{(2 r^{3}) \sqrt{2}} < 0$
Thus, for $x = r \sqrt{2}$ , the area of $△ ABC$ is maximum
The maximum value can be calculated by substituting $x = r \sqrt{2}$ in equation (i),
$\Rightarrow y = \sqrt{4 r^{2} - (r \sqrt{2})^{2}}$
$\Rightarrow y = \sqrt{4 r^{2} - 2 r^{2}} \Rightarrow y = \sqrt{2 r^{2}} \Rightarrow y = r \sqrt{2} = x$
i.e., the two sides of the $△ ABC$ are equal
Hence, the area of $△ ABC$ is maximum, when it is isosceles
Hence, proved.

Question:33

A metal box with a square base and vertical sides is to contain $1024 c m^{3}$ . The material for the top and bottom costs Rs $5 / c m^{2}$ and the material for the sides costs Rs $2.50 / c m^{2}$ . Find the least cost of the box.

Answer:

Given: A metal box with a square base and vertical sides is to contain $1024 c m^{3}$ . The material for the top and bottom costs Rs $5 / c m^{2}$ and the material for the sides costs Rs $2.50 / c m^{2}$
To find: the minimum cost of the box

.
Take x cm as the side of the square
Take y cm as the vertical side of the metal box
According to the given information in the question, the formula used volume for square base is
V=base × height

Due to its square base, the formula of the volume is
$V = x^{2} y$
This is equal to $1024 {cm}^{3}$ . So, volume becomes
$1024 {cm}^{3} = x^{2} y$

$\Rightarrow y = \frac{1024}{x^{2}} \dots \dots (i)$
Then we need to calculate the total area of the metal box.
Area of top and bottom $= 2 x^{2} c m^{2}$
The mentioned material for the top and bottom costs Rs $5 / c m^{2}$ , thus, the material cost for top and bottom becomes
Cost of top and bottom =Rs. 5( $2 x^{2}$ )
Area of one side of the metal box = xy cm
The total sides present in the metal box are 4, so
Thus, the total area of all the sides of the metal box = 4xy
The cost of the material for sides is Rs $2.50 / c m^{2}$
∴ Cost of all the sides of the metal box =Rs. 2.50(4xy)
The overall area of the metal box will be
$A = 2 x^{2} + 4 x y$
This will make the cost of the box to be
$C = 5 (2 x^{2}) + 2.50 (4 x y)$
Putting the value of y from equation (i) in the above equation,
$C = 5 (2 x^{2}) + 2.50 (4 x (\frac{1024}{x^{2}}))$

$\Rightarrow C = 10 x^{2} + \frac{10240}{x}$
Both the sides are differentiated with respect to x
$\Rightarrow C^{'} = \frac{d (10 x^{2} + \frac{10240}{x})}{d x}$
Using differentiation rule of sum, we get
$\Rightarrow C^{'} = \frac{d (10 x^{2})}{d x} + \frac{d (\frac{10240}{x})}{d x}$
$\Rightarrow C^{'} = 10 \frac{d (x^{2})}{d x} + 10240 \frac{d (x^{- 1})}{d x}$
Then using the derivative, we get
$\Rightarrow C^{'} = 10 \times 2 x + 10240 \times (- x^{- 2})$

$\Rightarrow C^{'} = 20 x - \frac{10240}{x^{2}} \dots (i i)$
Take c=0 to find the minimum value of x by apply second derivative test, so the above equation is equated with 0
$20 x - \frac{10240}{x^{2}} = 0$

$\Rightarrow 20 x = \frac{10240}{x^{2}}$

$\Rightarrow x^{3} = \frac{10240}{20}$
$\Rightarrow x^{3} = 512$
Solving this we get
$\Rightarrow x = 8$
Again, differentiating equation (ii) with respect to x,
$\Rightarrow C^{''} = \frac{d (20 x - \frac{10240}{x^{2}})}{d x}$
Using the differentiation rule of sum,
$\Rightarrow C^{''} = \frac{d (20 x)}{d x} - \frac{d (\frac{10240}{x^{2}})}{d x}$

$\Rightarrow C^{''} = 20 \frac{d (x)}{dx} - 10240 \frac{d (x^{- 2})}{dx}$

$\Rightarrow C^{''} = 20 + 20480 \times (- x^{- 3})$

$\Rightarrow C^{''} = 20 + \frac{20480}{x^{3}}$
At x=8, the above equation becomes,
$\Rightarrow {(C^{''})}_{x = 8} = 20 + \frac{20480}{(8)^{3}}$

$\Rightarrow {(C^{''})}_{x = 8} = 20 + \frac{20480}{512} > 0$
Now at x=8, $C^{'} (8) = 0$ and $C^{''} (8) > 0$ , so as per the second derivative test, x is a point of local minima and $C (8)$ will be minimum value of C.
Hence least cost becomes
$C_{x = 8} = 10 (8)^{2} + \frac{10240}{8}$

$\Rightarrow C_{y = 8} = 640 + 1280 = R S .1920$
Hence the least cost of the metal box is Rs. 1920

Question:34

The sum of the surface areas of a rectangular parallelepiped with sides x, 2x and $\frac{x}{3}$ and a sphere is given to be constant. Prove that the sum of their volumes is minimum, if x is equal to three times the radius of the sphere. Also find the minimum value of the sum of their volumes.
Answer:

Given: x, 2x and $\frac{x}{3}$ are the sides of a rectangular parallelepiped. The sum of the surface areas of a sphere and a rectangular parallelepiped is given to be constant.
To prove: if x is equal to three times the radius of the sphere, the sum of their volumes is minimum and find the minimum value of the sum of their volumes
Surface area of rectangular parallelepiped:
$S_{rp} = 2 (x (2 x) + 2 x (\frac{x}{3}) + (\frac{x}{3}) x)$

$\Rightarrow S_{rp} = 2 (2 x^{2} + \frac{2 x^{2}}{3} + \frac{x^{2}}{3})$
Let radius of sphere be r cm, then surface area is $S_{s} = 4 π r^{2}$
Now sum of the surface areas is,
$S = S_{rp} + S_{s} = 2 (2 x^{2} + \frac{2 x^{2}}{3} + \frac{x^{2}}{3}) + 4 π r^{2}$
$\Rightarrow S = 2 (\frac{6 x^{2} + 2 x^{2} + x^{2}}{3}) + 4 π r^{2}$
$\Rightarrow S = 6 x^{2} + 4 π r^{2} \dots (i)$
Now given that the sum of the surface areas is constant, so
$\frac{dS}{dr} = 0$
Now, differentiate (i) with respect to r and get
$\frac{d S}{d r} = \frac{d (6 x^{2} + 4 π r^{2})}{d r} = 0$
Apply differentiation rule of sum and get
$\Rightarrow \frac{d (6 x^{2})}{dr} + \frac{d (4 π r^{2})}{dr} = 0$
Take the constant terms out and get
$\Rightarrow 6 \frac{d (x^{2})}{dr} + 4 π \frac{d (r^{2})}{dr} = 0$
Apply derivative and get
$\Rightarrow 6 \times 2 x \frac{d (x)}{dr} + 4 π \times 2 r \frac{d (r)}{dr} = 0$

$\Rightarrow 12 x \frac{dx}{dr} + 8 π r = 0$

$\Rightarrow 12 x \frac{dx}{dr} = - 8 π r$

$\Rightarrow \frac{d x}{d r} = - \frac{8 π r}{12 x}$

$\Rightarrow \frac{d x}{d r} = - \frac{2 π r}{3 x} \dots \dots . . (i i)$
Let V denote the sum of volumes of both the shapes, so
$V = x \times 2 x \times \frac{x}{3} + \frac{4}{3} π r^{3}$
$\Rightarrow V = \frac{2}{3} x^{3} + \frac{4}{3} π r^{3} \dots \dots (i i i)$
The first derivative of volume must be equal to 0 for minima or maxima
$\frac{dV}{dr} = 0$
Differentiate (iii) with respect to r and get
$\frac{d V}{dr} = \frac{d (\frac{2}{3} x^{3} + \frac{4}{3} π r^{3})}{dr} = 0$
Apply differentiation rule of sum and get
$\Rightarrow \frac{d (\frac{2}{3} x^{3})}{dr} + \frac{d (\frac{4}{3} π r^{3})}{dr} = 0$
Take constant terms out and get
$\Rightarrow \frac{2}{3} \frac{d (x^{3})}{dr} + \frac{4}{3} π \frac{d (r^{3})}{dr} = 0$
Apply derivative and get
$\Rightarrow \frac{2}{3} \times 3 x^{2} \times \frac{d (x)}{dr} + \frac{4}{3} π \times 3 r^{2} \frac{d (r)}{dr} = 0$
$\Rightarrow 2 x^{2} \frac{dx}{dr} + 4 π r^{2} = 0$
Substitute value of $\frac{d x}{d r}$ from (ii) and get
$\Rightarrow 2 x^{2} (- \frac{2 π r}{3 x}) + 4 π r^{2} = 0$

$\Rightarrow - \frac{4 π r x}{3} + 4 π r^{2} = 0 \dots . . (i v)$

$\Rightarrow 4 π r^{2} = \frac{4 π r x}{3}$

$\Rightarrow r = \frac{x}{3}$

$\Rightarrow x = 3 r$
i.e., the radius of the sphere is 1/3 of x.
Hence proved
Now let’s find the second derivative value at x=3r.
Now, apply derivative with respect to r to (iv) and get
$\frac{d^{2} V}{{dr}^{2}} = \frac{d (- \frac{4 π r x}{3} + 4 {π r}^{2})}{dr}$
Apply differentiation rule of sum and get
$\Rightarrow \frac{d^{2} V}{{dr}^{2}} = \frac{d (- \frac{4 π rx}{3})}{dr} + \frac{d (4 {π r}^{2})}{dr}$
Take constant terms out and get
$\Rightarrow \frac{d^{2} V}{{dr}^{2}} = - \frac{4 π}{3} \frac{d (rx)}{dr} + 4 π \frac{d (r^{2})}{dr}$
The first part is applied the differentiation rule of product, so
$\Rightarrow \frac{d^{2} V}{{dr}^{2}} = - \frac{4 π}{3} (r \frac{dx}{dr} + x \frac{dr}{dr}) + 4 π \times 2 r \frac{d (r)}{dr}$ $\Rightarrow \frac{d^{2} V}{{dr}^{2}} = - \frac{4 π}{3} (r \frac{d x}{dr} + x) + 4 π \times 2 r$
Substitute value of $\frac{d r}{d x}$ from (ii) and get
$\Rightarrow \frac{d^{2} V}{{dr}^{2}} = - \frac{4 π}{3} (r (- \frac{2 π r}{3 x}) + x) + 4 π \times 2 r$ $\Rightarrow \frac{d^{2} V}{{dr}^{2}} = - \frac{4 π}{3} (- \frac{2 π r^{2}}{3 x} + x) + 8 π r$
Substitutex=3r and get
$\Rightarrow {(\frac{d^{2} V}{{dr}^{2}})}_{x = 3 r} = - \frac{4 π}{3} (- \frac{2 π r^{2}}{3 (3 r)} + 3 r) + 8 π r$ $\Rightarrow {(\frac{d^{2} V}{{dr}^{2}})}_{x = 3 r} = \frac{4 π}{3} (\frac{2 π r}{9} - 3 r) + 8 π r$ $\Rightarrow {(\frac{d^{2} V}{{dr}^{2}})}_{x = 3 r} = \frac{4 π}{3} (\frac{2 π r}{9} - 3 r + 6 r)$
$\Rightarrow {(\frac{d^{2} V}{{dr}^{2}})}_{x = 3 r} = \frac{4 π}{3} (\frac{π r}{3} + 3 r)$
It is positive;so V is minimum when $x = 3 r$ or $r = \frac{x}{3}$ , and the minimum value of Volume can be obtained by substituting $r = \frac{x}{3}$ in equation (iii), we get
$V_{r = \frac{x}{3}} = \frac{2}{3} x^{3} + \frac{4}{3} π r^{3}$

$\Rightarrow V_{r = \frac{x}{3}} = \frac{2}{3} (x)^{3} + \frac{4}{3} π {(\frac{x}{3})}^{3}$

$\Rightarrow V_{r = \frac{x}{3}} = \frac{2}{3} x^{3} + \frac{4}{3} π (\frac{x^{3}}{27})$

$\Rightarrow V_{r = \frac{x}{3}} = \frac{2 x^{3}}{3} [1 + \frac{2 π}{27}]$ $
Therefore, it is the minimum value of the sum of their volumes.

