Yearly Archives: 2022

Chapter 27 Direction Cosines and Direction Ratios Exercise Ex. 27.1

Question 1

If a line makes angles of 90°, 60° and 30° with the positive direction of x,y and z-axis respectively, find its direction cosines.Solution 1

Let l, m and n be the direction cosines of a line.

l = cos 90° = 0

Question 2

If a line has direction ratios 2, -1, -2, determine its direction cosines.Solution 2

Question 3

Find the direction cosines of the line passing through two points (-2, 4, -5) and (1, 2, 3).Solution 3

Question 4

Using direction ratios show that the points A (2, 3, -4), B (1, -2, 3) and C (3, 8, -11) are collinear.Solution 4

Question 5

Find the direction cosines of the sides of the triangle whose vertices are (3, 5, -4), (-1, 1, 2) and (-5, -5, -2)Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Find the acute angle between the lines whose direction ratios are proportional to 2:3:6 and 1:2:2.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)

Solution 16(i)

Question 16(ii)

Solution 16(ii)

Question 16(iii)

Solution 16(iii)

Question 16(iv)

Find the angle between the lines whose direction cosines are given by equations

2l + 2m – n = 0, mn + ln + lm = 0Solution 16(iv)

Chapter 26 Scalar Triple Product Exercise Ex. 26.1

Question 1(i)

Solution 1(i)

Question 1(ii)

Solution 1(ii)

Question 2(i)

Solution 2(i)

Question 2(ii)

Solution 2(ii)

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 4(i)

Solution 4(i)

Question 4(ii)

Solution 4(ii)

Question 4(iii)

Solution 4(iii)

Question 5(i)

Solution 5(i)

Question 5(ii)

Solution 5(ii)

Question 5(iii)

Solution 5(iii)

Question 5(iv)

Solution 5(iv)

Question 6

Solution 6

Question 7

Solution 7

Question 8

Show that the four points whose position vectors are  Coplanar.Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12(i)

Solution 12(i)

Question 12(ii)

Solution 12(ii)

Question 13

Solution 13

Chapter 25 Vector or Cross Product Exercise Ex. 25.1

Question 1

Solution 1

Question 2(i)

Solution 2(i)

Question 2(ii)

Solution 2(ii)

Question 3(i)

Solution 3(i)

Question 3(ii)

Solution 3(ii)

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7(i)

Solution 7(i)

Question 7(ii)

Solution 7(ii)

Question 8(i)

Solution 8(i)

Question 8(ii)

Solution 8(ii)

Question 8(iii)

Solution 8(iii)

Question 8(iv)

Solution 8(iv)

Question 9(i)

Solution 9(i)

Question 9(ii)

Solution 9(ii)

Question 9(iii)

Solution 9(iii)

Question 9(iv)

Solution 9(iv)

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(i)

Solution 27(i)

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

Using Vectors, find the area of the triangle with vertices: (i) A (1, 1, 2), B (2, 3, 5) and C (1, 5, 5) (ii) A (1, 2, 3), B(2, -1, 4) and C (4, 5, -1).Solution 34

Question 35

Solution 35

Chapter 24 Scalar Or Dot Product Exercise Ex. 24.1

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5 (i)

Solution 5 (i)

Question 5 (ii)

Solution 5 (ii)

Question 5 (iii)

Solution 5 (iii)

Question 5 (iv)

Solution 5 (iv)

Question 5 (v)

Solution 5 (v)

Question 6

Solution 6

Question 7(i)

Solution 7(i)

Question 7(ii)

Solution 7(ii)

Question 8 (i)

Solution 8 (i)

Question 8 (ii)

Solution 8 (ii)

Question 9

Solution 9

Question 10

Solution 10

Given that  are mutually perpendicular, so,

Question 11

Solution 11

Question 12

Show that the vector  is equally inclined with the coordinate axes.Solution 12

Question 13

Show that the vectors  are mutually perpendicualr unit vectors.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 two vector  are such that , then find the value of (3a – 5b) . (2a + 7b).Solution 29

Question 30(i)

Solution 30(i)

Question 30(ii)

Solution 30(ii)

Question 31(i)

Solution 31(i)

Question 31(ii)

Solution 31(ii)

Question 31(iii)

Solution 31(iii)

Question 32(i)

Solution 32(i)

Question 32(ii)

Solution 32(ii)

Question 32(iii)

Solution 32(iii)

Question 33(i)

Solution 33(i)

Question 33(ii)

Solution 33(ii)

