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0 jnvLudhiana Page 1 jnvLudhiana Page 2 SIMPLIFIED MATERIAL FORM jnvLudhiana Page 3 Relations and Functions Basic Concepts and Formulae: Relations: A relation in set A is a subset of A X A. We also write it as R = {(a, b) ∈ AXA / aRb} • Type of Relation : I. Reflexive Relation: A relation R in a set X is called reflexive if ( x, x ) ∈ R , II. ∀ x∈ X. Symmetric Relation: A relation R in a set X is called symmetric if ( x, y ) ∈ R ⇒ ( y , x ) ∈ R , III. ∀ x, y ∈ X . Transitive Relation: A relation R in a set X is called transitive if (x, y )∈ R and ( y, z )∈ R ⇒ (x, z )∈ R, ∀ x, y , z ∈ X . Equivalence relation: A relation which reflexive, symmetric and transitive is called equivalence relation. jnvLudhiana Page 4 • Function: Let X and Y are two non empty sets. The function is a rule or formula which associate each element x ∈ X f from X into Y with unique element y ∈ Y . Mathematically f : X → Y f (x ) = y. Here X is called domain of f and Y is called co domain of f . The set of images of of all the elements of X is called Range of f . • Types of Functions 1. One – One (Injective) Function: A function f : X → Y is said to be One-One if distinct elements of X has distinct images of Y. Mathematically, for ∀ x1 , x 2 ∈ X , f ( x1 ) = f (x 2 ) ⇒ x1 = x 2 . 2. Onto (Surjective) Function: A function f : X → Y is said to be Onto if every element of Y is the image of at least one element of X. Mathematically, for y ∈ Y ∃ x ∈ X such that f (x ) = y. Note that for onto function co domain is equal to the range set. 3. One to one (bijective) function: A function which is both one-one and onto is called one to one function. jnvLudhiana Page 5 • Composite function: Let f : X → Y and g : Y → Z are two functions. The composite of f and g , written as fog is defined as the function that fog : X → Z such fog ( y ) = f (g ( y )), ∀ y ∈ Y . Note that fog and gof are two different functions. • Invertible function: A f : X → Y function is said to be invertible if it satisfy the following equivalent conditions: 1. f must be one to one function. 2. ∃ g :Y → X such that gof = I x and fog = I y . The function g is called the inverse of f and is denoted by f −1 . • Binary Operations: Let A be a non empty set. Then binary operation * on A is a function from A × A to A. Mathematically * ( x . y ) = x * y . jnvLudhiana Page 6 • Properties of binary operations: Let 1. * is associative if x, y, z ∈ X . Then x * ( y * z ) = (x * y )* z . 2. * is commutative if x * y = y * x. 3. e ∈ X is called identity element of * , if 4. i∈ X is called inverse element of x w. r. t. * , if x * e = x = e * x. x * i = e = i * x. INVERSE TRIGONOMETRIC FUNCTIONS Basic Concepts and Formulae: jnvLudhiana Page 7 • Domain and Range of trigonometric functions: FUNCTION sin x cos x tan x cot x sec x cos ecx DOMAIN R R RANGE π R − x : x = (2n + 1) , n ∈ Z 2 R − {x : x = n π , n ∈ Z } π R − x : x = (2n + 1) , n ∈ Z 2 R − {x : x = n π , n ∈ Z } R [− 1, 1] [− 1, 1] R R − (− 1 , 1 ) R − (− 1 , 1 ) • Domain, Range and Principal value branch for inverse trigonometric functions: FUNCTION DOMAIN x [− 1, 1] −1 x [− 1, 1] tan −1 x R cot −1 x R sec −1 x R − (− 1 , 1 ) sin −1 cos Range(PRINCIPAL VALUE BRANCH) π π − 2 , 2 [0 , π ] π π − , 2 2 (0 , π ) [0, π ] − π 2 cos ec −1 x R − (− 1 , 1 ) π π − 2 , 2 Properties And Formulas of inverse trigonometrical function Set ‘A’ jnvLudhiana Page 8 Sin −1 (sin x ) = x , x ∈ ( − cos −1 (cos x ) = x −1 (tan x ) = x cot −1 (cot x ) = x sec −1 (sec x ) = x tan cos ce −1 Π Π , ) 2 2 . (cos ecx ) = x Set ‘B’ sin −1 1 = cos ec x , x ≥ 1orx ≤ 1 x −1 1 cos −1 = sec −1 x , x ≥ 1orx ≤ 1 x 1 tan −1 = cot −1 x x>0 x 1 cos ec −1 = sin −1 x x , sec 1 −1 = cos x x −1 1 cot −1 = tan −1 x x Set ‘C’ sin cos −1 (− x ) = −1 jnvLudhiana − sin (− x ) = π −1 x − cos −1 x Page 9 tan cos −1 (− x ) = ec −1 − tan (− x ) = −1 x − cos ec −1 sec −1 (− x ) = π − sec −1 x cot −1 (− x ) = π − cot −1 x x Set ‘D’ π sin −1 x + cos −1 x = sec −1 x + cos ec −1 x = tan −1 x + cot −1 x = 2 π 2 π 2 Set ‘E’ tan −1 x + tan −1 y = tan −1 x + y 1 − xy tan −1 x − tan −1 y = tan −1 x− y 1 + xy if xy <1 if xy>-1 2 2x −1 2 x −1 1 − x −1 sin = cos = tan = 2 tan −1x 2 2 2 1− x 1+ x 1 + x Set ‘F’ o sin −1 jnvLudhiana x + sin −1 y = sin −1 (x 1− y2 + y 1− x2 ) Page 10 ( o sin −1 x − sin −1 y = sin −1 x 1 − y 2 − y 1 − x 2 o cos −1 x + cos −1 y = cos −1 xy − 1 − y 2 1 − x 2 o cos −1 x − cos −1 y = cos −1 xy + 1 − y 2 1 − x 2 ) ( ) ( ) While writing inverse trigonometrical functions in simplest form, we can use the following substitution for a 2 − x 2 ; we substitute x=a sinө or x=a cosө for a 2 + x 2 ; we substitute x=a tanө or x=a cotө for x 2 − a 2 ; we substitute x=a secө or x=a cosecө Matrices Basic Concepts and Formulae: jnvLudhiana Page 11 1. Matrix : A set of mn numbers arranged in the form of a rectangular array of m rows and n columns is called an m × n matrix. 