Канал: s priya maths

10thmaths// unit -3// algebra//Matrices introduction

10thmaths// unit -3// algebra//Matrices introduction

11thmaths// unit 2 // basic algebra// example 2.1 to 2.3

11thmaths// unit 2 // basic algebra// example 2.1 to 2.3

11thmaths//unit-2//Basic algebra//exercise 2.1(full)

11thmaths//unit-2//Basic algebra//exercise 2.1(full)

10thmaths//Graph of variation//exercise 3.15(5)

10thmaths//Graph of variation//exercise 3.15(5)

10thmaths//unit-3//Graph of variations//exercise 3.15(6)

10thmaths//unit-3//Graph of variations//exercise 3.15(6)

11thmaths//unit2// basic algebra//theorem 2.1

11thmaths//unit2// basic algebra//theorem 2.1

11thmaths//unit2//basic algebra// introduction

11thmaths//unit2//basic algebra// introduction

12thmaths//unit -7//exercise 7.6(2(i))

12thmaths//unit -7//exercise 7.6(2(i))

12thmaths// unit -7// monotonicity of functions introduction

12thmaths// unit -7// monotonicity of functions introduction

11thmaths//vector algebra//exercise 8.2 (11)

11thmaths//vector algebra//exercise 8.2 (11)

12thmaths//complex numbers// example 2.1

12thmaths//complex numbers// example 2.1

12thmaths//complex numbers//introduction

12thmaths//complex numbers//introduction

12thmaths//Discrete mathematics//mathematical logic introduction

12thmaths//Discrete mathematics//mathematical logic introduction

12thmaths//applications of vector algebra//exercise 6.7(1)

12thmaths//applications of vector algebra//exercise 6.7(1)

12thmaths//applications of vector algebra//exercise 6.7(formulas)

12thmaths//applications of vector algebra//exercise 6.7(formulas)

10thmaths//Geometry//unit exercise-4 (4)

10thmaths//Geometry//unit exercise-4 (4)

10thmaths//Geometry//unit exercise-4 (3)

10thmaths//Geometry//unit exercise-4 (3)

10thmaths//Geometry//unit exercise -4 (2)

10thmaths//Geometry//unit exercise -4 (2)

10thmaths//Geometry//unit exercise-4(1)

10thmaths//Geometry//unit exercise-4(1)

12thmaths//Theory of equations//example 3.2

12thmaths//Theory of equations//example 3.2

12thmaths//Theory of equations//example 3.1

12thmaths//Theory of equations//example 3.1

12thmaths// Theory of equations//construct a quadratic equations of given roots

12thmaths// Theory of equations//construct a quadratic equations of given roots

10thmaths//algebra//exercise 3.2(2(iv,v,vi))

10thmaths//algebra//exercise 3.2(2(iv,v,vi))

10thmaths// algebra// exercise 3.2(2(i,ii,iii))

10thmaths// algebra// exercise 3.2(2(i,ii,iii))

12thmaths//probability distribution//exercise 11.2(2)

12thmaths//probability distribution//exercise 11.2(2)

12thmaths//probability distribution//types of random variables

12thmaths//probability distribution//types of random variables

10thmaths//practical Geometry//construction of triangles//exercise 4.2(15)

10thmaths//practical Geometry//construction of triangles//exercise 4.2(15)

10thmaths//practical Geometry//construction of triangles// exercise 4.2(12)

10thmaths//practical Geometry//construction of triangles// exercise 4.2(12)

12thmaths//applications of vector algebra//revision (part-2)

12thmaths//applications of vector algebra//revision (part-2)

12thmaths//applications of vector algebra//revision (part-1)

12thmaths//applications of vector algebra//revision (part-1)

10thmaths//algebra//exercise 3.14(4)

10thmaths//algebra//exercise 3.14(4)

10thmaths// Algebra//nature of roots introduction

10thmaths// Algebra//nature of roots introduction

10thmaths//algebra //exercise 3.2(1(iii))

10thmaths//algebra //exercise 3.2(1(iii))

10thmaths//algebra//exercise 3.2(1(ii))

10thmaths//algebra//exercise 3.2(1(ii))

10thmaths//construction of triangles//exercise 4.2(13)

10thmaths//construction of triangles//exercise 4.2(13)

