Chapter 21 Areas of bounded regions Exercise Ex. 21.1

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Thus, Required area = square unitsQuestion 8

Solution 8

Question 9

Solution 9

Question 11

Sketch the region {(x, y):9x² + 4y² = 36} and find the area enclosed by it, using integration.Solution 11

9x² + 4y² = 36

Area of Sector OABCO =

Area of the whole figure = 4 × Ar. D OABCO

= 6p sq. unitsQuestion 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

What dose this integral represent on the graph?.Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 10

and evaluate the area of the region under the curve and above the x-axis.Solution 10

Question 27

Find the area of the minor segment of the circle x² + y² = a² cut off by the line x =Solution 27

Question 28

Find the area of the region bounded by the curve x = at², y = 2at between the ordinates corresponding t = 1 and t = 2.Solution 28

Question 29

Find the area enclosed by the curve x = 3 cost,

y = 2 sin t.Solution 29

Chapter 21 Areas of Bounded Regions Exercise Ex. 21.2

Question 1

Solution 1

Question 2

Solution 2

Question 3

Find the area of the region bounded by x² = 4ay and its latusrectum.Solution 3

Question 4

Find the area of the region bounded by x² + 16y = 0 and its latusrectum.Solution 4

Question 5

Find the area of the region bounded by the curve ay² = x³, the y-axis and the lines y = a and y = 2a.Solution 5

Chapter 21 – Areas of Bounded Regions Exercise Ex. 21.3

Question 2

Find the area of the region common to the parabolas 4y² = 9x and 3x² = 16y.Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Find the area of the region between the circles x² + y² = 4 and (x – 2)² + y² = 4.Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23 (i)

Using Integration, find the area of the region bounded by the triangle whose vertices are (- 1, 2), (1, 5) and (3, 4).Solution 23 (i)

Equation of side AB,

Equation of side BC,

Equation of side AC,

Area of required region

= Area of EABFE + Area of BFGCB – Area of AEGCA

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 29

Solution 29

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solution 37

 Question 39

Solution 39

Question 40

Solution 40

Question 41

Solution 41

Question 42

Solution 42

Question 43

Solution 43

Question 44

Solution 44

Question 46

Solution 46

Question 47

Solution 47

Question 48

Solution 48

Question 49

Solution 49

Question 50

Solution 50

Question 1

Calculate the area of the region bounded by the parabolas y² = 6x and x² = 6y.Solution 1

Question 18

Find the area of the region bounded by y =, x = 2y + 3 in the first quadrant and x-axis.Solution 18

Question 24

Find the area of the bounded by y =and y = x.Solution 24

Question 28

Find the area enclosed by the curve y = -x² and the straight line x + y + 2 = 0.Solution 28

Question 30

Using the method of integration, find the area of the region bounded by the following lines: 3x – y – 3 = 0,

2x + y – 12 = 0, x – 2y – 1 = 0.Solution 30

Question 38

Find the area of the region enclosed by the parabola

 x² = y and the line y = x + 2.Solution 38

Question 51

Solution 51

Question 52

Solution 52

Chapter 21 Areas of Bounded Regions Exercise Ex. 21.4

Question 1

Find the area of the region between the parabola x = 4y – y² and the line x = 2y – 3.Solution 1

Question 2

Find the area bounded by the parabola x = 8 + 2y – y²; the y-axis and the lines y = -1 and y = 3.Solution 2

Question 3

Find the area bounded by the parabola y² = 4x and the line

y = 2x – 4.

i. By using horizontal strips

ii. By using vertical stripsSolution 3

Question 4

Find the area of the region bounded the parabola y² = 2x and straight line x – y = 4.Solution 4

Chapter 20 Definite Integrals Exercise Ex. 20.1

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Question 38

Solution 38

Question 39

Solution 39

Question 40

Solution 40

Question 41

Solution 41

Question 42

Solution 42

Question 43

Solution 43

Question 44

Evaluate the Integral in using substitution.