2. Elements of a Matrix : Each of the mn entries of m × n matrix is called an element or an entry of the matrix. 3. To define a matrix ,we must define its order and its elements. 4. Row matrix : A matrix having only one row is called a row matrix. Hence, Row matrix is of order 1 × n . 5. Column Matrix: A Matrix having only one column is called a column matrix. Hence , Column matrix is of order m × 1. 6. Square Matrix : A matrix in which the number of rows is equal to the number of columns is called a square matrix. [ ] 7. Diagonal Matrix : A square matrix A = aij m×n is called a diagonal matrix if all the elements except those in the leading diagonal , are zero i.e. aij = 0 , i [ ] 8. Scalar Matrix : Square matrix A = aij (ii) aij = C , C ≠ 0 m×n ≠ j . is called a scalar matrix if (i) aij = 0 , i ≠ j .and . 9. Null Matrix : A matrix whose all elements are zero is called a null matrix or Zero matrix. [ ] 10. Upper Triangular Matrix : A square matrix A= aij is called an upper triangular matrix if aij=0 for all i>j. jnvLudhiana Page 12 [ ] 11. Lower Triangular Matrix : A square matrix A= aij is called an lower triangular matrix if aij=0 for all i<j. [ ] 12. Equality of Matrices : Two matrices A = aij m×n [ ] and B = bij r×s are equal if (i) m=r , the number of rows in A equatls to number of rows in B. (ii) n=s , the number of columns in A equals to number of columns in B (iii) aij = bij , for all i = 1,2,3,….. and 1,2,3,….. j= 13. Addition of Matrices: Let A , B be two matrices , each of order m × n . Then, their sum A + B is a matrix of order m × n and is obtained by adding the corresponding elements of A and B. 14. Commutative : If A and B are two m × n matrices , then A+B=B+A i.e. matrix addition is commutative. 15. Associative : If A ,B,C are three matrices of the same order , then (A+B)+C=A+(B+C) 16. Existence of Identity : The null matrix is the identity element for matrix addition i.e. A+O = A = O+A [ ] 17. Existence of Inverse : For every matrix A = aij m×n [ ,there exist a matrix A = − aij ] m× n , denoted by –A such that A+(-A) =O =(-A) +A [ ] 18. Multiplication of a Matrix by a Scalar : Let A = aij m×n be an m × n matrix and k be any number called a scalar .Then the matrix obtained by multiplying every element of A by k is called the scalar multiple of A by k and is denoted by kA. jnvLudhiana Page 13 [ ] 19. Multiplication of Matrices : Let A= aij m×n [ ] and B= a jk n× p be two matrices , given in such way that the number of columns in A is equal to the number of rows in B . Then their product AB is an m × p matrix , given by AB = [cik ]m× p where cik = ai1b1k + ai 2 b2 k + ..... + ain bnk 20. If A ,B, C be three matrices of order m × n , n × p, p × q respectively , then (AB)C=A(BC). 21. If A ,B, C be three matrices of order m × n , n × p, n × p respectively , then AB+AC =A(B+C) 22. Integral power of Matrices exists only for Square matrices. 23. If A is any square matrix , we write A2 for A × A. 24. Transpose of a Matrix : Let A be an m × n . Then the n × m matrix, obtained by interchanging the rows and columns of A , is called the transpose of A and is denoted by A’. Thus 25. (i) if the order of A is m × n , then order of A’ os n × m. (ii) (i, j) th element of A = ( j, i )th element of A’. (A')'= A 26. (A+B)’=A’+B’ 27. (kB)’ = k B’ 28. (AB)’=B’A’ jnvLudhiana Page 14 29. A square matrix A = [ aij ] is said to be a symmetric matrix iff A’ = A i.e. aij = aji ∀ i,j. 30. A square matrix A = [ aij ] is said to be a skew symmetric matrix iff A’ = -A i.e. aij =- aji ∀ i,j. 31. Invertible Matrices : If A is a square matrix of order n and if there exists another square matrix of the same order n such that AB= BA = I ,then B is called the inverse of matrix A and denoted by A−1 32. Inverse of Matrix , if it exists , can be found by using either row or column operations but not both simultaneous. 