10thmaths//practical Geometry//exercise 4.2(11)//construction of triangles

10thmaths//practical Geometry//exercise 4.2(11)//construction of triangles

10thmaths//Algebra//GCDand LCM of polynomials//exercise 3.2(1(i))

10thmaths//Algebra//GCDand LCM of polynomials//exercise 3.2(1(i))

12thmaths//probability distribution//exercise 11.2(1)

12thmaths//probability distribution//exercise 11.2(1)

12thmaths//probability distribution//probability mass function introduction

12thmaths//probability distribution//probability mass function introduction

12thmaths//probability distribution//exercise 11.1 (5)

12thmaths//probability distribution//exercise 11.1 (5)

12thmaths//probability distribution//exercise 11.1(4)

12thmaths//probability distribution//exercise 11.1(4)

12thmaths//probability distribution//exercise 11.1(3)

12thmaths//probability distribution//exercise 11.1(3)

12thmaths//probability distribution//exercise 11.1(2)

12thmaths//probability distribution//exercise 11.1(2)

10thmaths//quadratic graphs//example 3.52

10thmaths//quadratic graphs//example 3.52

11thmaths//vector algebra//introduction

11thmaths//vector algebra//introduction

10thmaths//co-ordinate geometry// slope -introduction

10thmaths//co-ordinate geometry// slope -introduction

10thmaths//co-ordinate geometry//exercise 5.2(9(ii))

10thmaths//co-ordinate geometry//exercise 5.2(9(ii))

10thmaths//co-ordinate geometry //exercise 5.2(9(i))

10thmaths//co-ordinate geometry //exercise 5.2(9(i))

12thmaths//probability distribution//example 11.1

12thmaths//probability distribution//example 11.1

12thmaths//probability distribution//exercise 11.1 (1)

12thmaths//probability distribution//exercise 11.1 (1)

12thmaths//probability distribution//random variable introduction

12thmaths//probability distribution//random variable introduction

10thmaths//Mensuration//exercise 7.1 (1)

10thmaths//Mensuration//exercise 7.1 (1)

10thmaths//Mensuration//exercise 7.1(6)

10thmaths//Mensuration//exercise 7.1(6)

10thmaths//Mensuration//exercise 7.1(7)

10thmaths//Mensuration//exercise 7.1(7)

10thmaths//mensuration//formulae

10thmaths//mensuration//formulae

12thmaths//applications of matrices and determinants// JEE question

12thmaths//applications of matrices and determinants// JEE question

11thmaths//matrices and determinants//JEE question

11thmaths//matrices and determinants//JEE question

12thmaths ||Applications of vector algebra ||exercise 6.4(9)

12thmaths ||Applications of vector algebra ||exercise 6.4(9)

12thmaths ||Applications of vector algebra ||exercise 6.4 (8)

12thmaths ||Applications of vector algebra ||exercise 6.4 (8)

12thmaths || Applications of vector algebra ||exercise 6.4(7)

12thmaths || Applications of vector algebra ||exercise 6.4(7)

12thmaths||Applications of vector algebra || exercise 6.4(6)

12thmaths||Applications of vector algebra || exercise 6.4(6)

12thmaths ||Applications of vector algebra ||exercise 6.4(5(i))

12thmaths ||Applications of vector algebra ||exercise 6.4(5(i))

12thmaths ||Applications of vector algebra ||exercise 6.4(5(ii))

12thmaths ||Applications of vector algebra ||exercise 6.4(5(ii))

12thmaths ||Applications of vector algebra ||exercise 6.4(5(iii))

12thmaths ||Applications of vector algebra ||exercise 6.4(5(iii))

12thmaths ||Applications of vector algebra || Angle between two straight lines formula

12thmaths ||Applications of vector algebra || Angle between two straight lines formula

12thmaths ||Applications of vector algebra ||exercise 6.4(1)

12thmaths ||Applications of vector algebra ||exercise 6.4(1)

12thmaths ||Applications of vector algebra ||exercise 6.4(2)

12thmaths ||Applications of vector algebra ||exercise 6.4(2)

12thmaths ||Applications of vector algebra ||exercise 6.4(4)

12thmaths ||Applications of vector algebra ||exercise 6.4(4)

12thmaths ||Applications of vector algebra ||exercise 6.4(3)

12thmaths ||Applications of vector algebra ||exercise 6.4(3)