Question:35

The sides of an equilateral triangle are increasing at the rate of 2 cm/sec. The rate at which the area increases, when side is 10 cm is:
A. 10 $c m^{2} / s$
B. $\sqrt{3}$ $c m^{2} / s$
C. $10 \sqrt{3}$ $c m^{2} / s$
D. $\frac{10}{3}$ $c m^{2} / s$

Answer:

Let x cm be the side of the equilateral triangle, then the area of the triangle is
$A = \frac{\sqrt{3}}{4} x^{2}$

$\frac{d x}{d t} = 2 cm / \sec$
Also, the rate of side increasing at instant of time t is
Differentiate area with respect to time t and get
$\frac{d A}{dt} = \frac{d (\frac{\sqrt{3}}{4} x^{2})}{dt}$
Take the constants out and get,
$\frac{d A}{dt} = \frac{\sqrt{3}}{4} \frac{d (x^{2})}{dt}$
Apply the derivative and get
$\frac{d A}{dt} = \frac{\sqrt{3}}{4} \times 2 x \times \frac{dx}{dt}$
Substitute given value of $\frac{dx}{dt}$ and get
$\frac{d A}{dt} = \frac{\sqrt{3}}{4} \times 2 x \times 2$

$\Rightarrow \frac{d A}{dt} = \sqrt{3} x$
Now, put side $x = 10 cm$
$\frac{d A}{{dt}_{x = 10}} = \sqrt{3} \times 10$
Hence, the rate at which the area increases is $10 \sqrt{3} {cm}^{2} / s$ .
So the correct answer is option C.

Question:36

A ladder, 5 meter long, standing on a horizontal floor, leans against a vertical wall. If the top of the ladder slides downwards at the rate of 10 cm/sec, then the rate at which the angle between the floor and the ladder is decreasing when lower end of ladder is 2 metres from the wall is:
A. $\frac{1}{10}$ radian/sec
B. $\frac{1}{20}$ radian/sec
C. 20 radian/sec
D. 10 radian/sec

Answer:

Let 5m/500cm be the length of the ladder, which is the hypotenuse of the right triangle formed in the above figure.
Now let the angle between the ladder and the floor be β, so
$\sin β = \frac{opposite side}{hypotenuse}$

$\Rightarrow \sin β = \frac{x}{500}$
Differentiate both sides with respect to time t and get
$\Rightarrow \frac{d (\sin β)}{dt} = \frac{d (\frac{x}{500})}{dt}$
Apply derivatives and get
$\Rightarrow \cos β \frac{d β}{dt} = \frac{1}{500} \frac{dx}{dt}$
Now, the top of the ladder slides downwards at the rate of
$10 cm / s e c$
$\frac{d x}{d t} = 10 cm / \sec$
So the equation is
$\Rightarrow \cos β \frac{d β}{dt} = \frac{1}{500} \frac{dx}{dt} = \frac{1}{500} \times 10$

$\Rightarrow \cos β \frac{d β}{dt} = \frac{1}{50} \dots \dots (i i)$
Now,
$\cos β = \frac{adacent side}{hypotenuse}$
$\begin{aligned} Substitute (iii) in (ii) and get \\ \Rightarrow \cos β \frac{d β}{dt} = \frac{1}{50} \\ \Rightarrow \frac{y}{500} \times \frac{d β}{d t} = \frac{1}{50} \\ \Rightarrow \frac{d β}{dt} = \frac{1}{50} \times \frac{500}{y} \\ \Rightarrow \frac{d β}{dt} = \frac{10}{y} \\ \begin{matrix} Now when y = 200 cm, we get \end{matrix} \\ \Rightarrow {\frac{d β}{dt}}_{y = 200} = \frac{10}{y} \\ \Rightarrow {\frac{d β}{dt}}_{y = 200} = \frac{10}{200} \\ \Rightarrow {\frac{d β}{dt}}_{y = 200} = \frac{1}{20} \end{aligned}$
Hence the rate at which the angle between the floor and the ladder is decreasing is $\frac{1}{20}$ radian/sec
So the correct answer is option B.

Question:37

The curve $y = x^{\frac{1}{5}}$ has at (0, 0)
A. a vertical tangent (parallel to y-axis)
B. a horizontal tangent (parallel to x-axis)
C. an oblique tangent
D. no tangent

Answer:

Given $y = x^{\frac{1}{5}}$
Differentiate both sides with x and get
$\frac{d y}{d x} = \frac{d (x^{\frac{1}{5}})}{d x}$
Apply power rule and get
$\Rightarrow \frac{d y}{d x} = \frac{1}{5} \times x^{\frac{1}{5} - 1}$

$\Rightarrow \frac{d y}{d x} = \frac{1}{5} \times x^{- \frac{4}{5}}$

$\Rightarrow \frac{d y}{d x} = \frac{1}{5 x^{\frac{4}{5}}}$
Now at (0,0)
$\Rightarrow {(\frac{dy}{dx})}_{(0, 0)} = \frac{1}{5 x^{\frac{4}{5}}}$

$\Rightarrow {(\frac{d y}{d x})}_{(0, 0)} = \frac{1}{5 (0)^{\frac{4}{5}}}$

$\Rightarrow {(\frac{dy}{dx})}_{(0, 0)} = \infty$
So the curve $y = x^{\frac{1}{5}}$ at (0,0) has vertical tangent parallel to Y-axis.
Hence the correct answer is option A.

Question:38

The equation of normal to the curve $3 x^{2} - y^{2} = 8$ which is parallel to the line $x + 3 y = 8$ is
$A . 3 x - y = 8 B . 3 x + y + 8 = 0 C . x + 3 y \pm 8 = 0 D . x + 3 y = 0$

Answer:

Given the equation of the line is $3 x^{2} - y^{2} = 8$
Differentiate both sides with x and get
$\frac{d (3 x^{2} - y^{2})}{dx} = \frac{d (8)}{dx}$
Apply sum rule and 0 is the differentiation of constant, so
$\Rightarrow \frac{d (3 x^{2})}{dx} - \frac{d (y^{2})}{dx} = 0$
Take the constants out and get
$\Rightarrow 3 \frac{d (x^{2})}{dx} - \frac{d (y^{2})}{dx} = 0$
Apply power rule and get
$\Rightarrow 3 \times 2 (x^{2 - 1}) \times \frac{d (x)}{dx} - 2 (y^{2 - 1}) \times \frac{d (y)}{dx} = 0$

$\Rightarrow 6 x - 2 y \times \frac{dy}{dx} = 0$

$\Rightarrow 6 x = 2 y \times \frac{dy}{dx}$

$\Rightarrow \frac{dy}{dx} = \frac{6 x}{2 y} = \frac{3 x}{y}$
Hence, the slope of the given curve is provided.
Also, the slope of the normal to the curve is
$= - \frac{1}{\frac{d y}{d x}}$

$\Rightarrow= - \frac{1}{\frac{3 x}{y}} = (- \frac{y}{3 x}) \dots \dots (i)$
Now, $x + 3 y = 8$
$\Rightarrow 3 y = 8 - x$
After differentiating with respect to x
$\Rightarrow \frac{d (3 y)}{dx} = \frac{d (8 - x)}{dx}$

$\Rightarrow 3 \frac{dy}{dx} = - 1$
$\Rightarrow \frac{dy}{dx} = - \frac{1}{3}$
Therefore, the slope is $- \frac{1}{3}$
Now, because the normal to the curve is parallel to this line, that means the slope of the line must be equal to slope of the normal to the given curve,
$∴ (- \frac{y}{3 x}) = - \frac{1}{3}$
$\Rightarrow 3 y = 3 x \Rightarrow y = x$
Substitute the value of the given equation
$3 x^{2} - y^{2} = 8$

$\Rightarrow 3 x^{2} - (x)^{2} = 8$

$\Rightarrow 2 x^{2} = 8 \Rightarrow x^{2} = 4 \Rightarrow x = \pm 2$
When x=2, the equation is
$3 x^{2} - y^{2} = 8$

$\Rightarrow 3 (2)^{2} - y^{2} = 8$

$\Rightarrow 3 (4) - y^{2} = 8 \Rightarrow 12 - 8 = y^{2} \Rightarrow y^{2} = 4 \Rightarrow y = \pm 2$
When x=-2, the equation is
$3 x^{2} - y^{2} = 8$

$\Rightarrow 3 (- 2)^{2} - y^{2} = 8$

$\Rightarrow 3 (4) - y^{2} = 8 \Rightarrow 12 - 8 = y^{2} \Rightarrow y^{2} = 4 \Rightarrow y = \pm 2$
So, the points are $(\pm 2, \pm 2)$ at which normal is parallel to the given line.
And required equation at $(\pm 2, \pm 2)$ is
$y - (\pm 2) = - \frac{1}{3} [x - (\pm 2)]$

$\Rightarrow 3 (y - (\pm 2)) = - (x - (\pm 2))$

$\Rightarrow 3 y - (\pm 6) = - x + (\pm 2)$

$\Rightarrow x + 3 y - (\pm 6) - (\pm 2) = 0$

$\Rightarrow x + 3 y + (\pm 8) = 0$
Hence the equation of normal to the curve is $x + 3 y + (\pm 8) = 0$
So the correct answer is option C

Question:39

If the curve $a y + x^{2} = 7$ and $x^{3} = y$ , cut orthogonally at (1, 1), then the value of a is:
A. 1
B. 0
C. – 6
D. 6

Answer:

Given the fact that curve $a y + x^{2} = 7$ and $x^{3} = y$ , cut orthogonally at (1, 1)
Differentiate on both sides with x and get
$\frac{d (ay + x^{2})}{dx} = \frac{d (7)}{dx}$
Apply sum rule and also 0 is the derivative of the constant, so
$\Rightarrow \frac{d (ay)}{dx} + \frac{d (x^{2})}{dx} = 0$
Apply power rule and get
$\Rightarrow a \frac{dy}{dx} + 2 x = 0$ $\Rightarrow \frac{d y}{d x} = - \frac{2 x}{a}$
Putting (1,1)
$\Rightarrow {(\frac{d y}{d x})}_{(1, 1)} = - \frac{2 x}{a}$ $\Rightarrow {(\frac{dy}{dx})}_{(1, 1)} = - \frac{2 (1)}{a} = - \frac{2}{a} \dots \dots (i)$

$x^{3} = y$
Differentiate on both sides with x and get
$\frac{d (x^{3})}{dx} = \frac{d (y)}{dx}$
Apply power rule and get
$\Rightarrow 3 x^{2} = \frac{dy}{dx}$
Putting (1,1)
$\Rightarrow {(\frac{dy}{dx})}_{(1, 1)} = 3 x^{2}$

$\Rightarrow {(\frac{dy}{dx})}_{(1, 1)} = 3 (1)^{2} = 3 \dots \dots (i i)$
Both curves cut orthogonally at (1,1)
${(\frac{d y}{d x})}_{(1, 1)} \times {(\frac{d y}{d x})}_{(1, 1)} = - 1$
So from (i) and (ii), we get
$(- \frac{2}{a}) \times 3 = - 1$

$\Rightarrow - \frac{6}{a} = - 1$ $\Rightarrow a = 6$
Hence when the curves cut orthogonally at (1, 1), then the value of a is 6.
So the correct answer is option D.