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

Chapter 24 Scalar Or Dot Product Exercise Ex. 24.2

Question 1

Solution 1

Question 2

Prove that: If the diagonals of a quadrilateral bisect each other at right angles, then it is a rhombus.Solution 2

Question 3

(Pythagoras’s Theorem) Prove by vector method that in a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.Solution 3

Question 4

Prove by vector method that the sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of its sides.Solution 4

Question 5

Prove using vectors: The quadrilateral obtained by joining mid-points of adjacent sides of a rectangle is a rhombus.Solution 5

Question 6

Prove that the diagonals of a rhombus are perpendicular bisectors of each other.Solution 6

Question 7

Prove that the diagonals of a rectangle are perpendicular if and only if the rectangle is a square.Solution 7

Question 8

If AD is the median of D ABC, using vectors, prove that

AB² + AC² = 2 (AD² + CD²).Solution 8

Question 9

If the median to the base of a triangle is perpendicular to the base, then triangle is isosceles.Solution 9

Question 10

In a quadrilateral ABCD, prove that AB² + BC² + CD²+ DA² = AC² + BD² + 4 PQ², where P and Q are middle points of diagonals AC and BD. Solution 10

Chapter 23 Algebra of Vectors Exercise Ex. 23.1

Question 1(i)

Represent graphically a dispacement of 40 km, 30° east of north.Solution 1(i)

Question 1(ii)

Represent graphically a displacement of 50 km, south-eastSolution 1(ii)

Here, vector  represents the displacement of 50 km, south-east.Question 1(iii)

Represent graphically a displacement of 70 km, 40° north of west.Solution 1(iii)

Here, vector  represents the displacement of 70 km, 40° north of west.Question 2

Classify the following measures as scalars and vectors.

(i) 15 kg

(ii) 20 kg weight

(iii) 45°

(iv) 10 metres south-east

(v) 50 m/s²Solution 2

(i) 15 kg is a scalar quantity because it involves only

(ii) 20 kg weight is a vector quantity as it involves both magnitude and direction.

(iii) 45° is a scalar quantity as it involves only magnitude.

(iv) 10 metres south-east is a vector quantity as it involve direction.

(v) 50 m/s² is a scalar quantity as it involves magnitude of acceleration.Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Chapter 23 Algebra of Vectors Exercise Ex. 23.2

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

Chapter 23 Algebra of Vectors Exercise Ex. 23.3

Question 1

Find the position vector of a point R which divides the line joining the two points P and Q with position vectors

Solution 1

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Chapter 23 – Algebra of Vectors Exercise Ex. 23.4

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Prove by vector method that the internal bisectors of the angles of a triangle are concurrent.Solution 6

Chapter 23 – Algebra of Vectors Exercise Ex. 23.5

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Find |AB| in each case.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 12

Solution 12

Chapter 23 Algebra of Vectors Exercise Ex. 23.6

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

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

Chapter 23 Algebra of Vectors Exercise Ex. 23.7

Question 1

Solution 1

Question 2 (i)

Solution 2 (i)

Question 2 (ii)

Solution 2 (ii)

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

Using vectors, find the value of λ such that the points

 (λ, – 10, 3), (1 -1, 3) and (3, 5, 3) are collinear.Solution 13

Chapter 23 Algebra of Vectors Exercise Ex. 23.8

Question 1

Solution 1

Question 2 (i)

Solution 2 (i)

Question 2 (ii)

Solution 2 (ii)

Question 2 (iii)

Solution 2 (iii)

Question 2 (iv)

Solution 2 (iv)

Question 3 (i)

Solution 3 (i)

Question 3 (ii)

Solution 3 (ii)

Question 4

Solution 4

Question 5 (i)

Solution 5 (i)

Question 5 (ii)

Solution 5 (ii)

Question 6 (i)

Solution 6 (i)

Question 6 (ii)

Solution 6 (ii)

Question 7 (i)

Solution 7 (i)

Question 7 (ii)

Solution 7 (ii)

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Chapter 23 Algebra of Vectors Exercise Ex. 23.9

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 (i)

Solution 7 (i)

Question 7 (ii)

Solution 7 (ii)

Question 7 (iii)

Solution 7 (iii)

Question 8

Solution 8

Question 9

Solution 9

Question 10

If a unit vector  makes and angles  and an acute angle θ with , then find θ and hence, the components of .Solution 10

Question 11

Solution 11

Question 12

Solution 12

Chapter 22 Differential Equations Exercise Ex. 22.1

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 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Determine the order and degree of the following differential equations. State also whether they are linear or non-linear.