33. If after doing one or more than one elementary operation, we obtain all 0’s in one or more rows of the given matrix, then A−1 does not exist. Commit to Memory jnvLudhiana Page 15 1. A square matrix A = [ aij ] is said to be a symmetric matrix iff A’ = A i.e. aij = aji ∀ i,j. 2. A square matrix A = [ aij ] is said to be a skew symmetric matrix iff A’ = -A i.e. aij =- aji ∀ i,j. 3. (kB)’ = k B’ 4. (AB)’=B’A’ 5. ( AB ) −1 = B −1 A −1 6. ( A −1 )' = ( A' ) −1 7. ( A n ) −1 = ( A −1 ) n 8. If A and B are invertible matrices of same order , then ( AB ) −1 = B −1 A −1 9. If A and B are invertible matrices of same order , then adj(AB)=(adjA)(adjB) 10. ( A T ) −1 = ( A −1 ) T 11. Inverse of Matrix , if it exists , can be found by using either row or column operations but not both simultaneous . 12. If after doing one or more than one elementary operation, we obtain all 0’s in one or more rows of the given matrix, then jnvLudhiana A−1 does not exist. Page 16 Determinant Basic Concepts and Formulae a b 1. Determinant of a square matrix of order two : Let be a square matrix of order 2 , c d a b then det(A) = A = = ad-bc c d a 2. Determinant of a square matrix of order three : Let d g a b c e f d f d 3 , then det(A) = A = d e f = a -b +c h i g i g g h i b e h c f be a square matrix of order i e h 3. Area of triangle : Area of Triangle ABC whose vertices are A ( x 1 , y 1 ) , B ( x 2 , y 2 ) and x1 y1 1 1 C ( x3 , y3 ) is given by Area of triangle = x 2 y 2 1 2 x3 y 3 1 4. A Matrix A is said to be a singular matrix if A = 0 jnvLudhiana Page 17 5. Adjoint of a Matrix : The adjoint of a matrix A is the transpose of a matrix of cofactors of elements of matrix A. [ ] 6. Inverse of a square matrix : Let A = aij be asqaure matrix of oreder n . Then inverse of amatrix A is defined as A −1 and is given by A −1 = 1 adj( A) A Commit to Memory 1. Expanding a determinant along any row or column gives the same value. 2. A Matrix A is said to be a singular matrix if A = 0 3. A(adjA)=(adjA)A= A I 4. For matrices /a and /b of same order AB = A B 5. If A is a square matrix of order n , then adjA = A n −1 6. If A is a square matrix of order n , then kA = k n A 7. If A is an invertible matrix then (adjA)’ = adj(A’) jnvLudhiana Page 18 8. adj(adjA)= A 9. A−1 = n−2 A 1 A 10. Since area is a positive quantity , so always take the absolute value of determinant. 11. A System of equations is said to be consistent if its solution exists. CONTINUOUTY AND DIFFERENTIABILITY Basic Concepts and Formulae: 1. Let f be a real valued function on the subset of real numbers and let a be any point in the domain of f. Then f is said to be a continous at x = a if Lt x → a f ( x ) = f ( a ) Or Lt x → a + f ( x ) = f ( a ) = Lt x → a − f ( x ) 2. A function is discontinous at x = a if (i) f (a ) does not exist jnvLudhiana Page 19 (ii) Lt x→ a f (x ) does not exist (iii) Both exist but are not equal. 3. A function is said to be continuous if it is continous at every point in its domain. 4. If f and g are two continuous functions at a point a then (i)f + g is continuous at a (ii) f - g is continuous at a (iii) f . g is continuous at a (iv) f is continuous at a g (v) c. f is continuous at a, where c is a constant. 5. For a function y = f ( x ) , dy represent the instantaneous rate of change of y with respect to x. dx dy f ( x + h) − f ( x) = Lt h → 0 dx h where h is a very small increment in h. 6. List of some useful formulae FARMULAE: Derivatives (1) d x n+1 = x n dx n + 1 (2) d (sin x ) = cos x dx jnvLudhiana Page 20 (3) d (cos x ) = − sin x dx (4) d (tan x ) = sec 2 x dx (5) d (cot x ) = − cos ec 2 x dx (6) d (sec x ) = sec x + tan x dx (7) d (cos ecx ) = − cos ecx cot x dx (8) d 1 (sin −1 x) = dx 1− x2 (9) d 1 (cos −1 x) = − dx 1− x2 (10) d 1 (tan −1 x ) = dx 1+ x2 (11) d 1 (cot −1 ) = − dx 1+ x2 (12) d 1 (sec −1 x) = dx x x2 −1 jnvLudhiana Page 21 (13) d 1 (cos ec −1 x) = − dx x x2 −1 (14) d x e = ex dx (15) d 1 (log x ) = dx x 7. Product Rule – Let u and v be two functions of x, then d (u.v ) dv du =u +v dx dx dx 8. Quotient Rule – If u and v be two functions of x , then d u = dx v v du dv −u dx dx v2 9. Every differential function is continuous but converse may not be true. 10.Logarithmic Differentiation Logarithmic differentiation are used for differentiation of functions which consists of the product or quotients of a number of functions or the given function is of type [ f ( x)] g ( x) . In this method we take the logarithm on both the sides of the function and then differentiate it with respect to x jnvLudhiana Page 22 11.Parametric form : Sometimes we come across the function when both s and y are expressed in dy terms of another variable say t, i.e. x = f (t ) and y = g (t ) .This form is parametric and is found dx dy dy dx by applying the formula = ÷ dx dt dt 12. Rolle’s Theorem : If f(x) be a real valued function defined in a closed interval [a , b ] such that i) It is continuous in closed interval [a , b ] ii) It is differentiable in open interval ( a , b ) iii) f ( a ) = f ( b ) then there exist at least one real value c ∈ ( a , b ) such that f ' ( c ) = 0 13. Lagrange’s Mean Value Theorem : If f(x) be a real valued function defined in a closed interval [a , b ] such that i) It is continuous in closed interval [a , b ] ii) It is differentiable in open interval ( a , b ) then there exist at least one real value c ∈ ( a , b ) such that f ' ( c ) = f (b ) − f ( a ) b−a APPLICATION OF DERIVATIVES Basic Concepts and Formulae 1. For the curve y = f ( x ) , dy represent the slope of the tangent to the curve y = f ( x ) .At any dx dy point ( x 1 , y 1 ) it is represented by . dx ( x1 , y1 ) 2. If dy = 0 ,then tangent is parallel to x axis and vice versa. dx jnvLudhiana Page 23 3. If dy is not defined then tangent is parallel to y axis. dx 4. The equation of tangent to the curve y = f ( x ) at ( x1 , y1 ) is given by ( x1 , y1 ) is given by dy y − y1 = ( x − x1 ) dx ( x1 , y1 ) 5. The equation of normal to the curve y = f ( x ) at y − y1 = − 1 ( x − x1 ) dy dx ( x1 , y1 ) 6. A function is said to be increasing in an interval ( a,b ) if f ' ( x ) > 0 7. A function is said to be decreasing in an interval ( a,b ) if f ' ( x ) < 0 8. If ‘f’ be a function defined in the closed interval I and there exist a point ‘a’ in the interval I such that f ( a ) ≥ f ( x ) for all x ∈ I .Then a function is said to attain absolute maximum at x = a and f (a ) is absolute maximum value. 9. If ‘f’ be a function defined in the closed interval I and there exist a point ‘a’ in the interval I such that f ( a ) ≤ f ( x ) for all x ∈ I .Then a function is said to attain absolute minimum at x = a and f (a ) is absolute maximum value. 10.To find absolute maximum value and absolute minimum value for function f(x) Defined in [a , b] jnvLudhiana Page 24 i)Find solution of f ' ( x ) = 0 .Let these are x 1 , x2 , x3,……. ii)Find f ( a ), f ( x 1 ), f ( x 2 )......... ....., f ( b ) iii)Maximum and minimum values among f ( a ), f ( x 1 ), f ( x 2 )......... ....., f ( b ) are the absolute maximum and absolute minimum values and corresponding points are the points of absolute maxima and absolute minima. 11) For any function f ( x ) , all such points where f ' ( x ) = 0 are called the turning points. 12) To get a point of maxima for any function f ( x ) we must have f ' ( x ) = 0 and f ' ' ( x ) < 0 .In the same way for getting a point of minima for any function f ( x ) we must have f ' ( x ) = 0 and f ' ' ( x ) > 0 . Integrals and its applications Basic Concepts and Formulae: jnvLudhiana Page 25 1.Integration:- Integration is the process of finding the function whose derivatives is given. 2.Indefinite Integral:- Indefinite means not unique. Actually there exists infinitely many integrals of a function, which can be obtained by choosing arbitrary the value of C from the real numbers. 3.Methods of Integrations:- (a)Integration by substitution. (Used to integrate the function by suitable substitution) (b) Integration by Parts.(Used to integrate the product of two functions. The first function is taken as per order of ILATE ) ( c ) Integration by Partial Fraction. (Used to integrate the rational algebraic functions) 4.Important Therems:- jnvLudhiana Page 26 (a) The Indefinite Integral of an algebraic sum of two or more functions is equal to the algebraic sum of their integrals i.e. dx = dx + dx (b) A Constant term may be taken outside from the integral sign. i.e dx = dx where k is a constant. ( c ) If the numerator in an integral is the exact derivative of denominator then its integral is logarithmic of denominator. i.e dx = (d) +c dx = +c jnvLudhiana Page 27 ( e) Integral of the functions whose numerator is unity and denominator is a homogeneous function of First degree can be found easily by substitution of sin 2x = , cos 2x = and tan x = t (f) Integral of the functions whose numerator is unity and denominator is a homogeneous function of Second degree in sin x , cos x or both can be found easily by Dividing both numerator and dnominator by and then substituting tan x = t. DIFFERENTIAL EQUATION Basic Concepts and Formulae: Definition: A differential equation is an equation which involves unknown function and their derivatives w.r.t one or more independent variables. For a given function g, find a function f such that jnvLudhiana Page 28 dy = g (x) dx where y= f(x) ……………..i An equation of the form (1) is known as a differential equation. 001111 d2y dy + 9y = 0 −6 2 dx dx ……..