Question:40

If $y = x^{4} - 10$ and if x changes from 2 to 1.99, what is the change in y
A. 0.32
B. 0.032
C. 5.68
D. 5.968

Answer:

A)
Given $y = x^{4} - 10$
Differentiate on both sides with x and get
$\frac{d (y)}{dx} = \frac{d (x^{4} - 10)}{dx}$

Apply power rule and get
$\Rightarrow \frac{dy}{dx} = 4 x^{3}$
Now, value of x changes from 2 to 1.99, so the change in x is
$Δ x = 2 - 1.99 = 0.01$
So the change in y is,
$\Rightarrow Δ y = \frac{d y}{d x} \times Δ x$
Substitute corresponding values and get
$\Rightarrow Δ y = 4 x^{3} \times (0.01)$
Now at x=2, the change in y becomes
$\Rightarrow Δ y_{x = 2} = 4 (2)^{3} \times (0.01)$

$\Rightarrow Δ y_{x = 2} = 4 \times 8 \times (0.01)$

$\Rightarrow Δ y_{x = 2} = 0.32$
Therefore, change in y is 0.32.

Question:41

The equation of tangent to the curve $y (1 + x^{2}) = 2 - x,$ where it crosses x-axis is:
$A . x + 5 y = 2 B . x - 5 y = 2 C . 5 x - y = 2 D . 5 x + y = 2$

Answer:

A)
Given the equation of the curve is
$y (1 + x^{2}) = 2 - x$
Both the sides are differentiated with respect to x,
$\frac{d (y (1 + x^{2}))}{d x} = \frac{d (2 - x)}{d x}$
Using the power rule
$\Rightarrow y \cdot \frac{d (1 + x^{2})}{d x} + (1 + x^{2}) \cdot \frac{d y}{d x} = \frac{d (2 - x)}{d x}$
As the derivative of a constant is always 0 we get
$\Rightarrow y \cdot \frac{d (x^{2})}{d x} + (1 + x^{2}) \cdot \frac{d y}{d x} = \frac{d (- x)}{d x}$
Again, using the power rule
$\Rightarrow y \cdot 2 x + (1 + x^{2}) \cdot \frac{d y}{d x} = - 1$

$\Rightarrow (1 + x^{2}) \cdot \frac{d y}{d x} = 2 x y - 1$

$\Rightarrow \frac{d y}{d x} = \frac{2 x y - 1}{1 + x^{2}} \dots \dots (i)$

The mentioned curve passes through the x -axis, i.e., y=0
Thus, the curve equation becomes
$y (1 + x^{2}) = 2 - x$

$\Rightarrow 0 (1 + x^{2}) = 2 - x$

$\Rightarrow 0 = 2 - x \Rightarrow x = 2$
As the point of passing for the given curve is (2,0)
So the equation (i) at point (2,0) is,
$\Rightarrow {(\frac{d y}{d x})}_{(2, 0)} = \frac{2 x y - 1}{1 + x^{2}}$

$\Rightarrow {(\frac{d y}{d x})}_{(2, 0)} = \frac{2 (2) (0) - 1}{1 + (2)^{2}}$

$\Rightarrow {(\frac{d y}{d x})}_{(2, 0)} = \frac{0 - 1}{1 + 4}$ $\Rightarrow {(\frac{dy}{dx})}_{(2, 0)} = - \frac{1}{5}$
So, the slope of tangent to the curve is $- \frac{1}{5}$
Therefore, the equation of tangent of the curve passing through (2,0) is given by
$y - 0 = - \frac{1}{5} (x - 2)$
$\Rightarrow 5 y = - x + 2 \Rightarrow x + 5 y = 2$
Thus, the equation of tangent to the curve $y (1 + x^{2}) = 2 - x,$ , where it crosses x-axis is $x + 5 y = 2$ .
Hence, the correct option is option A.

Question:42

The points at which the tangents to the curve $y = x^{3} - 12 x + 18$ are parallel to x-axis are:
A. (2, -2), (-2, -34)
B. (2, 34), (-2, 0)
C. (0, 34), (-2, 0)
D. (2, 2), (-2, 34)

Answer:

D)
Given the equation of the curve is
$y = x^{3} - 12 x + 18$
Differentiating on both sides with respect to x, we get
$\frac{d (y)}{dx} = \frac{d (x^{3} - 12 x + 18)}{dx}$
Applying the sum rule of differentiation, we get
$\Rightarrow \frac{dy}{dx} = \frac{d (x^{3})}{dx} - 12 \frac{d (x)}{dx} + \frac{d (18)}{dx}$
We know derivative of a constant is 0,so above equation becomes
$\Rightarrow \frac{dy}{dx} = \frac{d (x^{3})}{dx} - 12 \frac{d (x)}{dx} + 0$
Applying the power rule we get
$\Rightarrow \frac{dy}{dx} = 3 x^{2} - 12 \dots (i)$
Thus, the slope of line parallel to the x -axis is given by
$\frac{dy}{dx} = 0$
So equating equation (i) to 0 we get
$3 x^{2} - 12 = 0$
$\Rightarrow 3 x^{2} = 12 \Rightarrow x^{2} = 4$

$\Rightarrow x = \pm 2$
When x=2, the given equation of curve becomes,
$y = x^{3} - 12 x + 18$

$\Rightarrow y = (2)^{3} - 12 (2) + 18$

$\Rightarrow y = 8 - 24 + 18 \Rightarrow y = 2$
When x=-2, the given equation of curve becomes,
$y = x^{3} - 12 x + 18$

$\Rightarrow y = (- 2)^{3} - 12 (- 2) + 18$

$\Rightarrow y = - 8 + 24 + 18 \Rightarrow y = 34$
Hence, the points at which the tangents to the curve $y = x^{3} - 12 x + 18$ are parallel to x-axis are (2, 2) and (-2, 34).
So, the correct option is option D.

Question:43

The tangent to the curve $y = e^{2 x}$ at the point (0, 1) meets x-axis at:
A. (0, 1)
B. $(- \frac{1}{2}, 0)$
C. (2, 0)
D. (0, 2)

Answer:

Given the equation of the curve is
$y = e^{2 x}$
Differentiating on both sides with respect to x, we get
$\frac{d (y)}{d x} = \frac{d (e^{2 x})}{d x}$
Applying the exponential rule of differentiation, we get
$\Rightarrow \frac{d y}{d x} = e^{2 x} \frac{d (2 x)}{d x}$

$\Rightarrow \frac{d y}{d x} = 2 e^{2 x} \dots . . (i)$
As it is given the curve has tangent at (0,1), so the curve passes through the point (0,1), so above equation at (0,1), becomes
$\Rightarrow {(\frac{dy}{dx})}_{(0.1)} = 2 e^{2 x}$

$\Rightarrow {(\frac{dy}{dx})}_{(0, 1)} = 2 e^{2 (0)}$

$\Rightarrow {(\frac{dy}{dx})}_{(0, 1)} = 2$
So, the slope of the tangent to the curve at point (0,1) is 2
Hence the equation of the tangent is given by
$y - 1 = 2 (x - 0) \Rightarrow y - 1 = 2 x \Rightarrow y = 2 x + 1$
It is given that the tangent to the curve $y = e^{2 x}$ at the point (0,1) meet x-axis i.e., y=0
So the equation on tangent becomes,
$\Rightarrow y = 2 x + 1 \Rightarrow 0 = 2 x + 1 \Rightarrow 2 x = - 1$
$\Rightarrow x = - \frac{1}{2}$
Hence, the required point is $(- \frac{1}{2}, 0)$
Therefore, the tangent to the curve $y = e^{2 x}$ at the point (0,1) meets x -axis at $(- \frac{1}{2}, 0)$
So, the correct option is option B.

Question:44

The slope of tangent to the curve $x = t^{2} + 3 t - 8, y = 2 t^{2} - 2 t - 5$ at the point (2, -1) is:
A. $\frac{22}{7}$
B. $\frac{6}{7}$
C. $- \frac{6}{7}$
D. $- 6$

Answer:

Curve of the given equation is $x = t^{2} + 3 t - 8, y = 2 t^{2} - 2 t - 5$
With respect to t, while differentiating on both sides, we get
$\frac{d (x)}{d t} = \frac{d (t^{2} + 3 t - 8)}{d t}$
After application of the sum rule of differentiation, we get
$\frac{d (x)}{dt} = \frac{d (t^{2})}{dt} + \frac{d (3 t)}{dt} + \frac{d (- 8)}{dt}$
Constant's derivative is 0, so above equation becomes
$\frac{d (x)}{dt} = \frac{d (t^{2})}{dt} + 3 \frac{d (t)}{dt} + 0$
Power Rule application leads to
$\frac{d (x)}{d t} = 2 t + 3 \dots \dots (i)$

$y = 2 t^{2} - 2 t - 5$
With respect to t, we differentiate on both side and get
$\frac{d (y)}{dt} = \frac{d (2 t^{2} - 2 t - 5)}{dt}$
Sum Rule application leads to
$\frac{d (y)}{dt} = \frac{d (2 t^{2})}{dt} - \frac{d (2 t)}{dt} + \frac{d (- 5)}{dt}$
The Constant's derivative is 0, so the equation becomes
$\frac{d (y)}{dt} = 2 \frac{d (t^{2})}{dt} - 2 \frac{d (t)}{dt} + 0$
Applying power rule
$\frac{d (y)}{d t} = 4 t - 2 \dots (i i)$
We know,
$\frac{d y}{d x} = \frac{\frac{d y}{d t}}{\frac{d x}{d t}}$
Substitute values from equation (i) and (ii)
$\frac{d y}{d x} = \frac{4 t - 2}{2 t + 3}$
The point through which the curve passes is (2,-1), now, substitute the same and get
$x = t^{2} + 3 t - 8$

$\Rightarrow 2 = t^{2} + 3 t - 8$

$\Rightarrow t^{2} + 3 t - 8 - 2 = 0$

$\Rightarrow t^{2} + 3 t - 10 = 0$
Split the middle term
$\Rightarrow t^{2} + 5 t - 2 t - 10 = 0$

$\Rightarrow t (t + 5) - 2 (t + 5) = 0$

$\Rightarrow (t + 5) (t - 2) = 0$

$\Rightarrow t + 5 = 0 o r t - 2 = 0$

$\Rightarrow t = - 5 o r t = 2 \dots \dots \dots . (i i i)$

$y = 2 t^{2} - 2 t - 5$

$\Rightarrow - 1 = 2 t^{2} - 2 t - 5$

$\Rightarrow 2 t^{2} - 2 t - 5 + 1 = 0$

$\Rightarrow 2 t^{2} - 2 t - 4 = 0$
Take 2 as common
$\Rightarrow t^{2} - t - 2 = 0$
Split the middle term again
$\Rightarrow t^{2} - 2 t + t - 2 = 0$

$\Rightarrow t (t - 2) + 1 (t - 2) = 0$

$\Rightarrow (t - 2) (t + 1) = 0$

$\Rightarrow (t - 2) = 0 o r (t + 1) = 0$

$\Rightarrow t = 2 o r t = - 1 \dots \dots . . (i v)$
In equation (iii) and (iv), 2 is common
So, t=2
So, the slope of the tangent at t=2 is as follows
${(\frac{d y}{d x})}_{t = 2} = \frac{4 t - 2}{2 t + 3}$

$\Rightarrow {(\frac{dy}{dx})}_{t = 2} = \frac{4 (2) - 2}{2 (2) + 3}$

$\Rightarrow {(\frac{d y}{d x})}_{t = 2} = \frac{8 - 2}{4 + 3}$

$\Rightarrow {(\frac{d y}{d x})}_{t = 2} = \frac{6}{7}$
Therefore, the slope of tangent at the point (2,-1) is $\frac{6}{7}$
So, the correct answer is option B.