Solution 44

Question 45

Solution 45

Question 46

Solution 46

Question 47

Solution 47

Question 48

Solution 48

Question 49

Solution 49

Question 50

Solution 50

Question 51

Solution 51

Question 52

Solution 52

Question 54

Solution 54

Question 55

Solution 55

Question 56

Solution 56

Let cosx =u , Then

Hence

Question 57

Solution 57

Question 58

Solution 58

Question 59

Solution 59

Question 60

Solution 60

Given :

Question 61

Solution 61

Question 62

Solution 62

Question 63

Solution 63

Question 64

Solution 64

We know , By reduction formula 

For n=2

For n=4

Hence

Note: Answer given at back is incorrect.Question 65

Solution 65

Using Integration By parts

Question 66

Solution 66

Question 67

Solution 67

Note: Answer given in the book is incorrect. Question 68

Solution 68

 =(1/4)log(2e)

Chapter 20 Definite Integrals Exercise Ex. 20.2

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Using Integration By parts

Hence

Question 25

Solution 25

Question 26

Solution 26

Question 27

Evaluate Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

?Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Question 38

Solution 38

Question 39

Solution 39

Question 40

Solution 40

Question 41

Solution 41

Question 42

Solution 42

Question 43

Solution 43

Question 44

Solution 44

Question 45

Solution 45

Question 46

Solution 46

Question 47

Solution 47

Question 48

Solution 48

Question 49

Solution 49

Question 50

Solution 50

Question 51

Solution 51

Question 52

Solution 52

Question 53

Solution 53

Question 54

Solution 54

Question 55

Solution 55

Question 56

Solution 56

Question 57

Solution 57

Question 58

Solution 58

Question 59

Solution 59

Question 60

Solution 60

Question 61

Solution 61

Question 62

Solution 62

Chapter 20 Definite Integrals Exercise Ex. 20.3

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

2x+3 is positive for x>-1.5 . Hence

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Evaluate the integral Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

For

Using Integration By parts

For

Using Integration By parts

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 27

Solution 27

[x]=0 for 0 < x

and [x]=1 for 1< x < 2

Hence

Question 28

Solution 28

Question 26

Evaluate the following integrals:

Solution 26

NOTE: Answer not matching with back answer.

Chapter 20 Definite Integrals Exercise Ex. 20.4A

Question 1

Solution 1

We know

Hence

We know

If

Then also

Hence

Question 2

Solution 2

We know

Hence

If

Then

Question 3

Solution 3

We know

Hence

If

Then

So

Question 4

Solution 4

We know

Hence

If

Then

Hence

Question 5

Solution 5

We know

Hence

If

Then

So

We know

If f(x) is even

If f(x) is odd

Here

f(x) is even, hence

Note: Answer given in the book is incorrect.Question 6

Solution 6

We know

Hence

If

Then

So

Question 7

Solution 7

We know

Hence

If

Then

So

Question 8

Solution 8

We know

Hence

If

Then

So

Note: Answer given in the book is incorrect. Question 9

Solution 9

If f(x) is even

If f(x) is odd

Here

 is odd and

 is even. Hence

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Chapter 20 Definite Integrals Exercise Ex. 20.4B

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

BQuestion 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Hence

Question 20

Solution 20

Question 21

Solution 21

Now

Let cosx=t

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25 (i)

Solution 25 (i)

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32 (i)

Solution 32 (i)

Question 33

Solution 33

Question 34

Evaluate the integral Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Question 38

Solution 38

We know

Also here

So

Hence

Question 39

Solution 39

Question 43

Solution 43

Question 44

Solution 44

Question 45

Solution 45

Question 46

Solution 46

Question 47

Solution 47

Question 48

Solution 48

Question 49

Solution 49

Chapter 20 Definite Integrals Exercise Ex. 20RE

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Evaluate the following integrals

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Evaluate the integral Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Evaluate the following integrals

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Question 38

Evaluate the following integrals

Solution 38

Question 39

Solution 39

Question 40

Solution 40

Question 41

Solution 41

Question 42

Solution 42

Question 43

Solution 43

Question 44

Solution 44

Question 45

Solution 45

Question 46

Solution 46

Question 47

Solution 47

Question 48

Solution 48

Question 49

Solution 49

Question 50

Solution 50

Question 51

Solution 51

Question 52

Solution 52

Question 53

Solution 53

Question 54

Solution 54

Question 55

Solution 55

Question 56

Solution 56

Question 57

Solution 57

Question 58

Solution 58

Question 59

Solution 59

Question 60

Solution 60

Question 61

Solution 61

Question 62

Solution 62

Question 63

Solution 63

Question 64

Solution 64

Question 65

Solution 65

Question 66

Solution 66

Question 67

Solution 67

Question 68

Solution 68

Question 69

Solution 69

Chapter 20 Definite Integrals Exercise Ex. 20.5

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Evaluate the following in tegrals as limit of sums