12thmaths ||Applications of vector algebra ||exercise 6.3 (7)

12thmaths ||Applications of vector algebra ||exercise 6.3 (7)

12thmaths ||Applications of vector algebra ||exercise 6.3 (6)

12thmaths ||Applications of vector algebra ||exercise 6.3 (6)

12thmaths ||Applications of vector algebra ||exercise 6.3(8)

12thmaths ||Applications of vector algebra ||exercise 6.3(8)

12thmaths ||Applications of vector algebra ||different forms of equations of a straightline

12thmaths ||Applications of vector algebra ||different forms of equations of a straightline

12thmaths ||Applications of vector algebra ||exercise 6.3 (5)

12thmaths ||Applications of vector algebra ||exercise 6.3 (5)

12thmaths ||Applications of vector algebra ||exercise 6.3 (4(ii))

12thmaths ||Applications of vector algebra ||exercise 6.3 (4(ii))

12thmaths ||Applications of vector algebra ||exercise 6.3 (4(i))

12thmaths ||Applications of vector algebra ||exercise 6.3 (4(i))

12thmaths ||Applications of vector algebra ||exercise 6.3(3)

12thmaths ||Applications of vector algebra ||exercise 6.3(3)

10thmaths ||Geometry ||exercise 4.4(11)||construction of a tangent using centre

10thmaths ||Geometry ||exercise 4.4(11)||construction of a tangent using centre

10thmaths ||Geometry ||exercise 4.4(12)||construction of a tangent using alternate segment theorem

10thmaths ||Geometry ||exercise 4.4(12)||construction of a tangent using alternate segment theorem

10thmaths ||Geometry ||construction of tangents (type -lll)||exercise4.4(13)

10thmaths ||Geometry ||construction of tangents (type -lll)||exercise4.4(13)

12thmaths ||Applications of vector algebra ||exercise 6.3(2)

12thmaths ||Applications of vector algebra ||exercise 6.3(2)

12thmaths ||Applications of vector algebra ||exercise 6.3(1(i,ii))

12thmaths ||Applications of vector algebra ||exercise 6.3(1(i,ii))

12thmaths || Applications of vector algebra ||vector triple product introduction

12thmaths || Applications of vector algebra ||vector triple product introduction

10thmaths ||Geometry ||construction of similar triangles ||example 4.11(type -II)

10thmaths ||Geometry ||construction of similar triangles ||example 4.11(type -II)

10thmaths ||Geometry ||construction of similar triangles ||example 4.10(type I)

10thmaths ||Geometry ||construction of similar triangles ||example 4.10(type I)

12thmaths ||Applications of vector algebra ||exercise 6.2(9)

12thmaths ||Applications of vector algebra ||exercise 6.2(9)

12thmaths ||Applications of vector algebra ||exercise 6.2(10)

12thmaths ||Applications of vector algebra ||exercise 6.2(10)

12thmaths ||Applications of vector algebra ||example 6.6(appollonius theorem)

12thmaths ||Applications of vector algebra ||example 6.6(appollonius theorem)

10thmaths ||Relations and functions ||unit exercise-1 (9)

10thmaths ||Relations and functions ||unit exercise-1 (9)

10thmaths ||Relations and functions ||unit exercise-1 (8)

10thmaths ||Relations and functions ||unit exercise-1 (8)

10thmaths ||Relations and functions ||unit exercise-1 (7)

10thmaths ||Relations and functions ||unit exercise-1 (7)

10thmaths ||Relations and functions ||unit exercise-1 (6)

10thmaths ||Relations and functions ||unit exercise-1 (6)

10thmaths ||Relations and functions ||unit exercise-1 (10)

10thmaths ||Relations and functions ||unit exercise-1 (10)

10thmaths ||Relations and functions ||unit exercise-1 (2)

10thmaths ||Relations and functions ||unit exercise-1 (2)

10thmaths || Relations and functions ||unit exercise-1 (3)

10thmaths || Relations and functions ||unit exercise-1 (3)

10thmaths ||Relations and functions ||unit exercise-1(4)

10thmaths ||Relations and functions ||unit exercise-1(4)

10thmaths ||Relations and functions ||unit exercise-1 (5)

10thmaths ||Relations and functions ||unit exercise-1 (5)

10thmaths ||Relations and functions ||unit exercise -1(1)