Question:45

The two curves $x^{3} - 3 x y^{2} + 2 = 0$ and $3 x^{2} y - y^{3} - 2 = 0$ intersect at an angle of
A. $\frac{π}{4}$
B. $\frac{π}{3}$
C. $\frac{π}{2}$
D. $\frac{π}{6}$

Answer:

Given the curve $x^{3} - 3 x y^{2} + 2 = 0$ and $3 x^{2} y - y^{3} - 2 = 0$
Differentiate on both the sides with respect to x
$\frac{d (x^{3} - 3 {xy}^{2} + 2)}{dx} = \frac{d (0)}{dx}$
Apply the sum rule and also 0 is the the derivative of the constant, so it becomes
$\Rightarrow \frac{d (x^{3})}{dx} - \frac{d (3 {xy}^{2})}{dx} + \frac{d (2)}{dx} = 0$
Apply power rule and get
$\Rightarrow 3 x^{2} - 3 \frac{d (x y^{2})}{d x} + 0 = 0$
Apply product rule and get
$\begin{aligned} \Rightarrow 3 x^{2} - 3 (x \frac{d (y^{2})}{d x} + y^{2} \frac{d x}{d x}) = 0 \\ \Rightarrow 3 x^{2} - 3 (x .2 y \frac{d (y)}{d x} + y^{2}) = 0 \\ \Rightarrow 3 x^{2} - 6 x y \frac{d y}{d x} - 3 y^{2} = 0 \\ \Rightarrow 3 x^{2} - 3 y^{2} = 6 x y \frac{d y}{d x} \\ \Rightarrow \frac{d y}{d x} = \frac{3 x^{2} - 3 y^{2}}{6 x y} = \frac{3 (x^{2} - y^{2})}{6 x y} \\ \Rightarrow \frac{dy}{dx} = \frac{x^{2} - y^{2}}{2 xy} \\ Let it be equal to m_{d_{k}} so \\ \Rightarrow m_{1} = \frac{x^{2} - y^{2}}{2 xy} . . (i) \\ 3 x^{2} y - y^{3} - 2 = 0 \end{aligned}$
Differentiate on both the sides with respect to x and get
$\frac{d (3 x^{2} y - y^{3} - 2)}{d x} = \frac{d (0)}{d x}$
Apply the sum rule and also 0 is the derivative of the constant, so
$\Rightarrow \frac{d (3 x^{2} y)}{dx} - \frac{d (y^{3})}{dx} - \frac{d (2)}{dx} = 0$
Apply power rule and get
$\Rightarrow 3 \frac{d (x^{2} y)}{d x} - 3 y^{2} \frac{d (y)}{d x} + 0 = 0$
Apply product rule and get
$\Rightarrow 3 (x^{2} \frac{d (y)}{d x} + y \frac{d (x^{2})}{d x}) - 3 y^{2} \frac{d (y)}{d x} = 0$

$\Rightarrow 3 x^{2} \frac{d y}{d x} + 3 y (2 x) - 3 y^{2} \frac{d y}{d x} = 0$
$\begin{aligned} \Rightarrow \frac{dy}{dx} (3 x^{2} - 3 y^{2}) + 6 xy = 0 \\ \Rightarrow \frac{d y}{d x} (3 x^{2} - 3 y^{2}) = - 6 x y \\ \Rightarrow \frac{d y}{d x} = - \frac{6 x y}{(3 x^{2} - 3 y^{2})} \\ Let it be equal to m_{2} \\ \Rightarrow m_{2} = - \frac{6 xy}{(3 x^{2} - 3 y^{2})} \dots \dots . \\ The multiply equation (i) and (ii) and get \\ m_{1} \cdot m_{2} = (\frac{x^{2} - y^{2}}{2 xy}) \cdot (- \frac{6 xy}{(3 x^{2} - 3 y^{2})}) \\ \Rightarrow m_{1} \cdot m_{2} = (- \frac{3 (x^{2} - y^{2})}{(3 x^{2} - 3 y^{2})}) \\ \Rightarrow m_{1} \cdot m_{2} = - 1 \end{aligned}$
Since the product of the slopes is -1, it means that both the curves are intersecting at right angle i.., they are making $\frac{π}{2}$ angle with each other.
So, the correct answer is option C.

Question:46

The interval on which the function $f (x) = 2 x^{3} + 9 x^{2} + 12 x - 1$ is decreasing is:
A. [-1, ∞)
B. [-2, -1]
C. (-∞, -2]
D. [-1, 1]

Answer:

Given $f (x) = 2 x^{3} + 9 x^{2} + 12 x - 1$
Apply first derivative and get
$\begin{aligned} f (x) = \frac{d (2 x^{3} + 9 x^{2} + 12 x - 1)}{d x} \\ Apply the sum rule of the differentiation and 0 is the derivative of the constant, so \\ \Rightarrow f (x) = \frac{d (2 x^{3})}{d x} + \frac{d (9 x^{2})}{d x} + \frac{d (12 x)}{d x} - \frac{d (1)}{d x} \end{aligned}$
Apply power rule and get
$\Rightarrow f^{'} (x) = 6 x^{2} + 18 x + 12 - 0$

$\Rightarrow f^{'} (x) = 6 (x^{2} + 3 x + 2)$
Split the middle term and get
$\Rightarrow f^{'} (x) = 6 (x^{2} + 2 x + x + 2)$

$\Rightarrow f^{'} (x) = 6 (x (x + 2) + 1 (x + 2))$

$\Rightarrow f^{'} (x) = 6 ((x + 2) (x + 1))$
Now $f^{'} (x) = 0$ gives
$x = - 1, - 2$
Three intervals are made when these points divide the real number line
$(- \infty, - 2), [- 2, - 1] a n d (- 1, \infty)$

(i) in the interval $(- \infty , -2), f'(x)>0

$∴ f (x)$ is increasing in $(- \infty, - 2)$

(ii) in the interval $[- 2, - 1], f^{'} (x) \leq 0$

$∴ f (x)$ is decreasing in $[- 2, - 1]$

(iii) in the interval $(- 1, \infty), f^{'} (x) > 0$

$∴ f (x)$ is increasing in $(- 1, \infty)$
So, the interval on which the function decreases is [-2, -1].
So, the correct answer is option B.

Question:47

Let the $f : R \to R$ be defined by $f (x) = 2 x + c o s x$ , then f(x) :
A. has a minimum at x = π
B. has a maximum, at x = 0
C. is a decreasing function
D. is an increasing function

Answer:

Given $f (x) = 2 x + c o s x$ if $f : R \to R$
Apply the first derivative and get
$\begin{aligned} f (x) = \frac{d (2 x + \cos x)}{d x} \\ Apply the sum rule and 0 is the derivative of the constant, so \\ \Rightarrow f (x) = \frac{d (2 x)}{d x} + \frac{d (\cos x)}{d x} \end{aligned}$
Apply power rule and get
$\Rightarrow f^{'} (x) = 2 - s i n x$
Now, 1 is the maximum value of sin x.
$S o f^{'} (x) > 0, \forall x$
So, function f is an increasing function.
So the correct answer is option D.

Question:48

y = $x (x - 3)^{2}$ decreases for the values of x given by :
A. $1 < x < 3$
B. $x < 0$
C. $x > 0$
D. $0 < x < \frac{3}{2}$

Answer:

Given $y = x (x - 3)^{2}$
$\Rightarrow y = x (x^{2} - 6 x + 9)$

$\Rightarrow y = x^{3} - 6 x^{2} + 9 x$
Apply first derivative and get
$\frac{d y}{d x} = \frac{d (x^{3} - 6 x^{2} + 9 x)}{d x}$
Apply sum rule of differentiation and get
$\Rightarrow \frac{dy}{dx} = \frac{d (x^{3})}{dx} - \frac{d (6 x^{2})}{dx} + \frac{d (9 x)}{dx}$
Apply power rule and get
$\Rightarrow \frac{dy}{dx} = 3 x^{2} - 12 x + 9$

$\Rightarrow \frac{d y}{d x} = 3 (x^{2} - 4 x + 3)$
Now, split middle term and get
$\Rightarrow \frac{dy}{dx} = 3 (x^{2} - 3 x - x + 3)$

$\Rightarrow \frac{d y}{d x} = 3 (x (x - 3) - 1 (x - 3))$

$\Rightarrow \frac{dy}{dx} = 3 ((x - 3) (x - 1))$
Now, $\frac{d y}{dx} = 0$ gives us
$x = 1, 3$
The points divide this real number line into three intervals
$(- \infty, 1), (1, 3) a n d (3, \infty) (i) i n t h e i n t e r v a l (- \infty, 1), f^{'} (x) > 0$

$∴ f(x) is increasing in (- \infty, 1) . . . (i i)$

$in the interval (1, 3), f^{'} (x) \leq 0$

$∴ f (x) is decreasing in (1, 3)$

$(i i i) in the interval (3, \infty), f^{'} (x) > 0 ∴ f(x) is increasing in (3, \infty)$
So, the interval on which the function decreases is $(1, 3)$ i.e., $1 < x < 3$
So the correct answer is option A.

Question:49

The function $f (x) = 4 s i n^{3} x - 6 s i n^{2} x + 12 s i n x + 100$ is strictly
A. increasing in $(π, \frac{3 π}{2})$
B. decreasing in $(\frac{π}{2}, π)$
C. decreasing in $(- \frac{π}{2}, \frac{π}{2})$
D. decreasing in $(0, \frac{π}{2})$

Answer:

Given $f (x) = 4 \sin^{3} x - 6 \sin^{2} x + 12 \sin x + 100$
Apply the first derivative and get
$f^{'} (x) = \frac{d (4 \sin^{3} x - 6 \sin^{2} x + 12 \sin x + 100)}{dx}$
Apply sum rule and get
$\Rightarrow f^{'} (x) = \frac{d (4 \sin^{3} x)}{dx} - \frac{d (6 \sin^{2} x)}{dx} + \frac{d (12 \sin x)}{dx} + 0$
Then apply power rule and get
$\Rightarrow f^{'} (x) = 12 \sin^{2} x \frac{d (\sin x)}{dx} - 12 \sin x \frac{d (\sin x)}{dx} + 12 \frac{d (\sin x)}{dx}$
Now apply the derivative,
$\Rightarrow f^{'} (x) = 12 \sin^{2} x (\cos x) - 12 \sin x (\cos x) + 12 (\cos x)$

$\Rightarrow f^{'} (x) = 12 \sin^{2} x \cos x - 12 \sin x \cos x + 12 \cos x$

$\Rightarrow f^{'} (x) = 12 \cos x (\sin^{2} x - \sin x + 1)$
Now, $1 - \sin x \geq 0$ and $\sin^{2} x \geq 0$
Hence $\sin^{2} x - s i n x + 1 \geq 0$
Therefore, $f^{'} (x) > 0,$ when $\cos x > 0,$ i.e $_{x}$ X $\in (\frac{- π}{2}, \frac{π}{2})$
Hence f(x) is increasing when $x \in (\frac{- π}{2}, \frac{π}{2})$
and $f^{'} (x) < 0$ , when $\cos x < 0,$ i.e. $x \in (\frac{π}{2}, \frac{3 π}{2})$
Hence f(x) is decreasing when $X \in (\frac{π}{2}, \frac{3 π}{2})$
Now $(\frac{π}{2}, π) \in (\frac{π}{2}, \frac{3 π}{2})$
Hence, f(x) is decreasing in $(\frac{π}{2}, π)$
So the correct answer is option B.