Solution 31

Question 32

Evaluate the following in tegrals as limit of sums

Solution 32

Question 33

Solution 33

Chapter 18 Maxima and Minima Ex 18.1

Question 1

Solution 1

Question 2

Find the maximum and minimum values, if any, without using derivatives of the following function given by f(x) = -(x-1)² + 2 on R.Solution 2

Question 3

Solution 3

Question 4

Find the maximum and minimum values, if any, without using derivatives of the following function given by h(x) = sin(2x) + 5 on R.Solution 4

Question 5

Find the maximum and minimum values, if any,  usingwithout derivatives of the following function given by  on R.Solution 5

Question 6

Solution 6

Question 7

Find the maximum and minimum values, if any, without using derivatives of the following function given by  on R.Solution 7

Question 8

Solution 8

Question 9

Solution 9

Chapter 18 Maxima and Minima Ex. 18.2

Question 1

Solution 1

Question 2

Find the local maxima and local minima, if any, of the following functions using first derivative test. Find also the local maximum and the local minimum values, as the case may be:

f(x) =x³ – 3xSolution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Find the local maxima and local minima, if any, of the following functions using first derivative test. Find also the local maximum and the local minimum values, as the case may be:

f(x) = sinx – cos x, 0 < x < 2πSolution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Find the local maxima and local minima, if any, of the following functions using first derivative test. Find also the local maximum and the local minimum values, as the case may be:

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Chapter 18 Maxima and Minima Ex. 18.3

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 1(v)

Solution 1(v)

Question 1(vi)

Solution 1(vi)

Question 1(vii)

Solution 1(vii)

Question 1(viii)

Solution 1(viii)

Question 1(ix)

Solution 1(ix)

Question 1(x)

Solution 1(x)

Question 1(xi)

Solution 1(xi)

Question 1(xii)

Solution 1(xii)

Question 2(i)

Solution 2(i)

Question 2(ii)

Solution 2(ii)

Question 2(iii)

Solution 2(iii)

Question 3

Solution 3

Question 4

Show that the function given by f(x)= has maximum value at x = e.Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

If f(x) = x³ + ax² + bx + c has a maximum at x = -1 and minimum at x = 3. Determine a, b and c.Solution 7

Question 8

Prove that  has maximum value at Solution 8

Given: 

Differentiating w.r.t x, we get

Take f'(x) = 0

Differentiating f'(x) w.r.t x, we get

At 

Clearly, f”(x) < 0 at 

Thus,  is the maxima.

Hence, f(x) has maximum value at .

Chapter 18 Maxima and Minima Ex. 18.4

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Find both the absolute maximum and absolute minimum of 3x⁴ – 8x³ + 12x² – 48x + 25 on the interval [0, 3] Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Chapter 18 Maxima and Minima Ex. 18.5

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Divide 15 into two parts such that the square of one multiplied with the cube of the other is maximum.Solution 4

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

A square piece of tin of side 18 cm is to made into a box without top, by cutting a square from each corner and folding up the flaps to form the box. What should be the side of the square to be cut off so that the volume of the box is the maximum possible? Also, find this maximum volume.Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

An isosceles triangle of vertical angle 2θ is inscribed in a circle radius a. show that the area of the triangle is maximum when Solution 22

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Question 38

Solution 38

Question 39

Solution 39

Question 40

Solution 40

Question 41

A given quantity of metal is to be cast into a half cylinder with a rectangular base and semicircular ends. Show that in order that the total surface area may be minimum, the ratio of the length of the cylinder to the diameter of its semi – circular ends is π : ( π + 2) Solution 41

Question 42

Solution 42

Question 43

Solution 43

Question 44

Solution 44

Question 45

Solution 45

Question 45

Solution 45

Question 5

Amongst all open (from the top) right circular cylindrical boxes of volume 125π cm³, find the dimensions of the box which has the least surface area.Solution 5

Let r and h be the radius and height of the cylinder.

Volume of cylinder  

 … (i)

Surface area of cylinder  

From (i), we get

Question 23

Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is  Solution 23

Let  be an isosceles triangle with AB = AC.