10thmaths ||Relations and functions ||unit exercise -1(1)

10thmaths ||algebra ||exercise 3.16(7) (graph)

10thmaths ||algebra ||exercise 3.16(7) (graph)

10thmaths ||algebra ||exercise 3.16(6)(graph)

10thmaths ||algebra ||exercise 3.16(6)(graph)

10thmaths ||Relations and functions ||exercise 1.5 (10)

10thmaths ||Relations and functions ||exercise 1.5 (10)

10thmaths ||Relations and functions ||exercise 1.5(9)

10thmaths ||Relations and functions ||exercise 1.5(9)

10thmaths|| Relations and functions ||exercise 1.5 (7)

10thmaths|| Relations and functions ||exercise 1.5 (7)

10thmaths ||Relations and functions ||exercise 1.5(8(ii))

10thmaths ||Relations and functions ||exercise 1.5(8(ii))

10thmaths ||Relations and functions ||exercise 1.5(8(i))

10thmaths ||Relations and functions ||exercise 1.5(8(i))

10thmaths ||Relations and functions ||exercise 1.5(8(iii))

10thmaths ||Relations and functions ||exercise 1.5(8(iii))

10thmaths ||algebra ||nature of roots (graphs)

10thmaths ||algebra ||nature of roots (graphs)

10thmaths ||algebra|| quadratic graphs||example 3.51

10thmaths ||algebra|| quadratic graphs||example 3.51

10thmaths ||Relations and functions ||exercise 1.5(5)

10thmaths ||Relations and functions ||exercise 1.5(5)

10thmaths|| Relations and functions ||exercise 1.5(6)

10thmaths|| Relations and functions ||exercise 1.5(6)

10thmaths ||Relations and functions || exercise 1.5 (4)

10thmaths ||Relations and functions || exercise 1.5 (4)

10thmaths ||Relations and functions ||exercise 1.5 (3)

10thmaths ||Relations and functions ||exercise 1.5 (3)

11thmaths||sets, Relations and functions ||example 1.3

11thmaths||sets, Relations and functions ||example 1.3

10thmaths ||Relations and functions ||special types of functions with graph

10thmaths ||Relations and functions ||special types of functions with graph

10thmaths ||Relations and functions ||exercise 1.5(1(i,ii))

10thmaths ||Relations and functions ||exercise 1.5(1(i,ii))

10thmaths ||Relations and functions ||exercise 1.5(2(i,ii))

10thmaths ||Relations and functions ||exercise 1.5(2(i,ii))

10thmaths ||Relations and functions || exercise1.5(1(iii,iv,v))

10thmaths ||Relations and functions || exercise1.5(1(iii,iv,v))

10thmaths ||Relations and functions ||example 1.24

10thmaths ||Relations and functions ||example 1.24

10thmaths ||Relations and functions ||example 1.23

10thmaths ||Relations and functions ||example 1.23

10thmaths ||Relations and functions ||example 1.22

10thmaths ||Relations and functions ||example 1.22

10thmaths ||Relations and functions ||example 1.20

10thmaths ||Relations and functions ||example 1.20

10thmaths ||Relations and functions ||example 1.21

10thmaths ||Relations and functions ||example 1.21

10thmaths ||Relations and functions ||example 1.19

10thmaths ||Relations and functions ||example 1.19

10thmaths ||Relations and functions ||composition of functions introduction

10thmaths ||Relations and functions ||composition of functions introduction

10thmaths ||Relations and functions ||exercise 1.4(9)

10thmaths ||Relations and functions ||exercise 1.4(9)

10thmaths ||Relations and functions ||exercise 1.4(10)

10thmaths ||Relations and functions ||exercise 1.4(10)

10thmaths ||Relations and functions ||example 1.18

10thmaths ||Relations and functions ||example 1.18

10thmaths ||Relations and functions ||exercise 1.4(8)

10thmaths ||Relations and functions ||exercise 1.4(8)

10thmaths||Relations and functions ||exercise 1.4(7(ii))

10thmaths||Relations and functions ||exercise 1.4(7(ii))

10thmaths ||Relations and functions ||exercise 1.4(11)

10thmaths ||Relations and functions ||exercise 1.4(11)

10thmaths ||Relations and functions ||exercise 1.4(12)