Question:50

Which of the following functions is decreasing on $(0, \frac{π}{2})$ .
A. sin2x
B. tan x
C. cos x
D. cos 3x

Answer:

(i) Let f(x)=sin 2x
Apply first derivative and get
f’(x)=2cos 2x
Put f’(x)=0, and get
2cos 2x =0
⇒ cos 2x=0
It is possible when
0≤x≤2π
Thus, sin 2x does not decrease or increase on $x \in (0, \frac{π}{2})$
(ii) Let f(x)=tan x
Apply first derivative and get
f’(x)= $s e c^{2} x$
Now. square of every number is always positive,
So, tan x is increasing function in $x \in (0, \frac{π}{2})$
(iii) Let f(x)=cos x
Apply first derivative and get
f’(x)=-sin x
But, sin x>0 for $x \in (0, \frac{π}{2})$
And -sin x<0 for $x \in (0, \frac{π}{2})$
Hence f’(x)<0 for $x \in (0, \frac{π}{2})$
⇒ cos x is strictly decreasing on $x \in (0, \frac{π}{2})$
(iv) Let f(x)=cos 3x
Apply first derivative and get
f’(x)=-3sin 3x
Put f’(x)=0, we get
-3sin 3x=0
⇒ sin 3x=0
Because sin θ=0 if θ=0, π, 2π, 3π
⇒ 3x=0,π, 2π, 3π
$\begin{aligned} \Rightarrow x = 0, \frac{π}{3}, \frac{2 π}{3}, π \\ x \in (0, \frac{π}{2}) \\ \Rightarrow x = \frac{π}{3} \\ since \\ x \in (0, \frac{π}{2}) \end{aligned}$
so we write it on number line as

Now, this point $x = \frac{π}{3}$ divides the interval $(0, \frac{π}{2})$ into 2 disjoint intervals.
i.e. $(0, \frac{π}{3})$ and $(\frac{π}{3}, \frac{π}{2})$
case 1 : for $x \in (0, \frac{π}{3})$
$0 < x < \frac{π}{3}$

$\Rightarrow 3 \times 0 < 3 x < \frac{π}{3} \times 3$

$0 < 3 x < π$
So where $x \in (0, \frac{π}{3}), 3 x \in (0, π) \dots . . (a)$
Also,
$\sin θ > 0$ for $θ \in (0, π)$

$\sin 3 x > 0$ for $3 x \in (0, π)$
From equation (a), we get
$\sin 3 x > 0$ for $x \in (0, \frac{π}{3})$
$\sin 3 x < 0$ for $x \in (0, \frac{π}{3})$
$\Rightarrow f^{'} (x) < 0$ for $x \in (0, \frac{π}{3})$

$\Rightarrow f (x)$ is strictly decreasing $(0, \frac{π}{3})$
Case 2: for $x \in (\frac{π}{3}, \frac{π}{2})$
Now $\frac{π}{3} < x < \frac{π}{2}$
$\Rightarrow 3 \times \frac{π}{3} < 3 x < \frac{π}{2} \times 3$

$\Rightarrow π < 3 x < \frac{3 π}{2}$ $x \in (\frac{π}{3}, \frac{π}{3}), 3 x \in (π, \frac{3 π}{2}) \dots (b)$
Also,
$\sin θ < 0$ in $3^{rd}$ quadrant $\sin θ < 0$ for $θ \in (0, π)$ $\sin θ < 0$ for $θ \in (π, \frac{3 π}{2})$

$\sin 3 x < 0$ for $3 x \in (π, \frac{3 π}{2})$
Equation (b) gives
$\sin 3 x < 0$ for $x \in (\frac{π}{3}, \frac{3 π}{2 \times 3})$ $- \sin 3 x > 0$ for $x \in (\frac{π}{3}, \frac{π}{2})$

$\Rightarrow f^{'} (x) < 0$ for $x \in (\frac{π}{3}, \frac{π}{2})$ $\Rightarrow f (x)$ is strictly increasing on $(\frac{π}{3}, \frac{π}{2})$
Hence, cos 3 x does not decrease or increase on $x \in (0, \frac{π}{2})$
So, the correct answer is option C i.e., $\cos x$ is decreasing in $(0, \frac{π}{2})$

Question:51

The function $f (x) = t a n x - x$
A. always increases
B. always decreases
C. never increases
D. sometimes increases and sometimes decreases.

Answer:

Given $f (x) = \tan x - x$
Apply first derivative and get
$f^{'} (x) = \frac{d (\tan x - x)}{dx}$
Apply sum rule and get
$\Rightarrow f^{'} (x) = \frac{d (\tan x)}{d x} - \frac{d (x)}{d x}$
Apply derivative,
$\Rightarrow f^{'} (x) = \sec^{2} x - 1$
Square of every number is always positive,
So $f^{'} (x) > 0 \forall x \in R$
So $f (x) = t a n x - x$ always increases.
So the correct answer is option A.

Question:52

If x is real, the minimum value of $x^{2} - 8 x + 17$ is
A. -1
B. 0
C. 1
D. 2

Answer:

Let $f (x) = x^{2} - 8 x + 17$
Apply first derivative and get
$\begin{aligned} f^{'} (x) = \frac{d (x^{2} - 8 x + 17)}{dx} \\ Apply sum rule and get \\ \Rightarrow f^{'} (x) = \frac{d (x^{2})}{d x} - 8 \frac{d (x)}{d x} + 0 \end{aligned}$
Apply derivative,
$\Rightarrow f^{'} (x) = 2 x - 8$

Put $f^{'} (x) = 0$ and get

$2 x - 8 = 0 \Rightarrow 2 x = 8 \Rightarrow x = 4$
Hence the minimum value of f(x) at x=4 is given by
$f (x) = x^{2} - 8 x + 17$

$\Rightarrow f (4) = 4^{2} - 8 (4) + 17$

$\Rightarrow f (4) = 16 - 32 + 17 \Rightarrow f (4) = 1$
So, if x is real, 1 is the minimum value of $x^{2} - 8 x + 17$
So the correct answer is option C.

Question:53

The smallest value of the polynomial $x^{3} - 18 x^{2} + 96 x$ in [0, 9] is
A. 126
B. 0
C. 135
D. 160

Answer:

Let $f (x) = x^{3} - 18 x^{2} + 96 x$
Apply first derivative and get
$\begin{aligned} f^{'} (x) = \frac{d (x^{3} - 18 x^{2} + 96 x)}{d x} \\ Apply sum rule and get \\ \Rightarrow f^{'} (x) = \frac{d (x^{3})}{d x} - 18 \frac{d (x^{2})}{d x} + 96 \frac{d (x)}{d x} \end{aligned}$
Apply derivative,
$\Rightarrow f^{'} (x) = 3 x^{2} - 36 x + 96$

Put $f^{'} (x) = 0$ , and get critical points

$3 x^{2} - 36 x + 96 = 0$

$\Rightarrow 3 (x^{2} - 12 x + 32) = 0$

$\Rightarrow x^{2} - 12 x + 32 = 0$
Split the middle term and get
$\Rightarrow x^{2} - 8 x - 4 x + 32 = 0$

$\Rightarrow x (x - 8) - 4 (x - 8) = 0$

$\Rightarrow (x - 8) (x - 4) = 0$

$\Rightarrow x - 8 = 0 o r x - 4 = 0$

$\Rightarrow x = 8 o r x = 4$

$\Rightarrow x \in [0, 9]$
Now we find the values of $f (x)$ at $x = 0, 4, 8, 9$
$f (x) = x^{3} - 18 x^{2} + 96 x$

$f (0) = 0^{3} - 18 (0)^{2} + 96 (0) = 0$

$f (4) = 4^{3} - 18 (4)^{2} + 96 (4) = 64 - 288 + 384 = 160$

$f (8) = 8^{3} - 18 (8)^{2} + 96 (8) = 512 - 1152 + 768 = 128$

$f (9) = 9^{3} - 18 (9)^{2} + 96 (9) = 729 - 1458 + 864 = 135$
Hence we find that 0 is the absolute minimum value of f(x) in [0,9] at x=0.
So the correct answer is option B.

Question:54

The function $f (x) = 2 x^{3} - 3 x^{2} - 12 x + 4$ , has
A. two points of local maximum
B. two points of local minimum
C. one maxima and one minima
D. no maxima or minima
Answer:

Let $f (x) = 2 x^{3} - 3 x^{2} - 12 x + 4$
Apply first derivative and get
$\begin{aligned} f^{'} (x) = \frac{d (2 x^{3} - 3 x^{2} - 12 x + 4)}{dx} \\ Apply sum rule and get \\ \Rightarrow f^{'} (x) = \frac{d (2 x^{3})}{dx} - 3 \frac{d (x^{2})}{d x} - 12 \frac{d (x)}{dx} + 0 \end{aligned}$
Apply derivative,
$\Rightarrow f^{'} (x) = 6 x^{2} - 6 x - 12$
Put $f^{'} (x) = 0$ , and get
$6 x^{2} - 6 x - 12 = 0$

$\Rightarrow 6 (x^{2} - x - 2) = 80$

$\Rightarrow x^{2} - x - 2 = 0$
Split middle term and get
$\Rightarrow x^{2} - 2 x + x - 2 = 0$

$\Rightarrow x (x - 2) + 1 (x - 2) = 0$

$\Rightarrow (x - 2) (x + 1) = 0$

$\Rightarrow x - 2 = 0 o r x + 1 = 0 \Rightarrow x = 2 o r x = - 1$
Now we find the values of $f (x)$ at $x = - 1, 2$
$f (x) = 2 x^{3} - 3 x^{2} - 12 x + 4$

$f (- 1) = 2 (- 1)^{3} - 3 (- 1)^{2} - 12 (- 1) + 4 = - 2 - 3 + 12 + 4 = 11$

$f (2) = 2 (2)^{3} - 3 (2)^{2} - 12 (2) + 4 = 16 - 12 - 24 + 4 = - 16$
Hence from above we find that the point of local maxima is x=-1 and 11 is the maximum value of f(x).
Whereas the point of local minima is x=2 and -16 is the minimum value of f(x).
So, the correct answer is option C.
Hence, the given function has 1 minima and 1 maxima.