Let  

Here, AO bisects 

Taking O as the centre of the circle, join OE, OF and OD such that

OE = OF = OD = r (radius)

Now, 

In 

Similarly, AF = r cot x

In 

As OB bisect  we have

In 

Similarly, BD = DC = CE = 

We have, perimeter of 

P = AB + BC + CA

 = AE + EC + BD + DC + AF + BF

Differentiating w.r.t x, we get

Taking 

As 

Therefore,  is an equilateral triangle.

Taking second derivative of P, we get

At 

Therefore, the perimeter is minimum when 

Least value of P 

Chapter 17 Increasing and Decreasing Functions Ex. 17.1

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Chapter 17 Increasing and Decreasing Functions Ex. 17.2

Question 1(i)

Find the intervals in which the following functions are increasing or decreasing:

10 – 6x – 2x²Solution 1(i)

Question 1(ii)

Find the intervals in which the following functions are increasing or decreasing:

x² + 2x – 5Solution 1(ii)

Question 1(iii)

Find the intervals in which the following functions are increasing or decreasing:

6 – 9x – x²Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 1(v)

Solution 1(v)

Question 1(vi)

Solution 1(vi)

Question 1(vii)

Solution 1(vii)

Question 1(viii)

Solution 1(viii)

Question 1(ix)

Solution 1(ix)

Question 1(xi)

Solution 1(xi)

Question 1(xii)

Solution 1(xii)

Question 1(xiii)

Solution 1(xiii)

Question 1(xiv)

Solution 1(xiv)

Question 1(xv)

Solution 1(xv)

Question 1(xvi)

Solution 1(xvi)

Question 1(xvii)

Solution 1(xvii)

Question 1(xviii)

Solution 1(xviii)

Question 1(xix)

Solution 1(xix)

Question 1(xx)

Solution 1(xx)

Question 1(xxi)

Solution 1(xxi)

Question 1(xxii)

Solution 1(xxii)

Question 1(xxiii)

Solution 1(xxiii)

Question 1(xxiv)

Solution 1(xxiv)

Question 1(xxv)

Find the values of x for which the function y = [x(x – 2)]² is increasing or decreasingSolution 1(xxv)

Question 1(xxvi)

Find the interval in which the following function is increasing or decreasing.

f(x) = 3x⁴– 4x³– 12x² + 5Solution 1(xxvi)

Question 1(xxvii)

Find the interval in which the following function is increasing or decreasing.

Solution 1(xxvii)

Question 1(xxviii)

Find the interval in which the following function is increasing or decreasing.

Solution 1(xxviii)

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Show that the function given by f(x) = sin x is

(a) increasing in (0, π/2)

(b) decreasing in (π/2, π)

(c) neither increasing nor decreasing in (0, π)Solution 7

Question 8

Prove that the function f given by f(x) = log sin x is increasing on  and decreasing on Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30(i)

Solution 30(i)

Question 31

Solution 31

Question 32

Solution 32

Question 33

Prove that the function f(x) = cos x is:

(i) strictly decreasing in (0, π)

(ii) strictly increasing in (π, 2π)

(iii) neither increasing nor decreasing in (0, 2π)Solution 33

Question 34

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Question 38

Solution 38

Question 39(i)

Find the interval in which f(x) is increasing or decreasing:

Solution 39(i)

Question 39(ii)

Find the interval in which f(x) is increasing or decreasing:

Solution 39(ii)

Question 39(iii)

Find the interval in which f(x) is increasing or decreasing:

Solution 39(iii)

Question 1(x)

Find the intervals in which the following functions are increasing or decreasing:

Solution 1(x)

Given: 

Differentiating w.r.t x, we get

Take f'(x) = 0

Clearly, f'(x) > 0 if x < -2 or x > -1

And, f'(x) < 0 if -2 < x < -1

Thus, f(x) increases on  and decreases on Question 1(xxix)

Find the intervals in which the following functions are increasing or decreasing:

Solution 1(xxix)

Given: 

Differentiating w.r.t x, we get

Take f'(x) = 0

The points x = 2, 4 and -3 divide the number line into four disjoint intervals namely 

Consider the interval 

In this case, x – 2 < 0, x – 4 < 0 and x + 3 < 0

Therefore, f'(x) < 0 when 

Thus the function is decreasing in 

Consider the interval 

In this case, x – 2 < 0, x – 4 < 0 and x + 3 > 0

Therefore, f'(x) > 0 when 

Thus the function is increasing in 

Now, consider the interval 

In this case, x – 2 > 0, x – 4 < 0 and x + 3 > 0

Therefore, f'(x) < 0 when 

Thus the function is decreasing in 

And now, consider the interval 

In this case, x – 2 > 0, x – 4 > 0 and x + 3 > 0

Therefore, f'(x) < 0 when 

Thus the function is increasing in Question 30(ii)

Prove that the following function is increasing on R:

Solution 30(ii)

Given: 

Differentiating w.r.t x, we get

Now, 

Hence, f(x) is an increasing function for all x.