10thmaths ||Relations and functions ||exercise 1.4(12)

10thmaths ||Relations and functions ||exercise 1.4(7(i))

10thmaths ||Relations and functions ||exercise 1.4(7(i))

10thmaths ||Relations and functions ||exercise 1.4(5)

10thmaths ||Relations and functions ||exercise 1.4(5)

10thmaths ||Relations and functions ||example 1.17

10thmaths ||Relations and functions ||example 1.17

10thmaths ||Relations and functions ||exercise 1.4(6)

10thmaths ||Relations and functions ||exercise 1.4(6)

10thmaths ||Relations and functions || special cases of functions

10thmaths ||Relations and functions || special cases of functions

10thmaths ||Relations and functions ||exercise 1.4(4)

10thmaths ||Relations and functions ||exercise 1.4(4)

10thmaths ||Relations and functions ||example 1.16

10thmaths ||Relations and functions ||example 1.16

10thmaths ||Relations and functions ||example 1.14

10thmaths ||Relations and functions ||example 1.14

10thmaths ||Relations and functions ||example 1.12

10thmaths ||Relations and functions ||example 1.12

10thmaths ||Relations and functions ||example 1.13

10thmaths ||Relations and functions ||example 1.13

10thmaths ||Relations and functions ||example 1.15

10thmaths ||Relations and functions ||example 1.15

10thmaths ||Relations and functions || Types of functions and horizontal line test

10thmaths ||Relations and functions || Types of functions and horizontal line test

10thmaths ||Relations and functions ||exercise 1.4(3)

10thmaths ||Relations and functions ||exercise 1.4(3)

10thmaths ||Relations and functions ||example1.10

10thmaths ||Relations and functions ||example1.10

10thmaths ||Relations and functions ||exercise 1.4(1)

10thmaths ||Relations and functions ||exercise 1.4(1)

10thmaths ||Relations and functions ||example 1.11

10thmaths ||Relations and functions ||example 1.11

10thmaths ||Relations and functions ||exercise 1.4(2)

10thmaths ||Relations and functions ||exercise 1.4(2)

10thmaths ||Relations and functions || Representation of function and vertical line test

10thmaths ||Relations and functions || Representation of function and vertical line test

10thmaths ||Relations and functions ||exercise 1.3(9)

10thmaths ||Relations and functions ||exercise 1.3(9)

10thmaths ||Relations and functions ||exercise 1.3(10)

10thmaths ||Relations and functions ||exercise 1.3(10)

10thmaths ||Relations and functions ||exercise 1.3(8)

10thmaths ||Relations and functions ||exercise 1.3(8)

10thmaths ||Relations and functions ||exercise 1.3(6)

10thmaths ||Relations and functions ||exercise 1.3(6)

10thmaths ||Relations and functions ||exercise 1.3(7)

10thmaths ||Relations and functions ||exercise 1.3(7)

10thmaths ||Relations and functions ||exercise 1.3(5)

10thmaths ||Relations and functions ||exercise 1.3(5)

10thmaths ||Relations and functions ||exercise 1.3(3)

10thmaths ||Relations and functions ||exercise 1.3(3)

10thmaths ||Relations and functions ||exercise 1.3(2)

10thmaths ||Relations and functions ||exercise 1.3(2)

10thmaths ||Relations and functions ||exercise 1.3(1)

10thmaths ||Relations and functions ||exercise 1.3(1)

10thmaths ||Relations and functions ||exercise 1.3(4)

10thmaths ||Relations and functions ||exercise 1.3(4)

12thmaths ||Applications of vector algebra || example 6.2

12thmaths ||Applications of vector algebra || example 6.2

12thmaths ||applications of vector algebra ||example 6.4

12thmaths ||applications of vector algebra ||example 6.4

12thmaths ||applications of vector algebra|| example 6.1

12thmaths ||applications of vector algebra|| example 6.1

10thmaths ||Relations and functions ||example 1.9

10thmaths ||Relations and functions ||example 1.9

10thmaths ||Relations and functions ||example 1.6

10thmaths ||Relations and functions ||example 1.6

10thmaths ||Relations and functions ||example 1.7

10thmaths ||Relations and functions ||example 1.7

10thmaths ||Relations and functions ||example 1.8

10thmaths ||Relations and functions ||example 1.8

12thmaths ||Applications of vector algebra ||exercise 6.2(8)