Question:55

The maximum value of sin x cos x is
A. $\frac{1}{4}$
B. $\frac{1}{2}$
C. $\sqrt{2}$
D. $2 \sqrt{2}$

Answer:

Let $f (x) = \sin x \cos x$
$\sin 2 x = 2 \sin x \cos x$
$\Rightarrow f (x) = \frac{1}{2} \sin 2 x$
Apply first derivative and get
$f^{'} (x) = \frac{d (\frac{1}{2} \sin 2 x)}{dx}$

$\Rightarrow f^{'} (x) = \frac{1}{2} \frac{d (\sin 2 x)}{d x}$
Apply derivative,
$\Rightarrow f^{'} (x) = \frac{1}{2} \cdot \cos 2 x \cdot \frac{d (2 x)}{d x}$
$\Rightarrow f^{'} (x) = \frac{1}{2} \cdot \cos 2 x \cdot 2$

$\Rightarrow f (x) = \cos 2 x \dots \dots (i)$
Put $f^{'} (x) = 0$ and get
$\cos 2 x = 0$

$\Rightarrow \cos 2 x = \cos \frac{π}{2}$
Equate the angles and get
$\Rightarrow 2 x = \frac{π}{2}$ $\Rightarrow x = \frac{π}{4}$
Now we find second derivative by deriving equation (i) and get
$f^{''} (x) = \frac{d (\cos 2 x)}{dx}$
Apply derivative,
$\Rightarrow f^{''} (x) = - \sin 2 x \cdot \frac{d (2 x)}{dx}$

$\Rightarrow f^{'} (x) = - \sin 2 x .2$

$\Rightarrow f^{'} (x) = - 2 \sin 2 x$
Now we find the value of $f^{''} (x)$ at $x = \frac{π}{4},$ we get
$\Rightarrow {(f^{'} (x))}_{x = \frac{π}{4}} = - 2 \sin 2 x$ $\Rightarrow {(f^{'} (x))}_{x = \frac{π}{4}} = - 2 \sin 2 (\frac{π}{4})$

$\Rightarrow {(f^{'} (x))}_{x = \frac{π}{4}} = - 2 \sin (\frac{π}{2})$
But $\sin (\frac{π}{2}) = 1$ so above equation becomes
$\Rightarrow {(f^{'} (x))}_{x = \frac{π}{4}} = - 2 \times 1 = - 2 < 0$
Hence at $x = \frac{π}{4}$ , $f (x)$ is maximum and $\frac{π}{4}$ is the point of maxima.
Now we will find the maximum value of $\sin x \cos x$ by substituting $x = \frac{π}{4},$ in $f (x),$ we get
$f (x) = \sin x \cos x$

$f (\frac{π}{4}) = \sin (\frac{π}{4}) \cos (\frac{π}{4})$

$\Rightarrow f (\frac{π}{4}) = \frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}}$

$\Rightarrow f (\frac{π}{4}) = \frac{1}{2}$
So, maximum value of $\sin x \cos x$ is $\frac{1}{2}$
So, the correct answer is option B.

Question:56

At $x = \frac{5 π}{6}$ , $f (x) = 2 s i n 3 x + 3 c o s 3 x$ is:
A. maximum
B. minimum
C. zero
D. neither maximum nor minimum

Answer:

Given $f (x) = 2 \sin 3 x + 3 \cos 3 x$
Apply first derivative and get
$f^{'} (x) = \frac{d (2 \sin 3 x + 3 \cos 3 x)}{dx}$
Apply sum rule and take the constant terms out and get
$\Rightarrow f^{'} (x) = 2 \frac{d (\sin 3 x)}{d x} + 3 \frac{d (\cos 3 x)}{d x}$
Apply derivative,
$\Rightarrow f^{'} (x) = 2 . \cos 3 x \cdot \frac{d (3 x)}{d x} + 3 \cdot (- \sin 3 x) \cdot \frac{d (3 x)}{d x}$
$\begin{aligned} \Rightarrow f^{'} (x) = 2 \cdot \cos 3 x \cdot 3 - 3 \cdot \sin 3 x \cdot 3 \\ \Rightarrow f^{'} (x) = 6 \cos 3 x - 9 \sin 3 x \dots \dots (i) \\ x = \frac{5 π}{6} \\ Now we find the value of f^{'} (x) at \frac{5 π}{6} and get \\ \Rightarrow (f^{'} (x))_{x = \frac{5 π}{6}} = 6 \cos 3 x - 9 \sin 3 x \\ \Rightarrow (f^{'} (x))_{x = \frac{5 π}{6}} = 6 \cos (\frac{5 π}{2}) - 9 \sin (\frac{5 π}{2}) \\ Splitting \frac{5 π}{2} as 2 π + \frac{π}{2} \\ \Rightarrow (f^{'} (x))_{x = \frac{5 π}{6}} = 6 \cos (2 π + \frac{π}{2}) - 9 \sin (2 π + \frac{π}{2}) \\ Also, \cos (2 π + θ) = \cos θ and \sin (2 π + θ) = \sin θ \\ \Rightarrow (f^{'} (x))_{x = \frac{5 π}{6}} = 6 \cos (\frac{π}{2}) - 9 \sin (\frac{π}{2}) \end{aligned}$
$\cos (\frac{π}{2}) = 0 \sin (\frac{π}{2}) = 1 \Rightarrow (f^{'} (x))_{x = \frac{5 π}{6}} = 6 (0) - 9 (1) = - 9$
And we found f’(x) at $x = \frac{5 π}{6}$ not equal to 0.
So $x = \frac{5 π}{6}$ cannot be point of minima or maxima.
Hence, $f (x) = 2 \sin 3 x + 3 \cos 3 x$ at $x = \frac{5 π}{6}$ is not minima nor maxima.
So, the correct answer is option D.

Question:57

Maximum slope of the curve $y = - x^{3} + 3 x^{2} + 9 x - 27$ is:
A. 0
B. 12
C. 16
D. 32

Answer:

Given equation of curve is $y = - x^{3} + 3 x^{2} + 9 x - 27$
Apply first derivative and get
$\frac{d y}{d x} = \frac{d (- x^{3} + 3 x^{2} + 9 x - 27)}{d x}$
Apply sum rule and $0$ is the differentiation of the constant term, so
$\frac{d y}{d x} = - \frac{d (x^{3})}{d x} + 3 \frac{d (x^{2})}{d x} + 9 \frac{d (x)}{d x} - 0$
Apply power rule and get
$\frac{d y}{d x} = - 3 x^{2} + 3 \cdot (2 x) + 9.1$

$\Rightarrow \frac{dy}{dx} = - 3 x^{2} + 6 x + 9 \dots \dots \dots (i)$
Hence, it is the slope of the curve.
Now to find out the second derivative of the given curve, we will differentiate equation (i) once again
$\frac{d^{2} y}{{dx}^{2}} = \frac{d (- 3 x^{2} + 6 x + 9)}{dx}$
Apply sum rule and 0 is the differentiation of the constant term so
$\frac{d^{2} y}{d x^{2}} = - 3 \frac{d (x^{2})}{d x} + 6 \frac{d (x)}{d x} + 0$
Apply power rule and get
$\frac{d^{2} y}{d x^{2}} = - 3 (2 x) + 6.1$

$\Rightarrow \frac{d^{2} y}{d x^{2}} = - 6 x + 6 = - 6 (x - 1) \dots (i i)$
Now we will find the critical point by equating the second derivative to 0, we get
-6(x-1) =0
⇒ x-1=0
⇒ x=1
Now, to find out the third derivative of the given curve, we will differentiate equation (ii) once again
$\frac{d^{3} y}{d x^{3}} = \frac{d (- 6 x + 6)}{d x}$
Apply sum rule and 0 is the differentiation of the constant term, so
$\frac{d^{3} y}{d x^{3}} = - 6 \frac{d (x)}{d x} + 0$
Apply power rule and get
$\frac{d^{3} y}{d x^{3}} = - 6 < 0$
Hence, maximum slope is at $x = 1$
Now, substitute $x = 1$ in (i), and get
${(\frac{d y}{d x})}_{x = 1} = - 3 x^{2} + 6 x + 9$ ${(\frac{d y}{d x})}_{x = 1} = - 3 (1)^{2} + 6 (1) + 9 = - 3 + 6 + 9 = 12$
Therefore, 12 is the maximum slope of the curve $y = - x^{3} + 3 x^{2} + 9 x - 27$ .
So, the correct answer is option B.

Question:58

$f (x) = x^{x}$ has a stationary point at
A. x = e
B. $x = \frac{1}{e}$
C. x = 1
D. $x = \sqrt{e}$

Answer:

Given equation is $f (x) = x^{x}$
Let $y = x^{x}$ ………(i)
Take logarithm on both side
$\log y = \log (x^{x})$
⇒ log y=x log x
Apply first derivative and get
$\frac{d (\log y)}{d x} = \frac{d (x \log x)}{d x}$
Apply product rule and get
$\frac{d (\log y)}{dx} = x \frac{d (\log x)}{d x} + \log x \cdot \frac{d (x)}{d x}$
Apply first derivative and get
$\frac{1}{y} \cdot \frac{d y}{d x} = x \cdot \frac{1}{x} + \log x \cdot 1$

$\Rightarrow \frac{d y}{d x} = y (1 + \log x)$
Substitute value of y from (i) and get
$\Rightarrow \frac{d y}{d x} = x^{x} (1 + \log x)$
Now we find the critical point by equating (i) to 0 and get
$x^{x} (1 + l o g x) = 0$

$\Rightarrow 1 + l o g x = 0 a s x^{x}$ cannot be equal to 0
$\Rightarrow l o g x = - 1$

But $- 1 = l o g e^{- 1}$

$\Rightarrow l o g x = l o g e^{- 1}$
Equate the terms and get
$x = e^{- 1}$
$\Rightarrow X = \frac{1}{e}$
Therefore f(x) has a stationary point at $X = \frac{1}{e}$
So, the correct answer is option B.

Question:59

The maximum value of ${(\frac{1}{x})}^{x}$ is:
A. $e$
B. $e^{e}$
C. $e^{\frac{1}{e}}$
D. ${(\frac{1}{e})}^{\frac{1}{e}}$

Answer:

Let $y = {(\frac{1}{x})}^{x}$
Take logarithm on both side
$\log y = \log {(\frac{1}{x})}^{x}$

$\Rightarrow \log y = x \log \frac{1}{x}$
Applying first derivative and get
$\frac{d (\log y)}{dx} = \frac{d (x \log \frac{1}{x})}{dx}$
Apply product rule and get
$\frac{d (\log y)}{dx} = x \frac{d (\log \frac{1}{x})}{dx} + \log \frac{1}{x} \cdot \frac{d (x)}{dx}$
Applying first derivative and get
$\begin{aligned} \frac{1}{y} \cdot \frac{d y}{d x} = x \cdot \frac{1}{\frac{1}{x}} \cdot \frac{d (\frac{1}{x})}{d x} + \log \frac{1}{x} \cdot 1 \\ \frac{1}{y} \cdot \frac{d y}{d x} = x^{2} \cdot \frac{d (x^{- 1})}{d x} + \log \frac{1}{x} \\ \frac{1}{y} \cdot \frac{d y}{d x} = x^{2} \cdot (- 1) \cdot x^{- 1 - 1} + \log \frac{1}{x} \\ \frac{1}{y} \cdot \frac{d y}{d x} = x^{2} \cdot (- 1) \cdot x^{- 2} + \log \frac{1}{x} \\ \frac{1}{y} \cdot \frac{d y}{d x} = - 1 + \log \frac{1}{x} \\ \Rightarrow \frac{d y}{d x} = y (- 1 + \log \frac{1}{x}) \\ Substitute value of y from (i) and get \\ \Rightarrow \frac{d y}{d x} = {(\frac{1}{x})}^{x} \cdot (- 1 + \log \frac{1}{x}) \end{aligned}$
Now we find critical point by equating (i) to 0
$\begin{aligned} {(\frac{1}{x})}^{x} \cdot (- 1 + \log \frac{1}{x}) = 0 \\ \Rightarrow (- 1 + \log \frac{1}{x}) = 0 {(\frac{1}{x})}^{x} can't be equal to 0 \\ \Rightarrow (\log \frac{1}{x}) = 1 \\ But 1 = \log e^{1} \\ \Rightarrow (\log \frac{1}{x}) = \log e \\ Equate the terms \\ \Rightarrow \frac{1}{x} = e \\ \Rightarrow x = \frac{1}{e} \end{aligned}$
Therefore f(x) has a stationary point at $x = \frac{1}{e}$ .
i.e the maximum value of $f (\frac{1}{e}) = e^{\frac{1}{e}}$
So, the correct answer is option C.