Chapter 16 Tangents and Normals Ex. 16.1

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 1(v)

Find the slopes of the tangent and the normal to the curve x = a(θ – sinθ), y =a(1 + cos θ) at θ = .Solution 1(v)

Question 1(vi)

Solution 1(vi)

Question 1(vii)

Solution 1(vii)

Question 1(viii)

Solution 1(viii)

Question 1(ix)

Solution 1(ix)

Question 1(x)

Solution 1(x)

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Chapter 16 Tangents and Normals Ex. 16.2

Question 1

Solution 1

Question 2

Solution 2

Question 3(i)

Solution 3(i)

Question 3(ii)

Find the equations of the tangent and normal to the given curves at the indicated points:

y=x⁴ – 6x³ + 13x² – 10x + 5 at (x = 1)Solution 3(ii)

Question 3(iii)

Solution 3(iii)

Question 3(iv)

Solution 3(iv)

Question 3(v)

Solution 3(v)

Question 3(vi)

Solution 3(vi)

Question 3(vii)

Solution 3(vii)

Question 3(viii)

Solution 3(viii)

Question 3(ix)

Solution 3(ix)

Question 3(x)

Solution 3(x)

Question 3(xi)

Solution 3(xi)

Question 3(xii)

Solution 3(xii)

Question 3(xiii)

Solution 3(xiii)

Question 3(xiv)

Solution 3(xiv)

Question 3(xv)

Solution 3(xv)

Question 3(xvi)

Find the equation of the normal to curve y² = 4x at the point (1, 2) and also find the tangent.Solution 3(xvi)

The equation of the given curve is y² = 4x . Differentiating with respect to x, we have: 

Question 3(xix)

Find the equations of the tangent and the normal to the given curves at the indicated points:

Solution 3(xix)

Question 4

Solution 4

Question 5(i)

Solution 5(i)

Question 5(ii)

Solution 5(ii)

From (A)

Equation of tangent is

Question 5(iii)

Solution 5(iii)

Question 5(iv)

Solution 5(iv)

Question 5(v)

Solution 5(v)

Question 5(vi)

Find the equations of the tangent and the normal to the following curves at the indicated points:

X = 3 cosθ – cos³θ , y = 3 sinθ – sin³θSolution 5(vi)

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 21

Find the equation of the tangents to the curve 3x² – y² = 8, which passes through the point (4/3, 0).Solution 21

Question 3(xvii)

Find the equations of the tangent and the normal to the following curves at the indicated points:

Solution 3(xvii)

Given equation curve is 

Differentiating w.r.t x, we get

Slope of tangent at  is

Slope of normal will be

Equation of tangent at  will be

Equation of normal at  is

Question 3(xviii)

Find the equations of the tangent and the normal to the following curves at the indicated points:

Solution 3(xviii)

Given equation curve is 

Differentiating w.r.t x, we get

Slope of tangent at  is

Slope of normal will be

Equation of tangent at  will be

Equation of normal at  is

Question 20

At what points will tangents to the curve  be parallel to x-axis? Also, find the equations of the tangents to the curve at these points.Solution 20

Given equation curve is 

Differentiating w.r.t x, we get

As tangent is parallel to x-axis, its slope will be m = 0

As this point lies on the curve, we can find y

Or

So, the points are (3, 6) and (2, 7).