Solution 27

The order of a differential equation is the order of the highest order derivative appearing in the equation.

The degree of a differential equation is the degree of the highest order derivative.

Consider the given differential equation

In the above equation, the order of the highest order derivative is 1.

So the differential equation is of order 1.

In the above differential equation, the power of the highest order derivative is 3.

Hence, it is a differential equation of degree 3.

Since the degree of the above differential equation is 3, more than one, it is a non-linear differential equation.

Chapter 22 – Differential Equations Exercise Ex. 22.2

Question 1

Solution 1

Question 2

Solution 2

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 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

Form the differential equation having y = (sin^-1x)² + A cos ^-1 x + B, where A and B are arbitrary constants, as its general solution.Solution 14

Question 15(i)

Solution 15(i)

Question 15(ii)

Solution 15(ii)

Question 15(iii)

Solution 15(iii)

Question 16(i)

Solution 16(i)

Question 16(ii)

Solution 16(ii)

Question 16(iii)

Solution 16(iii)

Question 16(iv)

Represent the following family of curves by forming the corresponding differential equation (a,b being parameters):

x² + (y – b)² = 1Solution 16(iv)

Question 16(v)

Solution 16(v)

Question 16(vi)

Solution 16(vi)

Question 16(vii)

Solution 16(vii)

Question 16(viii)

Solution 16(viii)

Question 16(ix)

Solution 16(ix)

Question 16(x)

Solution 16(x)

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Chapter 22 – Differential Equations Exercise Ex. 22.3

Question 1

Solution 1

Question 2

Solution 2

Question 3

show that y = ae²x + be^-x is a solution of the differential equation Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Verify that y =  + b is a solution of the differential equation Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Show that y = e^x(A cos x + B sin x) is the solution of the differential equation

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

Show that y = e^-x + ax + b is solution of the differential equation Solution 20

Question 21(i)

For the following differential equation verify that the accompanying function is a solution in the mentioned domain (a, b are parameters) Solution 21(i)

Question 21(ii)

Solution 21(ii)

Question 21(iii)

Solution 21(iii)

Question 21(iv)

Solution 21(iv)

Question 21(v)

Solution 21(v)

Chapter 22 – Differential Equations Exercise Ex. 22.4

Question 1

Solution 1

Question 2

Solution 2

Question 3

For the following initial value problem verify that the accompanying function is a solution:

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 22 – Differential Equations Exercise Ex. 22.5

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

Solve the following differential equation:

(sin x + cos x)dy + (cos x – sin x) dx = 0Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solve the following differential equation:

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solve the following differential equation

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

solve the following differential equation

Solution 26

Chapter 22 – Differential Equations Exercise Ex. 22.6

Question 1

Solve the following differential equation:

Solution 1

Question 2

Solve the following differential equation:

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Chapter 22 – Differential Equations Exercise Ex. 22.7

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

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solve the following differential equation:

Solution 35

Question 36

Solve the following differential equation:

Solution 36

Question 37(i)

Solution 37(i)

Question 37(ii)

Solve the following differential equation:

Solution 37(ii)

Question 38(i)

Solution 38(i)

Question 38(ii)

Solution 38(ii)

Question 38(iii)

ye^x/y dx = (xe^x/y + y²) dy, y ¹ 0Solution 38(iii)

Question 38(iv)

(1 + y²) tan^-1 x dx + 2y (1 + x²)dy = 0Solution 38(iv)

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(i)

Solution 45(i)

Question 45(ii)

Solution 45(ii)

Question 45(iii)

Solution 45(iii)

Question 45(iv)

Solution 45(iv)

Question 45(v)

Solution 45(v)

Question 45(vi)

Solve the following initial value problem

=1 + x² + y² + x²y², y(0) = 1Solution 45(vi)

Question 45(vii)

Solve the following initial value problem

Solution 45(vii)

Question 45(viii)

Solution 45(viii)

Question 45(ix)

Solution 45(ix)

Question 46

Solution 46

Question 47

Solution 47

Question 48

Solution 48

Question 49

Find the particular solution of e= x + 1, given that y = 3 when x = 0.Solution 49

Question 50

Solution 50

Question 51

Solution 51

Question 52

Find the equation of a curve passing through the point (0,0) and whose differential equation is Solution 52

Question 53

Solution 53

Question 54

The volume of spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6 units. Find the radius of balloon after after t seconds.Solution 54

Question 55

in a bank,principal increases continuously at the rate of r% per  year. Find The value of r if Rs 100 doubles itself in 10 years (log_e 2 = 0.6931).Solution 55

Let p, t and represent the principal, time, and rate of interest respectively.