Question:60

Fill in the blanks in each of the following
The curves $y = 4 x^{2} + 2 x - 8$ and $y = x^{3} - x + 13$ touch each other at the point_____.

Answer:

Given the first curve is $y = 4 x^{2} + 2 x - 8$
Applying first derivative and get
$\frac{d y}{d x} = \frac{d (4 x^{2} + 2 x - 8)}{d x}$
Apply sum rule and 0 is the differentiation of the constant term is 0,so
$\frac{d y}{d x} = 4 \frac{d (x^{2})}{d x} + 2 \frac{d (x)}{d x} + 0$
Apply power rule and get
$\frac{d y}{d x} = 8 x + 2$
This is the slope of the first curve; let $m_{1}$ be equal to this.
$\Rightarrow m_{1} = 8 x + 2 \dots \dots (0)$
The second curve is $y = x^{3} - x + 13$
Applying first derivative and get
$\frac{d y}{d x} = \frac{d (x^{3} - x + 13)}{d x}$
Apply sum rule and 0 is the differentiation of the constant term, so
$\begin{aligned} \frac{d y}{d x} = \frac{d (x^{3})}{d x} - \frac{d (x)}{d x} + 0 \\ Apply power rule and get \\ \frac{d y}{d x} = 3 x^{2} - 1 \end{aligned}$
This is the slope of the second curve; let $m_{2}$ be equal to this.
$\Rightarrow m_{2} = 3 x^{2} - 1 \dots \dots (i i)$
Now the slopes must be equal, because they touch each other, i.e.,
$m_{1} = m_{2}$
$\Rightarrow 8 x + 2 = 3 x^{2} - 1$

$\Rightarrow 3 x^{2} - 8 x - 1 - 2 = 0$

$\Rightarrow 3 x^{2} - 8 x - 3 = 0$
Split middle term and get
$\Rightarrow 3 x^{2} + x - 9 x - 3 = 0$

$\Rightarrow x (3 x + 1) - 3 (3 x + 1) = 0$

$\Rightarrow (3 x + 1) (x - 3) = 0$

$\Rightarrow (3 x + 1) = 0 o r (x - 3) = 0$
$\Rightarrow x = - \frac{1}{3}$ or $x = 3$
Substitute $x = - \frac{1}{3}$ in both the equations and get
For first curve, $y = 4 x^{2} + 2 x - 8$
$\Rightarrow y = 4 {(- \frac{1}{3})}^{2} + 2 (- \frac{1}{3}) - 8$

$\Rightarrow y = 4 (\frac{1}{9}) - \frac{2}{3} - 8$

$\Rightarrow y = \frac{4 - 2 \times 3 - 8 \times 9}{9} = - \frac{74}{9}$
For second curve, $y = x^{3} - x + 13$
$\Rightarrow y = {(- \frac{1}{3})}^{3} - (- \frac{1}{3}) + 13$
$\begin{aligned} \Rightarrow y = - \frac{1}{27} + \frac{1}{3} + 13 \\ \Rightarrow y = \frac{- 1 + 1 \times 9 + 13 \times 27}{27} = \frac{359}{27} \\ Thus at \\ x = - \frac{1}{3} curves don't touch \end{aligned}$
Substitute x=3 in both equations
For first curve, $y = 4 x^{2} + 2 x - 8$
$\Rightarrow y = 4 (3)^{2} + 2 (3) - 8$

$\Rightarrow y = 4 (9) + 6 - 8 = 34$
For second curve, $y = x^{3} - x + 13$
$\Rightarrow y = (3)^{3} - (3) + 13$

$\Rightarrow y = 27 - 3 + 13 = 37$
Hence at x=3 both curves don't touch
So, the curves $y = 4 x^{2} + 2 x - 8$ and $y = x^{3} - x + 13$ do not touch each other.

Question:61

Fill in the blanks in each of the following
The equation of normal to the curve $y = t a n x$ at (0, 0) is ________.

Answer:

Given curve is $y = t a n x$
Apply first derivative and get
$\frac{dy}{dx} = \frac{d (\tan x)}{dx}$

$\Rightarrow \frac{d y}{d x} = \sec^{2} x$
It is the slope of tangent
Substitute (0,0) in slope and get
${(\frac{d y}{d x})}_{(0, 0)} = \sec^{2} x$

$\Rightarrow {(\frac{dy}{dx})}_{(0, 0)} = \sec^{2} 0 = 1$
Hence, -1 is the slope of the normal to the curve at (0,0)
Hence the equation is
$y - 0 = - 1 (x - 0)$

$\Rightarrow y = - x$ or $x + y = 0$

Question:62

Fill in the blanks in each of the following
The values of a for which the function $f (x) = s i n x - a x + b$ increases on R are ______.

Answer:

Given $f (x) = s i n x - a x + b$
Apply first derivative and get
$f^{'} (x) = \frac{d (\sin x - ax + b)}{dx}$
Apply sum rule and 0 is the differentiation of the constant term, so
$f^{'} (x) = \frac{d (\sin x)}{dx} - a \frac{d (x)}{dx} + 0$
Apply first derivative and get
$f^{'} (x) = \cos x - a$
Also f(x) increases on R
$\Rightarrow f^{'} (x) \geq 0 \forall x ϵ R$

$\Rightarrow \cos x - a \geq 0 \forall x ϵ R$

$\Rightarrow \cos x \geq a \forall x ϵ R$
This is possible when $a \leq - 1$
Hence $a \in (- \infty, - 1]$
The values of a increases on $R$ are $(- \infty, - 1] .$

Question:63

Fill in the blanks in each of the following
The function $f (x) = \frac{2 x^{2} - 1}{x^{4}}, x > 0$ decreases in the interval _______.

Answer:

Given $f (x) = \frac{2 x^{2} - 1}{x^{4}}, x > 0$
After applying derivative, we get
$f^{'} (x) = \frac{d (\frac{2 x^{2} - 1}{x^{4}})}{d x}$
Apply quotient rule and 0 is the differentiation of the constant term, so
$f^{'} (x) = \frac{x^{4} \cdot \frac{d (2 x^{2} - 1)}{d x} - (2 x^{2} - 1) \cdot \frac{d (x^{4})}{d x}}{{(x^{4})}^{2}}$

$\Rightarrow f^{'} (x) = \frac{1}{x^{8}} (x^{4} \cdot \frac{d (2 x^{2} - 1)}{d x} - (2 x^{2} - 1) \cdot \frac{d (x^{4})}{d x})$

$\Rightarrow f^{'} (x) = \frac{1}{x^{8}} (x^{4} \cdot (4 x) - (2 x^{2} - 1) \cdot (4 x^{3}))$

$\Rightarrow f^{'} (x) = \frac{1}{x^{8}} (4 x^{5} - (2 x^{2}) \cdot (4 x^{3}) - (4 x^{3}))$

$\Rightarrow f^{'} (x) = \frac{1}{x^{8}} (4 x^{5} - 8 x^{5} - (4 x^{3}))$

$\Rightarrow f^{'} (x) = \frac{x^{3}}{x^{8}} (4 x^{2} - 8 x^{2} - 4)$

$\Rightarrow f^{'} (x) = \frac{1}{x^{5}} (- 4 x^{2} - 4)$

$\Rightarrow f^{'} (x) = \frac{- 4}{x^{5}} (x^{2} - 1)$
Equate this with 0 and get
$f^{'} (x) = 0$

$\Rightarrow \frac{- 4}{x^{5}} (x^{2} - 1) = 0$

$\Rightarrow x^{2} - 1 = 0$

$\Rightarrow x^{2} = 1$

$\Rightarrow x = \pm 1$
$(- \infty, - 1), (- 1, 0), (0, 1)$ and $(1, \infty)$ are the intervals formed by these two critical numbers
(i) in the interval $(- \infty, - 1), f^{'} (x) > 0$
$∴ f (x)$ is increasing in $(- \infty, - 1)$
(ii) in the interval (-1,0), $f^{'} (x) \leq 0$
$∴ f (x)$ is decreasing in (-1,0)
(iii) in the interval (0,1), $f^{'} (x) > 0$
$∴ f (x)$ is increasing in $(0, 1)$
(iv) in the interval $(1, \infty), f^{'} (x) < 0$
$∴ f (x)$ is decreasing in $(1, \infty)$
$f (x) = \frac{2 x^{2} - 1}{x^{4}}, x > 0$
Therefore, the function $f (x) = \frac{2 x^{2} - 1}{x^{4}}, x > 0$ decreases in the interval $(1, \infty)$ .

Question:64

Fill in the blanks in each of the following
The least value of the function $f (x) = a x + \frac{b}{x}$ $(a > 0, b > 0, x > 0)$ is ______.

Answer:

Given $f (x) = a x + \frac{b}{x}$
After applying the derivative
$f^{'} (x) = \frac{d (a x + \frac{b}{x})}{d x}$
Apply sum rule and get
$f^{'} (x) = \frac{d (ax)}{dx} + \frac{d (\frac{b}{x})}{dx}$
Apply quotient rule on second part and get
$f^{'} (x) = a + \frac{x \cdot \frac{d (b)}{d x} - b \cdot \frac{d (x)}{d x}}{(x)^{2}}$

$\Rightarrow f^{'} (x) = a + \frac{x \cdot 0 - b \cdot 1}{x^{2}}$

$\Rightarrow f^{'} (x) = a - \frac{b}{x^{2}}$
Equate it with 0 and get
$f^{'} (x) = 0$

$\Rightarrow a - \frac{b}{x^{2}} = 0$

$\Rightarrow a = \frac{b}{x^{2}}$

$\Rightarrow x^{2} = \frac{b}{a}$

$\Rightarrow x = \sqrt{\frac{b}{a}} ($ as $x > 0)$
Now given by second derivative,
Apply derivative and get
$f^{''} (x) = \frac{d (a - \frac{b}{x^{2}})}{dx}$
Apply sum rule and get
$f^{''} (x) = \frac{d (a)}{dx} - \frac{d (\frac{b}{x^{2}})}{dx}$
Apply quotient rule on the second part and get
$f^{''} (x) = 0 - \frac{x^{2} \cdot \frac{d (b)}{d x} - b \cdot \frac{d (x^{2})}{dx}}{{(x^{2})}^{2}}$

$\Rightarrow f^{''} (x) = \frac{x^{2} \cdot 0 - b \cdot (2 x)}{x^{4}}$

$\Rightarrow f^{''} (x) = \frac{2 x b}{x^{4}} = \frac{2 b}{x^{3}}$
Now, equate it with $x = \sqrt{\frac{b}{a}}$
$(f^{''} (x)) x = \sqrt{\frac{b}{a}} = \frac{2 b}{x^{3}}$

$\Rightarrow {(f^{''} (x))}_{x = \sqrt{\frac{b}{2}}} = \frac{2 b}{{(\sqrt{\frac{b}{a}})}^{3}}$

$\Rightarrow {(f^{''} (x))}_{x = \sqrt{\frac{b}{2}}} = \frac{2 ab}{b} = 2 a > 0$
The least value of f(x) is
$\begin{array}{l} f (\sqrt{\frac{b}{a}}) = ax + \frac{b}{x} \\ \Rightarrow f (\sqrt{\frac{b}{a}}) = a (\sqrt{\frac{b}{a}}) + \frac{b}{(\sqrt{\frac{b}{a}})} \\ \Rightarrow f (\sqrt{\frac{b}{a}}) = a (\sqrt{\frac{b}{a}}) + b \sqrt{\frac{a}{b}} \\ \Rightarrow f (\sqrt{\frac{b}{a}}) = \frac{ab + ab}{\sqrt{ab}} \\ \Rightarrow f (\sqrt{\frac{b}{a}}) = \frac{2 ab}{\sqrt{ab}} \end{array}$
Multiply and divide by $\sqrt{ab}$ and get
$\Rightarrow f (\sqrt{\frac{b}{a}}) = \frac{2 ab}{\sqrt{ab}} \times \frac{\sqrt{ab}}{\sqrt{ab}}$