Equation of tangent at (3, 6) is

y – 6 = 0 (x – 3)

y – 6 = 0

Equation of tangent at (2, 7) is

y – 7 = 0 (x – 2)

y – 7 = 0

Chapter 16 Tangents and Normals Ex. 16.3

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 1(v)

Find the angle of intersection of the folloing curves

Solution 1(v)

Question 1(vi)

Solution 1(vi)

Question 1(vii)

Solution 1(vii)

Question 1(viii)

Solution 1(viii)

Question 1 (ix)

Find the angle of intersection of the following curves:

Y = 4 – x² and y = x²Solution 1 (ix)

Question 2(i)

Solution 2(i)

Question 2(ii)

Solution 2(ii)

Question 2(iii)

Solution 2(iii)

Question 3(i)

Solution 3(i)

Question 3(ii)

Solution 3(ii)

Question 3(iii)

Solution 3(iii)

Question 4

Solution 4

Question 5

Solution 5

Question 6

Prove that the curves xy = 4 and x² + y² = 8 touch each other.Solution 6

Question 7

Prove that the curves y² = 4x and x² + y² – 6x + 1 = 0 touch each other at the point (1, 2)Solution 7

Question 8(i)

Solution 8(i)

Question 8(ii)

Solution 8(ii)

Question 9

Solution 9

Question 10

Solution 10

Chapter 15 Mean Value Theorems Ex 15.1

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 1(v)

Solution 1(v)

Question 1(vi)

Solution 1(vi)

Question 2(i)

Solution 2(i)

Question 2(iii)

Solution 2(iii)

Question 2(iv)

Solution 2(iv)

Question 2(v)

Solution 2(v)

Question 2(vi)

Solution 2(vi)

Question 2(vii)

Solution 2(vii)

Question 2(viii)

Solution 2(viii)

Question 3(i)

Solution 3(i)

Question 3(ii)

Solution 3(ii)

Question 3(iii)

Solution 3(iii)

Question 3(iv)

Solution 3(iv)

Question 3(v)

Solution 3(v)

Question 3(vi)

Solution 3(vi)

Question 3(vii)

Solution 3(vii)

Here,

Question 3(viii)

Solution 3(viii)

Question 3(ix)

Solution 3(ix)

Question 3(x)

Solution 3(x)

Question 3(xi)

Solution 3(xi)

Question 3(xii)

Solution 3(xii)

Question 3(xiii)

Solution 3(xiii)

Question 3(xiv)

Solution 3(xiv)

Question 3(xv)

Solution 3(xv)

Question 3(xvi)

Solution 3(xvi)

Question 3(xvii)

Solution 3(xvii)

Question 3(xviii)

Verify Rolle’s theorem for function f(x) = sin x – sin 2x on [0, pi] on the indicated intervals.Solution 3(xviii)

Question 7

Solution 7

x = 0 then y = 16

Therefore, the point on the curve is (0, 16) Question 8(i)

Solution 8(i)

x = 0, then y = 0

Therefore, the point is (0, 0)Question 8(ii)

Solution 8(ii)

Question 8(iii)

Solution 8(iii)

x = 1/2, then y = – 27

Therefore, the point is (1/2, – 27)Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 2(ii)

Verify Rolle’s theorem for each of the following functions on the indicated intervals:

Solution 2(ii)

Given function is 

As the given function is a polynomial, so it is continuous and differentiable everywhere.

Let’s find the extreme values

Therefore, f(2) = f(6).

So, Rolle’s theorem is applicable for f on [2, 6].

Let’s find the derivative of f(x)

Take f'(x) = 0

As 4 ∈ [2, 6] and f'(4) = 0.

Thus, Rolle’s theorem is verified.

Chapter 15 Mean Value Theorems Ex. 15.2

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 1(v)

Solution 1(v)

Question 1(vi)

Solution 1(vi)

Question 1(vii)

Solution 1(vii)

Question 1(viii)

Solution 1(viii)

Question 1(ix)

Solution 1(ix)

Question 1(x)

Solution 1(x)

Question 1(xi)

Solution 1(xi)

Question 1(xii)

Solution 1(xii)

Question 1(xiii)

Solution 1(xiii)

Question 1(xiv)

Solution 1(xiv)

Question 1(xv)

Solution 1(xv)

Question 1(xvi)

Solution 1(xvi)

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Chapter 14  Differentials, Errors and Approximations Ex 14.1

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9(ii)

Solution 9(ii)

Question 9(iii)

Solution 9(iii)

Question 9(iv)

Solution 9(iv)

Question 9(v)

Solution 9(v)

Question 9(vi)

Solution 9(vi)

Question 9(vii)

Solution 9(vii)

Question 9(viii)

Solution 9(viii)

Question 9(ix)

Solution 9(ix)

Question 9(x)

Using differentials, find the approximate values of the following:

log₁₀10.1, it being given that log₁₀e = 0.4343.Solution 9(x)