It is given that the principal increases continuously at the rate of r% per year.

Integrating both side, we get:

Question 56

Solution 56

Question 57

Solution 57

..Question 58

Solution 58

Chapter 22 – Differential Equations Exercise Ex. 22.8

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

Solve the following differential equation.

Solution 11

Chapter 22 – Differential Equations Exercise Ex. 22.9

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solve the following differential equation:

Solution 4

Question 5

Solve the following differential equation:

Solution 5

Question 6

Solve the following initial value problem

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

Solve the following initial value problem

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solve the following initial value poblem

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

Solve the following differential equation:

Solution 35

Question 36(i)

Solution 36(i)

Question 36(ii)

Solution 36(ii)

Question 36(iii)

Solve the following initial value problem

Solution 36(iii)

Question 36(iv)

Solution 36(iv)

Question 36(v)

Solution 36(v)

Question 36(vi)

Solution 36(vi)

Question 36(vii)

Solution 36(vii)

Question 36(viii)

Solution 36(viii)

Question 36(ix)

Solve the following initial value problem

Solution 36(ix)

Question 37

Solution 37

Question 38

Solution 38

Question 39

Solution 39

Chapter 22 – Differential Equations Exercise Ex. 22.10

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

Solve the following differential equation:

Solution 30

Question 31

Solve the following differential equation:

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solution 35

Question 36(i)

Solution 36(i)

Question 36(ii)

Solution 36(ii)

Question 36(iii)

Solution 36(iii)

Question 36(iv)

Solution 36(iv)

Question 36(v)

Solution 36(v)

Question 36(vi)

Solution 36(vi)

Question 36(vii)

Solution 36(vii)

Question 36(viii)

Solution 36(viii)

Question 36(ix)

Solution 36(ix)

Question 36(x)

Solution 36(x)

Question 36(xi)

Solution 36(xi)

Question 36(xii)

Solution 36(xii)

Question 37(i)

Solution 37(i)

Question 37(ii)

Solution 37(ii)

Question 37(iii)

Solution 37(iii)

Question 37(iv)

Solution 37(iv)

Question 37(v)

Solve the following initial value problem:

Solution 37(v)

Question 37(vi)

Solution 37(vi)

Question 37(vii)

Solution 37(vii)

Question 37(viii)

Solve the following initial value problem

Solution 37(viii)

Question 37(ix)

Solution 37(ix)

Question 37(x)

Solution 37(x)

Question 37(xi)

Solution 37(xi)

Question 37(xii)

dy = cos x (2 – y cosec x) dxSolution 37(xii)

Question 38

Solution 38

Question 39

Solution 39

Question 40

Solve the differential equation

Solution 40

Question 41

Solution 41

Chapter 22 – Differential Equations Exercise Ex. 22.11

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

Chapter 22 – Differential Equations Exercise Ex. 22RE

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 2

Solution 2

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 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

Solve the following differential equation:

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 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(i)

Solution 64(i)

Question 64(ii)

Solution 64(ii)

Question 64(iii)

Solution 64(iii)

Question 64(iv)

Solution 64(iv)

Question 64(v)

Solution 64(v)

Question 64(vi)

Solution 64(vi)

Question 65(i)

Solution 65(i)

Question 65(ii)

Solution 65(ii)

Question 65(iii)

Solution 65(iii)

Question 66(i)

Solution 66(i)

Question 66(ii)

Solution 66(ii)

Question 66(iii)

Solution 66(iii)

Question 66(iv)

Solution 66(iv)

Question 66(v)

Solution 66(v)

Question 66(vi)

Solution 66(vi)

Question 66(vii)

Solution 66(vii)

Question 66(viii)

Solution 66(viii)

Question 66(ix)

Solution 66(ix)

Question 66(x)

Solution 66(x)

Question 66(xi)

Solution 66(xi)

Question 66(xii)

Solution 66(xii)

Question 66(xiii)

Solution 66(xiii)

Question 66(xiv)

Solution 66(xiv)

Question 66(xv)

Solution 66(xv)

Question 67(i)

Solution 67(i)

Question 67(ii)

Solution 67(ii)

Question 67(iii)

Solution 67(iii)

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 75

Solution 75

Question 76

Solution 76

Question 77

Solution 77

Question 78

Solution 78

Question 79

Solution 79

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