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![maths - Starisrics Complefe Formulas 1. Mean (Arithmetic Mean) 2. Median (a) Individual Series () Individudl Series Ex n+lth observation] Mean Median 2 (b) Discrete Series (b) Discrete Series Efx Value corresponding 1o Median Mean Zf 2 cumuldtive frequency 2 (c) Continuous Series Direct Method (c) Continuous Series Efx Mean [(ڈ-4)]xh Zf Median + (d) Assumed Mean Method] ऋ = a + (Efd 1 = lower limit of median class] Where: Zf frequency before cf = cumulafive Where: d = _ 0 median class Step (e) Deviation Method f = frequency of median class Zfd` h = clqss width Xh x =a + Zf Q Where: d = 4. Empirical Relation Mode = 3 Median २ Mean 3. Mode (a) Discrete Individual Series frequency of modal class f1 frequently Value occurring Mode frequency of preceeding class most fo frequency of succeeding class (b) Continuous Series f2 fol Mode = [ + Xh 2f1 fo - f2 6. Standard Deviation] Sfandard Deviafion (o) = V Variance 5. Variance] V02 () Individudl Series E( 7. Coefficient of Variation (CV) 02 డ (b) Discrete Series CV = X 100 Ef( ~) 02 ٤f 8. Comparison of Two Data Sets Step Deviation Method (c) Gredter CV = Gredter varidbility [Zfd2 Efal X h२ 02 consisfency Smaller CV = More Zf Zf Starisrics Complefe Formulas 1. Mean (Arithmetic Mean) 2. Median (a) Individual Series () Individudl Series Ex n+lth observation] Mean Median 2 (b) Discrete Series (b) Discrete Series Efx Value corresponding 1o Median Mean Zf 2 cumuldtive frequency 2 (c) Continuous Series Direct Method (c) Continuous Series Efx Mean [(ڈ-4)]xh Zf Median + (d) Assumed Mean Method] ऋ = a + (Efd 1 = lower limit of median class] Where: Zf frequency before cf = cumulafive Where: d = _ 0 median class Step (e) Deviation Method f = frequency of median class Zfd` h = clqss width Xh x =a + Zf Q Where: d = 4. Empirical Relation Mode = 3 Median २ Mean 3. Mode (a) Discrete Individual Series frequency of modal class f1 frequently Value occurring Mode frequency of preceeding class most fo frequency of succeeding class (b) Continuous Series f2 fol Mode = [ + Xh 2f1 fo - f2 6. Standard Deviation] Sfandard Deviafion (o) = V Variance 5. Variance] V02 () Individudl Series E( 7. Coefficient of Variation (CV) 02 డ (b) Discrete Series CV = X 100 Ef( ~) 02 ٤f 8. Comparison of Two Data Sets Step Deviation Method (c) Gredter CV = Gredter varidbility [Zfd2 Efal X h२ 02 consisfency Smaller CV = More Zf Zf - ShareChat](https://cdn4.sharechat.com/bd5223f_s1w/compressed_gm_40_img_956284_1cb90884_1772443275863_sc.jpg?tenant=sc&referrer=user-profile-service%2FrequestType50&f=863_sc.jpg "maths - Starisrics Complefe Formulas 1. Mean (Arithmetic Mean) 2. Median (a) Individual Series () Individudl Series Ex n+lth observation] Mean Median 2 (b) Discrete Series (b) Discrete Series Efx Value corresponding 1o Median Mean Zf 2 cumuldtive frequency 2 (c) Continuous Series Direct Method (c) Continuous Series Efx Mean [(ڈ-4)]xh Zf Median + (d) Assumed Mean Method] ऋ = a + (Efd 1 = lower limit of median class] Where: Zf frequency before cf = cumulafive Where: d = _ 0 median class Step (e) Deviation Method f = frequency of median class Zfd` h = clqss width Xh x =a + Zf Q Where: d = 4. Empirical Relation Mode = 3 Median २ Mean 3. Mode (a) Discrete Individual Series frequency of modal class f1 frequently Value occurring Mode frequency of preceeding class most fo frequency of succeeding class (b) Continuous Series f2 fol Mode = [ + Xh 2f1 fo - f2 6. Standard Deviation] Sfandard Deviafion (o) = V Variance 5. Variance] V02 () Individudl Series E( 7. Coefficient of Variation (CV) 02 డ (b) Discrete Series CV = X 100 Ef( ~) 02 ٤f 8. Comparison of Two Data Sets Step Deviation Method (c) Gredter CV = Gredter varidbility [Zfd2 Efal X h२ 02 consisfency Smaller CV = More Zf Zf Starisrics Complefe Formulas 1. Mean (Arithmetic Mean) 2. Median (a) Individual Series () Individudl Series Ex n+lth observation] Mean Median 2 (b) Discrete Series (b) Discrete Series Efx Value corresponding 1o Median Mean Zf 2 cumuldtive frequency 2 (c) Continuous Series Direct Method (c) Continuous Series Efx Mean [(ڈ-4)]xh Zf Median + (d) Assumed Mean Method] ऋ = a + (Efd 1 = lower limit of median class] Where: Zf frequency before cf = cumulafive Where: d = _ 0 median class Step (e) Deviation Method f = frequency of median class Zfd` h = clqss width Xh x =a + Zf Q Where: d = 4. Empirical Relation Mode = 3 Median २ Mean 3. Mode (a) Discrete Individual Series frequency of modal class f1 frequently Value occurring Mode frequency of preceeding class most fo frequency of succeeding class (b) Continuous Series f2 fol Mode = [ + Xh 2f1 fo - f2 6. Standard Deviation] Sfandard Deviafion (o) = V Variance 5. Variance] V02 () Individudl Series E( 7. Coefficient of Variation (CV) 02 డ (b) Discrete Series CV = X 100 Ef( ~) 02 ٤f 8. Comparison of Two Data Sets Step Deviation Method (c) Gredter CV = Gredter varidbility [Zfd2 Efal X h२ 02 consisfency Smaller CV = More Zf Zf - ShareChat")