(4) Note :We use following notation 2 3 dy = y ′ d y = y ′′ d y = y ′′′ dx dx 2 dx 3 etc. Order of differential equation The order of the differential equation is the order of the highest order derivatives occurring in the differential equation d2y dy −6 + 9y = 0 2 dx dx d ′ ′′y d ′′y −7 + 11 y = 0 d x ′′′ d x ′′ Here The order of the differential equation first is 2 and The order of the differential equation second is 3. jnvLudhiana Page 29 Degree of differential equation The Degree of the differential equation is the index of the highest order derivatives which appears in the differential equation after making it free from negative and fractional power. d2y dy −6 + 9y = 0 2 dx dx d ′ ′′y d ′′y −7 + 11 y = 0 d x ′′′ d x ′′ Here The degree of the differential equation first is 1 and The order of the differential equation second is 1. General and particular solution of the differential equation Formulate of differential equation 1. Order of the differential equation is equal to the number of independent arbitrary constants in the equation. jnvLudhiana Page 30 2. Satisfied by the given equation. 3. Free from arbitrary constants. Equation with Variable Separable An equation whose variable are separable can be put into the form f 1 ( x ) dx + f 2 ( x ) dy = 0 Integrating the general soluation ∫ f 1 ( x ) dx + ∫ f 2 ( x ) dy = C where C is arbitrary constant. Homogeneous Differential Equation Definition : A differential equation is an equation which involves unknown faction and their of the form dy = f ( x , y ) dx g ( x , y ) Where f(x,y) and g(x,y) are homogeneous fuction Like jnvLudhiana Page 31 dy x 3 − 3x 2 y = dx 3xy 2 − y 3 (x 3 ) ( ) − 3 xy 2 dx = y 3 − 3 x 2 y dy dy 3xy = dx 3xy − y 2 To solve homogeneous function 1. Put y= vx 2. Find derivatives dy = v + x dv dx dx 3. Substituting value of y and dy dx in the given equation 4. After that it will be reducible to variable separable and we know how to solve it in the previous method Linear Differential Equation (of first order) Definition: A Linear differential equation is an equation which is form jnvLudhiana Page 32 dy dx + Py = Q Where P& Q are fuction of x only Like dy 2 + y = 3x dx x dy cos x + y = 3 x cos x dx sin x cos x 2 dy + y cos x = sin x dx To solve homogeneous equation 1. Make the equation in the form dy dx + Py = Q 2. Find the value of P & Q 3. Find integral factor i.e I .F . = e ∫ Pdx 4. FIND THE SOLUTION pdx Pdx ∫ ∫ ye = ∫Qe dx+c VECTORS Basic Concepts and Formulae: jnvLudhiana Page 33 • Section Formulae for vectors: The position vector of a point C dividing a ρline segment joining the points A and B whose position vectors are aρ and b respectively in the ratio m:n ρ ρ na + mb . 1} internally is given by m+n ρ ρ mb − na 2} externally is given by . m−n Dot Product between vectors: • ρ 1} If aρ and b are any two vectors and θ ( 0≤θ≤π ) is the angle between ρ ρ ρρ them, a ⋅ b = a b cos θ . ρ ) ρ ρ ρ ) ) 2} Let a = a1i + a 2 j + a 3 k and b = b1i + b2 j + b3 k then a ⋅ b = a1b1 + a 2 b2 + a 3 b3 3} If ρ a 2 ρ and b are any two vectors and θ is the angle between them then ρ ρ a ⋅b cos θ = ρ ρ . ab 4} Two vectors ρ a ρ ρ ρ and b are perpendicular then a ⋅ b = 0 . • Projectionρof a vector: Let aρ and b are any vector. Then Projection of ρ aρ ⋅ bρ b is ρ . b • ρ a in the direction of Cross Product between vectors:ρ 1} If θ is the angle between two vectors aρ and b then their cross product is ρ ρ ρρ ) a × b = a b sin θ n , Where Containing ρ a ρ ) n is a unit vector perpendicular to the plane ρ and b such that aρ , b and ) n form right handed system Of coordinate axes. jnvLudhiana Page 34 i j b1 a2 b2 ρ ) ) ) ρ ρ ρ 2} Let a = a1i + a 2 j + a 3 k and b = b1i + b2 j + b3 k , then a × b = a1 k a3 . b3 3} Note that the cross product of two vectors is a vector perpendicular to both of them. 4} Relation between ρ a ρ ρ a ⋅ b ρ2 ρ2 ρ ρ and a × b : we have aρ × b = a 2 b 2 − aρ ⋅ b . 5}. Area of a parallelogram whose adjacent sides are represented by ρ and b is ρ ρ axb . ρ 6} Area of a parallelogram whose diagonals are represented by aρ and b is 1 ρ ρ a xb . 2 ρ ρ a xb ρ 7} Area of a triangle whose two sides are represented by aρ and b is . 