$\Rightarrow f (\sqrt{\frac{b}{a}}) = 2 \sqrt{a b}$
Therefore, the least value of function $f (x) = a x + \frac{b}{x} (a > 0, b > 0, x > 0)$ is $2 \sqrt{a b}$

Subtopics in NCERT Exemplar Class 12 Maths Solutions Chapter 6

The sub-topics that are covered in this chapter are:

Introduction
Rate of change of quantities
Increasing and decreasing functions
Normals and Tangents
Approximations
Maxima and minima
Minimum and maximum value of the function is a closed interval

NCERT Exemplar Class 12 Maths Solutions

NCERT Exemplar for Class 12 Mathematics Chapter 1 Relations and Functions

NCERT Exemplar for Class 12 Mathematics Chapter 2 Inverse Trigonometric Functions

NCERT Exemplar for Class 12 Mathematics Chapter 3 Matrices

NCERT Exemplar for Class 12 Mathematics Chapter 4 Determinants

NCERT Exemplar for Class 12 Mathematics Chapter 5 Continuity and Differentiability

NCERT Exemplar for Class 12 Mathematics Chapter 7 Integrals

NCERT Exemplar for Class 12 Mathematics Chapter 8 Applications of Integrals

NCERT Exemplar for Class 12 Mathematics Chapter 9 Differential Equations

NCERT Exemplar for Class 12 Mathematics Chapter 10 Vector Algebra

NCERT Exemplar for Class 12 Mathematics Chapter 11 Three dimensional Geometry

NCERT Exemplar for Class 12 Mathematics Chapter 12 Linear Programming

NCERT Exemplar for Class 12 Mathematics Chapter 13 Probability

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Importance of Solving NCERT Exemplar Class 12 Maths Chapter 6 Solutions:

In NCERT Exemplar Class 12 Mathematics solutions chapter 6, the students will learn in detail about the derivatives and their fundamentals and applications. Several topics are covered in this, which has significance in higher calculus and even other subjects as it is a very common topic in exams.
In NCERT exemplar Class 12 Maths solutions chapter 6, the students will come face to face with several topics like decreasing and increasing functions, minima and maxima, approximations, normal, tangents, change of quantities and its rate which are very commonly asked in the exams.

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NCERT solutions for class 12 maths - Chapter-wise

Chapter 1	NCERT Solutions for Class 12 Maths Chapter 1 Relations and Functions
Chapter 2	NCERT solutions for class 12 maths chapter 2 Inverse Trigonometric Functions
Chapter 3	NCERT solutions for class 12 maths chapter 3 Matrices
Chapter 4	NCERT solutions for class 12 maths chapter 4 Determinants
Chapter 5	NCERT solutions for class 12 maths chapter 5 Continuity and Differentiability
Chapter 6	NCERT solutions for class 12 maths chapter 6 Application of Derivatives
Chapter 7	NCERT solutions for class 12 maths chapter 7 Integrals
Chapter 8	NCERT solutions for class 12 maths chapter 8 Application of Integrals
Chapter 9	NCERT solutions for class 12 maths chapter 9 Differential Equations
Chapter 10	NCERT solutions for class 12 maths chapter 10 Vector Algebra
Chapter 11	NCERT solutions for class 12 maths chapter 11 Three Dimensional Geometry
Chapter 12	NCERT solutions for class 12 maths chapter 12 Linear Programming
Chapter 13	NCERT solutions for class 12 maths chapter 13 Probability

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Frequently Asked Questions (FAQs)

1. What is the best way to prepare for an exam from NCERT books?

First of all, you should read the chapters well for the exam. You must also go through the questions after the end of each chapter.

2. Who has prepared the solutions?

The Class 12 Maths NCERT exemplar solutions chapters 6 are prepared by us who are experts in mathematics. The solutions are prepared after understanding and reference from professional books.

3. Can I download the solutions for this chapter?

Yes, you can click the NCERT exemplar Class 10 Maths solutions chapter 1 pdf download in the given link on the page.

4. Are these solutions helpful for board examinations?

Yes, the NCERT exemplar Class 12 solutions for Mathematics chapter 6 will provide you with the ability to ace the board exams if you utilise it properly.

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Questions related to CBSE Class 12th

Have a question related to CBSE Class 12th ?

Can I change my board from CBSE to CHSE Odisha in Class 12th?

Changing from the CBSE board to the Odisha CHSE in Class 12 is generally difficult and often not ideal due to differences in syllabi and examination structures. Most boards, including Odisha CHSE , do not recommend switching in the final year of schooling. It is crucial to consult both CBSE and Odisha CHSE authorities for specific policies, but making such a change earlier is advisable to prevent academic complications.

Read Complete Answer

Manasvini Gupta 30 January,2025

quartely question paper for mathematics cbse

Hello there! Thanks for reaching out to us at Careers360.

Ah, you're looking for CBSE quarterly question papers for mathematics, right? Those can be super helpful for exam prep.

Unfortunately, CBSE doesn't officially release quarterly papers - they mainly put out sample papers and previous years' board exam papers. But don't worry, there are still some good options to help you practice!

Have you checked out the CBSE sample papers on their official website? Those are usually pretty close to the actual exam format. You could also look into previous years' board exam papers - they're great for getting a feel for the types of questions that might come up.

If you're after more practice material, some textbook publishers release their own mock papers which can be useful too.

Let me know if you need any other tips for your math prep. Good luck with your studies!

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Jefferson 25 September,2024

i have taken admission in clg after my campartment exam in chemistry along with the neet 2025 preperation but unfortunately i again got campartment after doing all this preparation i got 15 marks only now I can't able to think what should I do next.

It's understandable to feel disheartened after facing a compartment exam, especially when you've invested significant effort. However, it's important to remember that setbacks are a part of life, and they can be opportunities for growth.

Possible steps:

Re-evaluate Your Study Strategies:
- Identify Weak Areas: Pinpoint the specific topics or concepts that caused difficulties.
- Seek Clarification: Reach out to teachers, tutors, or online resources for additional explanations.
- Practice Regularly: Consistent practice is key to mastering chemistry.
Consider Professional Help:
- Tutoring: A tutor can provide personalized guidance and support.
- Counseling: If you're feeling overwhelmed or unsure about your path, counseling can help.
Explore Alternative Options:
- Retake the Exam: If you're confident in your ability to improve, consider retaking the chemistry compartment exam.
- Change Course: If you're not interested in pursuing chemistry further, explore other academic options that align with your interests.
Focus on NEET 2025 Preparation:
- Stay Dedicated: Continue your NEET preparation with renewed determination.
- Utilize Resources: Make use of study materials, online courses, and mock tests.
Seek Support:
- Talk to Friends and Family: Sharing your feelings can provide comfort and encouragement.
- Join Study Groups: Collaborating with peers can create a supportive learning environment.

Remember: This is a temporary setback. With the right approach and perseverance, you can overcome this challenge and achieve your goals.

I hope this information helps you.

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Kanishka kaushikii 17 September,2024

i scored 84 percent in cbse 2 years ago.Am i eligible for medhavi scholarship if give 12th again after 2 drop years and will got 75+ in mpboard

Hi,

Qualifications:
Age: As of the last registration date, you must be between the ages of 16 and 40.
Qualification: You must have graduated from an accredited board or at least passed the tenth grade. Higher qualifications are also accepted, such as a diploma, postgraduate degree, graduation, or 11th or 12th grade.
How to Apply:
Get the Medhavi app by visiting the Google Play Store.
Register: In the app, create an account.
Examine Notification: Examine the comprehensive notification on the scholarship examination.
Sign up to Take the Test: Finish the app's registration process.
Examine: The Medhavi app allows you to take the exam from the comfort of your home.
Get Results: In just two days, the results are made public.
Verification of Documents: Provide the required paperwork and bank account information for validation.
Get Scholarship: Following a successful verification process, the scholarship will be given. You need to have at least passed the 10th grade/matriculation scholarship amount will be transferred directly to your bank account.

Scholarship Details:

Type A: For candidates scoring 60% or above in the exam.

Type B: For candidates scoring between 50% and 60%.

Type C: For candidates scoring between 40% and 50%.

Cash Scholarship:

Scholarships can range from Rs. 2,000 to Rs. 18,000 per month, depending on the marks obtained and the type of scholarship exam (SAKSHAM, SWABHIMAN, SAMADHAN, etc.).

Since you already have a 12th grade qualification with 84%, you meet the qualification criteria and are eligible to apply for the Medhavi Scholarship exam. Make sure to prepare well for the exam to maximize your chances of receiving a higher scholarship.

Hope you find this useful!

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Yuvan 30 August,2024

i upload 12th board result screenshot in ncweb form from the cbse site so it is valid

hello mahima,

If you have uploaded screenshot of your 12th board result taken from CBSE official website,there won,t be a problem with that.If the screenshot that you have uploaded is clear and legible. It should display your name, roll number, marks obtained, and any other relevant details in a readable forma.ALSO, the screenshot clearly show it is from the official CBSE results portal.

hope this helps.

Read Complete Answer

Ashvadha 12 July,2024

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Central Board of Secondary Education Class 12th Examination

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Option 1) $0.34\; J$	Option 2) $0.16\; J$
Option 3) $1.00\; J$	Option 4) $0.67\; J$

Option 1) $2,000 \; J - 5,000\; J$	Option 2) $200 \, \, J - 500 \, \, J$
Option 3) $2\times 10^{5}J-3\times 10^{5}J$	Option 4) $20,000 \, \, J - 50,000 \, \, J$

Option 1) $K/2\,$	Option 2) $\; K\;$
Option 3) $zero\;$	Option 4) $K/4$

Option 1) $11.2\, L\, H_{2(g)}$ at STP is produced for every mole $HCL_{(aq)}$ consumed	Option 2) $6L\, HCl_{(aq)}$ is consumed for ever $3L\, H_{2(g)}$ produced
Option 3) $33.6 L\, H_{2(g)}$ is produced regardless of temperature and pressure for every mole Al that reacts	Option 4) $67.2\, L\, H_{2(g)}$ at STP is produced for every mole Al that reacts .

Option 1) 0.02	Option 2) 3.125 × 10^-2
Option 3) 1.25 × 10^-2	Option 4) 2.5 × 10^-2

Option 1) decrease twice	Option 2) increase two fold
Option 3) remain unchanged	Option 4) be a function of the molecular mass of the substance.

Option 1) Molality	Option 2) Weight fraction of solute
Option 3) Fraction of solute present in water	Option 4) Mole fraction.

Option 1) twice that in 60 g carbon	Option 2) 6.023 × 10²²
Option 3) half that in 8 g He	Option 4) 558.5 × 6.023 × 10²³

Option 1) less than 3	Option 2) more than 3 but less than 6
Option 3) more than 6 but less than 9	Option 4) more than 9

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NCERT Exemplar Class 12 Maths Solutions Chapter 6 Applications Of Derivatives

NCERT Exemplar Class 12 Maths Solutions

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JEE Main Important Mathematics Formulas

Importance of Solving NCERT Exemplar Class 12 Maths Chapter 6 Solutions:

NCERT solutions for class 12 maths - Chapter-wise

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