Question 9(xi)

Solution 9(xi)

Question 9(xii)

Solution 9(xii)

Question 9(xiii)

Solution 9(xiii)

Question 9(xiv)

Solution 9(xiv)

Question 9(xv)

Solution 9(xv)

Question 9(xvi)

Solution 9(xvi)

Question 9(xvii)

Solution 9(xvii)

Question 9(xviii)

Solution 9(xviii)

Question 9(xix)

Solution 9(xix)

Question 9(xx)

Solution 9(xx)

Question 9(xxi)

Solution 9(xxi)

Question 9(xxii)

Solution 9(xxii)

Question 9(xxiii)

Solution 9(xxiii)

Question 9(xxiv)

Solution 9(xxiv)

Question 9(xxv)

Solution 9(xxv)

Question 9(xxvi)

Using differentials, find the approximate values of the following:

Solution 9(xxvi)

Question 9(xxvii)

Using differentials, find the approximate values of (3.968)^3/2Solution 9(xxvii)

Question 9(xxviii)

Using differentials, find the approximate values of the following:

(1.999)⁵Solution 9(xxviii)

Question 9(xxix)

Using differentials, find the approximate values of the following:

Solution 9(xxix)

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 9(i)

Using differentials, find the approximate values of the following:

Solution 9(i)

Consider 

Let x = 25 and 

Also,

Chapter 13 Derivative as a Rate Measurer Ex 13.1

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Find the rate of change of the volume of a cone with respect to the radius of its base.Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

The money to be spent for the welfare of the employees of a firm is proportional to the rate of change of its total revenue (Marginal revenue). If the total revenue (in rupees) received from the sale of x units of a product is given by R(x) = 3x² + 36x + 5, find the marginal revenue, when x=5, and write which value does the question indicate.Solution 10

Chapter 13 Derivative as a Rate Measurer Ex. 13.2

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

The radius of a spherical soap bubble is increasing at the rate of 0.2 cm/sec. Find the rate of increase of its surface area, when the radius is 7 cm.Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16(i)

Find an angle θ, which increases twice as fast as its cosine.Solution 16(i)

Question 16(ii)

Find the angle θ

Whose rate of increase is twice the rate of decrease of its consine.Solution 16(ii)

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Chapter 12 Higher Order Derivatives Ex 12.1

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 1(iii)

Find the second order derivative of log(sinx).Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 1(v)

Solution 1(v)

Question 1(vi)

Solution 1(vi)

Question 1(vii)

Solution 1(vii)

Question 1(viii)

Solution 1(viii)

Question 1(ix)

Solution 1(ix)

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

If , find Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Question 38

Solution 38

Question 39

Solution 39

Question 40

Solution 40

Question 41

Solution 41

Question 42

Solution 42

Question 43

Solution 43

Question 44

Solution 44

Question 45

Solution 45

Question 46

Solution 46

Question 47

Solution 47

Question 49

Solution 49

Question 50

Solution 50

Question 51

Solution 51

Question 52

Solution 52

Question 53

Solution 53

Question 9

If  prove that  and  Solution 9

Given: 

Differentiating ‘x’ w.r.t  we get

Differentiating ‘y’ w.r.t  we get

Dividing (ii) by (i), we get

 … (iii)

Differentiating above equation w.r.t x, we get

Hence,  Question 48

If  find  Solution 48

Given: 

Differentiating ‘x’ w.r.t t, we get

Differentiating ‘y’ w.r.t t, we get

Dividing (ii) by (i), we get

Differentiating above equation w.r.t x, we get

Hence, 

Chapter 11 Differentiation Ex 11.1

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Differentiate f(x)=x²e^x from first principles.Solution 8

Question 9

Solution 9

Question 10

Solution 10

Chapter 11 Differentiation Exercise Ex. 11.2

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Question 38

Solution 38

Question 39

Solution 39

Question 40

Solution 40

Question 41

Solution 41

Question 42

Solution 42

Question 43

Solution 43

Question 44

Solution 44

Question 45

Solution 45

Question 46

Solution 46

Question 47

Solution 47

Question 48

Solution 48

Question 49

Solution 49

Question 50

Solution 50

Question 51

Solution 51

Question 52

Solution 52

Question 53

Differentiate the following functions with respect to x:

Solution 53

Question 54

Solution 54

Question 55

Solution 55

Question 56

Solution 56

Question 57

Solution 57

Question 58

Solution 58

Question 59

Solution 59

Question 60

Solution 60

Question 61

Solution 61

Question 63

Solution 63

Question 64

Solution 64

Question 65

Solution 65

Question 66

Solution 66

Question 67

Solution 67

Question 68

Solution 68

Question 69

Solution 69

Question 70

Solution 70

Question 71

Solution 71

Question 72

Solution 72

Question 73

Solution 73

Question 74

Solution 74

Question 62

If  prove that Solution 62

Given: 

Differentiating w.r.t x, we get

Hence,  Question 75

If  find Solution 75

Given: 

Question 76

If  then find Solution 76

Given: 

Chapter 11 Differentiation Exercise Ex. 11.3

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37(i)

Solution 37(i)

Question 37(ii)

Solution 37(ii)

Question 38

show that dy/dx is independent of x.Solution 38

Question 39

Solution 39

Question 40

Solution 40

Question 41

Solution 41

Question 42

Solution 42

Question 43

Solution 43

Question 44

Solution 44

Question 45

If y = tan^-1 Solution 45

Question 46

Solution 46

Question 47

Solution 47

Question 12

Differentiate the following function with respect to x:

Solution 12

Let 

Question 48

If  then find Solution 48

Given: ………. (i)

Let 

From (i), we get

Chapter 11 Differentiation Exercise Ex. 11.4

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

, 

.Solution 27

Question 28

Solution 28

Question 30

Solution 30

Question 31

Solution 31

Question 29

If find  at x =1, Solution 29

Given: 

Differentiating w.r.t x. we get

When x =1 and  we get

Chapter 11 Differentiation Exercise Ex. 11.5

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18(i)

Solution 18(i)

Question 18(ii)

Solution 18(ii)

Question 18(iii)

Solution 18(iii)

Question 18(iv)

Solution 18(iv)

Question 18(v)

Solution 18(v)

Question 18(vi)

Solution 18(vi)

Question 18(vii)

Solution 18(vii)

Question 18(viii)

Solution 18(viii)

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 29(i)

Solution 29(i)

Question 29(ii)

Solution 29(ii)

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 37

Solution 37

Question 38

Solution 38

Question 39

Solution 39

Question 40

Solution 40

Question 41

Solution 41

Question 42

Solution 42

Question 43

Solution 43

Question 44

Solution 44

Question 45

Solution 45

Question 46

Solution 46

Question 47

Solution 47

Question 48

Solution 48

Question 49

Solution 49

Question 50

Solution 50

Question 51

Solution 51

Question 52

Solution 52

Question 53

Solution 53

Question 54

Solution 54

Question 55

Solution 55

Question 56

Solution 56

Question 57

Solution 57

Question 58

Solution 58

Question 59

Solution 59

Question 60

Solution 60

Question 61

Solution 61

Question 28

Find  when Solution 28

Given: 

Let 

Differentiating ‘u’ w.r.t x, we get

Differentiating ‘v’ w.r.t x, we get

From (i), (ii) and (iii), we get

Question 62

If  find Solution 62

Given: 

Let 

Taking log on both the sides of equation (i), we get

Taking log on both the sides of equation (ii), we get

Differentiating (iii) w.r.t x, we get

Using (iv) and (v), we have

Chapter 11 Differentiation Exercise Ex. 11.6

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Chapter 11 Differentiation Exercise Ex. 11.7

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

If  find  when  Solution 29

Given: 

Differentiate ‘x’ w.r.t , we get

Differentiate ‘y’ w.r.t , we get

Dividing (ii) by (i), we get

At 

Chapter 11 Differentiation Exercise Ex. 11.8

Question 2

Solution 2

Question 3

Solution 3

Question 4(i)

Solution 4(i)

Question 4(ii)

Solution 4(ii)

Question 5(i)

Solution 5(i)

Question 5(ii)

Solution 5(ii)

Question 5(iii)

Solution 5(iii)

Question 6

Solution 6

Question 7(i)

Solution 7(i)

Question 7(ii)

Solution 7(ii)

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 1

Differentiate  with respect to Solution 1

We need to find 

Let 

So, we need to find 

Question 21

Differentiate  with respect to Solution 21

We need to find 

Let 

Differentiating ‘u’ and ‘v’ w.r.t x, we get

Dividing (i) by (ii), we get