2 ρρ ρ axb 8} ρ ρ is a unit vector which is perpendicular to both aρ and b . axb 9} Two vectors ρ a ρ ρ ρ and b are parallel then axb = 0 . ρ • Scalar triple product: If aρ , b and ρ c are any three vectors then scalar a1 a2 ρ ρ ρ ρ ρ ρ triple product denoted as a ⋅ (b × c ) and a • (b × c ) = b1 b2 c1 c2 a3 b3 . c3 THREE DIMENSIONAL GEOMETRY Basic Concepts and Formulae: jnvLudhiana Page 35 The direction cosines of line with direction ratios a, b, c are • a b c . , , 2 2 2 2 2 2 2 2 2 a +b +c a +b +c a +b +c • Also if the line makes angles α , β , and γ respectively with the coordinate axis then the direction cosines are cos α , cos β and cos γ respectively. • Angle between two lines whose direction ratios are a1 , b1 , c1 and a 2 , b2 , c 2 is a1a2 + b1b2 + c1c2 cosθ = 2 2 2 2 2 (a 21 + b1 + c1 )(a2 + b2 + c2 ) 1} If the lines are perpendicular then a1 a 2 + b1b2 + c1c 2 = 0 2} If the lines are parallel then a1 b c = 1 = 1 . a 2 b2 c 2 TABLE OF EQUATION AND FORMULAE: Topic Equation of straight line passing through a point and parallel to a given vector Equation of straight line passing through two given point Shortest distance between two skew lines whose equations are ρ ρ ρ r = a1 + λ b1 and ρ ρ ϖ r = a 2 + µb2 jnvLudhiana Vector form ρ ρ ρ r = a + λb ρ ρ ρ ρ r = a + λ (b − a ) ρ ρ ρ ρ ( a 2 − a1 ) ⋅ (b1 × b2 ) ρ ρ b1 × b2 Cartesian form x − x1 y − y1 z − z1 = = a b c =λ x − x1 y − y1 z − z2 = = x 2 − x1 y 2 − y1 z 2 − z1 x 2 − x1 a1 a2 y 2 − y1 b1 b2 z 2 − z1 c1 c2 (b1c 2 − b2 c1 ) 2 + (c1 a 2 − c 2 a1 ) 2 + ( a1b2 − a 2 b1 ) 2 Page 36 ρ ρ ρ Shortest distance ( a 2 − a1 ) × b ρ between two b parallel planes whose equations ρ ρ ρ are r = a1 + λb ρ ρ and r = a 2 + µb ρ r . nˆ = d Equation of plane in normal form ρ Equation of a (rρ − aρ).N = 0 plane passing through a given point with position vector ρ a and perpendicular toρ a given vector N ρ Equation of (rρ − aρ).[(b − aρ)× (cρ − aρ)] plane passing =0 through three non collinear points with position vectors ρ ------------------ lx + my + nz = d A( x − x1 ) + B ( y − y1 ) + C ( z − z1 ) = 0 x − x1 y − y1 z − z1 x2 − x1 x3 − x1 y 2 − y1 y3 − y1 z 2 − z1 = 0 z3 − z1 ρ ρ a , b and c Intercept form of plane with a, b and c as x, y and z intercepts Equation of a plane through the line of intersection of two planes whose vector equations are ρ ρ r ⋅ n1 = d1 and --------- ρ ρ (r ⋅ n1 − d1 ) + ρ ρ λ ( r ⋅ n2 − d 2 ) = 0 x y z + + =1 a b c A1 x + B1 y + C1 z + D1 + λ ( A2 x + B2 y + C2 z + D2 ) = 0 ρ ρ r ⋅ n2 = d 2 Distance of a point (x1 , y1 , z1 ) from plane jnvLudhiana ----------- Ax1 + By1 + Cz1 − d A2 + B 2 + C 2 Page 37 Ax + By + Cz = d Distance of a point with position vector from plane ϖ a ρ ρ a⋅n −d ρ n ----------- ρ ρ r ⋅n = d. SOME OTHER CONCEPTS RELATED TO LINE AND PLANE: • • Angle between two straight lines whose vector equations are ρ ρ ρ ρ ρ ρ ρ ϖ b1 ⋅ b2 r = a1 + λ b1 and r = a 2 + µb2 is cos θ = ρ ρ . b1 b2 Equation of a plane which is perpendicular to any vector and which is at a distance p from the origin is rρ ⋅ nρ = d where ρ n ρ d = n p. • • Equation of a plane in Cartesian form is Ax + By + Cz = Angle between two planes whose vector equations are ρ ρ r ⋅ n1 = d1 and d. ρ ρ n1 ⋅ n2 ρρ r .n2 = d 2 is cosθ = ρ ρ . n1 n2 • ρ ρ ρ r = a + λb and Angle between a line and a plane whose vector equations are ρ ρ r ⋅n = d ρ ρ b ⋅n is sin θ = ρ ρ . . b n ρ ρ ρ ρ Distance between two parallel planes r ⋅ n = d1 and r ⋅ n = d 2 is • d1 − d 2 ρ . n jnvLudhiana Page 38 • Equation of a plane passing through two given points (x1 , y1 , z1 ) and (x 2 , y 2 , z 2 ) and perpendicular to a plane Ax + By + Cz x − x1 y − y1 z − z1 x 2 − x1 A y 2 − y1 B z 2 − z1 = 0. C is Equation of a plane passing through two given points • (x1 , y1 , z1 ) and (x 2 , y 2 , z 2 ) and parallel to a line • = d x − x1 y − y1 z − z1 x 2 − x1 a y 2 − y1 b z 2 − z1 = 0. c x − x1 y − y1 z − z1 = = is a b c Equation of a plane passing through a given point (x1 , y1 , z1 ) and perpendicular to two planes A1 x + B1 y + C1 z = d1 and A2 x + B 2 y + C 2 z = d 2 is x − x1 y − y1 A1 A2 B1 B2 z − z1 C1 = 0. C2 Equation of a plane passing through a given point (α , β , γ ) and • parallel to two lines x − x1 = y − y1 = z − z1 and x − x 2 = y − y 2 = z − z 2 is a1 x −α y−β z −γ a1 a2 b1 b2 c1 c2 jnvLudhiana b1 c1 a2 b2 c2 = 0. Page 39 LINEAR PROGRAMMING Basic Concepts and Formulae: In simple language Linear Programming Problem ( LPP ), stand for optimization ( to maximize or minimize ) a certain thing ( which may be profit, cost, use of resources, use of workers, usability of machines etc.) with respect to certain restrictions. The thing to optimize is known as objective function and the restrictions are known as constraints. In this chapter we shall learn to form an LPP and solve it graphically and also the types of LPP. It is very useful in formation of certain policies in corporate world. It is a very useful Mathematics tool in hands of Businessmen, by which they can maximize the profit or minimize the losses/ costs/ expenditure where the affecting variables are two in