12thmaths ||Applications of vector algebra ||exercise 6.2(8)

12thmaths ||Applications of vector algebra ||exercise 6.2(7)

12thmaths ||Applications of vector algebra ||exercise 6.2(7)

10thmaths ||Relations and functions ||exercise 1.2(5)

10thmaths ||Relations and functions ||exercise 1.2(5)

10thmaths ||Relations and functions ||exercise 1.2(4(ii))

10thmaths ||Relations and functions ||exercise 1.2(4(ii))

10thmaths ||Relations and functions ||exercise 1.2(4(i))

10thmaths ||Relations and functions ||exercise 1.2(4(i))

10thmaths ||Relations and functions ||exercise 1.2(3)

10thmaths ||Relations and functions ||exercise 1.2(3)

10thmaths ||Relations and functions ||exercise 1.2(2)

10thmaths ||Relations and functions ||exercise 1.2(2)

12thmaths ||Applications of vector algebra ||basics revision

12thmaths ||Applications of vector algebra ||basics revision

10thmaths ||Relations and functions ||exercise 1.2(1)

10thmaths ||Relations and functions ||exercise 1.2(1)

10thmaths ||Relations and functions |example 1.5

10thmaths ||Relations and functions |example 1.5

10thmaths ||Relations and functions |example 1.4

10thmaths ||Relations and functions |example 1.4

10thmaths ||Relations and functions ||Relations introduction

10thmaths ||Relations and functions ||Relations introduction

10thmaths ||Relations and functions |example 1.2

10thmaths ||Relations and functions |example 1.2

10thmaths ||Relations and functions |example 1.3(ii)

10thmaths ||Relations and functions |example 1.3(ii)

10thmaths ||Relations and functions |example 1.3(i)

10thmaths ||Relations and functions |example 1.3(i)

10thmaths ||Relations and functions ||exercise 1.1(7(i))

10thmaths ||Relations and functions ||exercise 1.1(7(i))

10thmaths ||Relations and functions ||exercise 1.1(7(ii))

10thmaths ||Relations and functions ||exercise 1.1(7(ii))

10thmaths//relations and functions//exercise 1.1(6(ii))

10thmaths//relations and functions//exercise 1.1(6(ii))

10thmaths//relations and functions//exercise 1.1(6(iii))

10thmaths//relations and functions//exercise 1.1(6(iii))

10thmaths//Relations and functions// exercise 1.1(5)

10thmaths//Relations and functions// exercise 1.1(5)

10thmaths /Relations and functions |example 1.1

10thmaths /Relations and functions |example 1.1

10thmaths/relations and functions//exercise 1.1(6(i))

10thmaths/relations and functions//exercise 1.1(6(i))

10thmaths ||Relations and functions exercise 1.1(3)

10thmaths ||Relations and functions exercise 1.1(3)

10thmaths ||Relations and functions |exercise 1.1(2)

10thmaths ||Relations and functions |exercise 1.1(2)

10thmaths ||Relations and functions//exercise 1.1(4)

10thmaths ||Relations and functions//exercise 1.1(4)

10thmaths ||Relations and functions ||exercise 1.1(1)

10thmaths ||Relations and functions ||exercise 1.1(1)

10thmaths| Relations and functions//ordered pair and cartesian product introduction

10thmaths| Relations and functions//ordered pair and cartesian product introduction

12thmaths |applications of vector algebra ||exercise 6.2(6)

12thmaths |applications of vector algebra ||exercise 6.2(6)

12thmaths |applications of vector algebra ||exercise 6.2(5)

12thmaths |applications of vector algebra ||exercise 6.2(5)

12thmaths |applications of vector algebra ||exercise 6.2(4)

12thmaths |applications of vector algebra ||exercise 6.2(4)

12thmaths |applications of vector algebra ||exercise 6.2(3)

12thmaths |applications of vector algebra ||exercise 6.2(3)

12thmaths |applications of vector algebra ||exercise 6.2(2)

12thmaths |applications of vector algebra ||exercise 6.2(2)

12thmaths |applications of vector algebra ||exercise 6.2(1)

12thmaths |applications of vector algebra ||exercise 6.2(1)

12thmaths |applications of vector algebra || scalar triple product introduction

12thmaths |applications of vector algebra || scalar triple product introduction