counting. Basic Concepts - Meaning of Linear Programming Problem (LPP) – jnvLudhiana Page 40 A linear Programming Problem is one that is concerned with finding the optimal value (i.e. maximum or minimum value) of a linear function (called objective function ) of several variables (in XII class syllabus we are restricted to only two variables say x and y ), subject to the conditions that the variables are non-negative and satisfy a set of linear inequalities ( called linear constraints). The term linear implies that all the mathematical relations used in the problem are linear relations while the programming refers to the method of determining a particular programme or plan of action. The general LPP is given by Max. Z (Min. Z) = ax + by ,………………(1) Subject to the constraints: a1 x + b1 y ≤ (≥)c1 a x + b y ≤ (≥)c 2 2 2 a3 x + b3 y ≤ (≥)c3 ……………………….(2) .......... .......... a n x + bn y ≤ (≥)cn x ≥ 0 , andy ≥ 0 . ……………………………(3) Here (1) is called objective function, (2) contains the constraints and (3) is called non negativity condition. Types of Linear Programming Problem (LPP) – jnvLudhiana Page 41 Manufacturing problems- In these problems, we determine the number of units of different products which should be produced and sold by a firm when each product requires a fixed manpower, machine hours, labor hour per unit of product, warehouse space per unit of the output etc., in order to make maximum profit. Diet problems -In these problems, we determine the amount of different kinds of constituents/nutrients which should be included in a diet so as to minimize the cost of the desired diet such that it contains a certain minimum amount of each constituent/nutrients. Transportation problems -In these problems, we determine a transportation schedule in order to find the cheapest way of transporting a product from plants/factories situated at different locations to different markets. Steps to solve a LPP using Graphical Method - 1. Construct the LPP using given word problem. 2. Write the auxiliary equations, by merely changing the sign of inequality to equality in all the constraints. 3. Write the graph table of all the linear equations formed in the step two. 4. Using graph table draw the graphs of all auxiliary equation, and find the solution region. jnvLudhiana Page 42 5. (Bounded Region) If solution region is bounded, then find its corner points say A, B, C,…etc. 6. Find the value of Z at A, B, C,……. Then the max.(min.) value of Z will represent the solution. It is called the optimal value and the corresponding pont will gives the value of x and y. 7. (Un-Bounded Region) If in step 5, the solution is not bounded i.e. not a closed region, then again find the corner point and find the value of Z on these values, say m is the largest/ maximum value of Z and n in the minimum value of Z, then check whether the region given by Z ≥ m or Z ≤ n have any common point with solution region or not. In case it has any common point with solution region then the corresponding max. or min. solution does not exists. In general in this case the maximum solution does not exist, and minimum solution may or may not exist. jnvLudhiana Page 43 Probability Basic Concepts and Formulae: Till now you are well familiar with the topic “Probability”. All of us know about the basic concepts of probability i.e. a trial, random experiment, outcomes of an experiments, events, complimentary events, exhaustive events and mutually events etc. In this chapter we shall discuss the concept of conditional probability of an event, given that another event has occurred, which will be helpful in understanding the Multiplication theorem of probability, Independence of events, Total probability theorem and hence Baye’s theorem. We shall also learn the concept of a Random Variable ( rv ) and Distribution of a random variable. At last we shall study the concept of discrete probability distribution called Binomial Distribution. Conditional Probability- The probability of occurrence an event E, given that event F has occurred, is called Conditional Probability of event E w.r.t. F, written as P(E/F) and is given by jnvLudhiana Page 44 P(E/F)= P (E ∩ F ) , provided P(F) ≠ 0. P( F ) The probability of occurrence an event F, given that event E has occurred, is called Conditional Probability of event F w.r.t. E, written as P(F/E) and is given by P(F/E)= P (E ∩ F ) , provided P(E) ≠ 0. P( E ) Properties Conditional Probability- • The Conditional Probability of an event E, given that event F has already occurred, is always greater than or equal to 0 and less than or equal to 1, i.e. 0 ≤ P ( E / F ) ≤ 1. • P(S/F) = 1, where S represent the sample space. • If A, B and F are the events if the sample space S, then P ( A ∪ B / F ) = P ( A / F ) + P ( B / F ) − P ( A ∩ B / F ). and P ( A ∪ B / F ) = P ( A / F ) + P ( B / F ). if A and B are mutually disjoint, i.e. A ∩ B = φ . Multiplication Theorem of Probability- jnvLudhiana Page 45 If E and F are two events given experiment, whose sample space is S, then the probability of simultaneous happening of E and F is given by P ( E ∩ F ) = P ( E ) P ( F / E ), P ( E ) ≠ 0 . OR P ( E ∩ F ) = P ( F ) P ( E / F ), P ( F ) ≠ 0 . Multiplication rule of probability for more than two Events- If E, F and G are three events of given experiment, whose sample space is S, then the probability of simultaneous happening of E, F and G is given by P ( E ∩ F ∩ G ) = P ( E ) P ( F / E ) P ( G / E ∩ F ). Independent Events- If E and F are two events of given experiment(s), then these are said to be independent if happening of one does not affect the probability of other. i.e. either P ( E / F ) = P ( E ), P ( F ) ≠ 0 . Or P ( F / E ) = P ( F ), P ( E ) ≠ 0 . If E and F are two events of same experiment, then these are said to be independent if P ( E ∩ F ) = P ( E ). P ( F ). jnvLudhiana Page 46 Multiplication Theorem of Probability For Independent Events - If E and F are two independent events of given experiment, whose sample space is S, then the probability of simultaneous happening of E and F is given by P ( E ∩ F ) = P ( E ) P ( F ). Partition of a Sample Space – The collection of subsets E1 , E 2 , E3 ,.......E n , of a sample space S of an experiment, is said to be a partition of the sample space S, if 1. P ( E i ) φ 0.∀i = 1,2,3,......n. 2. Ei ∩ E j = φ.∀i ≠ j, andi, j = 1,2,3,......n. 3. E1 ∪ E 2 ∪ E 3 ....... ∪ E n = S . In other words the collection of mutually disjoint and exhaustive events is called a partition of the given sample space. jnvLudhiana Page 47 Theorem of Total Probability – Let the events E1 , E 2 , E3 ,.......E n , form a partition of the sample space S, of an experiment. If A is any event of S, then n P ( A ) = P ( E 1 ) P ( A / E 1 ) + P ( E 2 ) P ( A / E 2 ) + .......... ...... P ( E n ) P ( A / E n ) = Σ P ( E i ) P ( A / E i ) i =1 Baye’s Theorem- Let E1, E2, E3, ……….., En be a partition of sample space S. If A is any random event of S, with P(A) > 0, then P ( E i / A) = P ( E i ).P ( A / E i ) n ;1 ≤ i ≤ n. ∑ P ( E i ).P ( A / E i ) i =1 Random Variable (rv) – jnvLudhiana Page 48 A random variable is a variable which takes real values depending upon the outcomes of an experiment. In other words we say that a rv is a function on the sample space of an experiment to the set of real numbers. Here in this class we deal with only discrete rv. A discrete random variable is a variable which takes integral values depending upon the outcomes of the experiment. Probability Distribution of rv – jnvLudhiana Page 49 The probability distribution of a rv X is given by X : P(X) : x1 x2 P(x1) P(x2) ………………xn …………….P(xn) Where the rv X takes possible values x1 ,x2, x3, ………, xn, with probabilities p(x1), n p(x2), p(x3), …………….p(xn) respectively, and p ( x i ) > 0 , ∑ p ( x i ) = 1, i = 1, 2 ,3,........, n. i =1 Mean, Variance and Standard Deviation of a rv – Let X be a rv whose possible values are x1 ,x2, x3, ………, xn occur with probabilities p(x1), p(x2), p(x3), …………….p(xn) respectively. Then Mean of rv written by E(X), variance of X written by Var(X), are given by jnvLudhiana Page 50 µ = E ( X ) = ∑ x i p ( x i ). n Var ( X ) = σ = ∑ ( x i − µ ) 2 p ( x i ) = E ( X − µ ) 2 2 x i =1 n S tan dardDeviat ion = Var ( X ) = ∑ (x i − µ ) 2 p ( xi ) . i =1 Bernoulli Trials – Trials of a random experiment are called Bernoulli Trials, if they satisfy the following conditions: 1. There should be a finite number of trials. 2. The trials should be independent. 3. Each trial has exactly two outcomes: called success or failure. 4. The probability of success (failure) remains same in each trial. jnvLudhiana Page 51 Binomial Distribution – Let we have n Bernoulli trials, with p as probability of success and q as probability of failure. Let X be a rv denotes the number of successes, then the probability of x n! successes is given by P ( X = x ) = p x q n − x = n C x p x q n − x ; x = 1, 2,3,........ n.( q = 1 − p ) x! ( n − x )! This type of rv distribution is called, binomial distribution. It is written as X~B(n, p), X is called binomial variate. Here in this case E(X) = np, and Var(X) = npq. (in syllabus without proof). jnvLudhiana Page 52