![maths - Linedr Inequdlities Complete Formulqs 2. LINEAR INEQUALITY IN ONE VARIABLE INEQUALITIES BASIC SYMBOLS General form: Greater than: ax + b > 0 Less thon: ax + b < 0 Greater than or equal toः > b > 0 ax Less than or equal to: < b<0 X 3. RULES FOR SOLVING INEQUALITIES 4. SOLUTION SET Adding or subtracting the same number on both The solution of an inequality is] the set of dll volues of the sides does not change the inequality sign. variable that satisfy it Multiplying or dividing both sides by a positive Inequdlity sign remdins the sume number Multiplying or dividing both sides by a negative Inequality sign is reversed number >2 6. LINEAR INEQUALITIES IN 5. GRAPHICAL REPRESENTATION ON NUMBER LINE TWO VARIABLES Open circle (o) for or General form: Closed circle for < or > by < c 5 X by > c shows direction of solution ax Arrow by < c X by > c aX SOLUTION OF LINEAR INEQUALITY IN 8. GRAPHICAL RULES TWO VARIABLES Dotted line for < or Steps: Solid line for < or 2 Replace inequality with equality: ax by = @ Shaded region represents solution set Draw the boundary line Choose a test point (usually origin) Check which region satisfies the inequality Shade the required region | 10. IMPORTANT POINTS 9. SYSTEM OF LINEAR INEQUALITIES Inequalities have infinitely many solutions The common shaded region satisfying all Graphical method is compulsory for two] inequdlities is the solution variables Boundary line may or may not be included) Linedr Inequdlities Complete Formulqs 2. LINEAR INEQUALITY IN ONE VARIABLE INEQUALITIES BASIC SYMBOLS General form: Greater than: ax + b > 0 Less thon: ax + b < 0 Greater than or equal toः > b > 0 ax Less than or equal to: < b<0 X 3. RULES FOR SOLVING INEQUALITIES 4. SOLUTION SET Adding or subtracting the same number on both The solution of an inequality is] the set of dll volues of the sides does not change the inequality sign. variable that satisfy it Multiplying or dividing both sides by a positive Inequdlity sign remdins the sume number Multiplying or dividing both sides by a negative Inequality sign is reversed number >2 6. LINEAR INEQUALITIES IN 5. GRAPHICAL REPRESENTATION ON NUMBER LINE TWO VARIABLES Open circle (o) for or General form: Closed circle for < or > by < c 5 X by > c shows direction of solution ax Arrow by < c X by > c aX SOLUTION OF LINEAR INEQUALITY IN 8. GRAPHICAL RULES TWO VARIABLES Dotted line for < or Steps: Solid line for < or 2 Replace inequality with equality: ax by = @ Shaded region represents solution set Draw the boundary line Choose a test point (usually origin) Check which region satisfies the inequality Shade the required region | 10. IMPORTANT POINTS 9. SYSTEM OF LINEAR INEQUALITIES Inequalities have infinitely many solutions The common shaded region satisfying all Graphical method is compulsory for two] inequdlities is the solution variables Boundary line may or may not be included) - ShareChat](https://cdn4.sharechat.com/bd5223f_s1w/compressed_gm_40_img_56656_303e7e3b_1772347754829_sc.jpg?tenant=sc&referrer=user-profile-service%2FrequestType50&f=829_sc.jpg "maths - Linedr Inequdlities Complete Formulqs 2. LINEAR INEQUALITY IN ONE VARIABLE INEQUALITIES BASIC SYMBOLS General form: Greater than: ax + b > 0 Less thon: ax + b < 0 Greater than or equal toः > b > 0 ax Less than or equal to: < b<0 X 3. RULES FOR SOLVING INEQUALITIES 4. SOLUTION SET Adding or subtracting the same number on both The solution of an inequality is] the set of dll volues of the sides does not change the inequality sign. variable that satisfy it Multiplying or dividing both sides by a positive Inequdlity sign remdins the sume number Multiplying or dividing both sides by a negative Inequality sign is reversed number >2 6. LINEAR INEQUALITIES IN 5. GRAPHICAL REPRESENTATION ON NUMBER LINE TWO VARIABLES Open circle (o) for or General form: Closed circle for < or > by < c 5 X by > c shows direction of solution ax Arrow by < c X by > c aX SOLUTION OF LINEAR INEQUALITY IN 8. GRAPHICAL RULES TWO VARIABLES Dotted line for < or Steps: Solid line for < or 2 Replace inequality with equality: ax by = @ Shaded region represents solution set Draw the boundary line Choose a test point (usually origin) Check which region satisfies the inequality Shade the required region | 10. IMPORTANT POINTS 9. SYSTEM OF LINEAR INEQUALITIES Inequalities have infinitely many solutions The common shaded region satisfying all Graphical method is compulsory for two] inequdlities is the solution variables Boundary line may or may not be included) Linedr Inequdlities Complete Formulqs 2. LINEAR INEQUALITY IN ONE VARIABLE INEQUALITIES BASIC SYMBOLS General form: Greater than: ax + b > 0 Less thon: ax + b < 0 Greater than or equal toः > b > 0 ax Less than or equal to: < b<0 X 3. RULES FOR SOLVING INEQUALITIES 4. SOLUTION SET Adding or subtracting the same number on both The solution of an inequality is] the set of dll volues of the sides does not change the inequality sign. variable that satisfy it Multiplying or dividing both sides by a positive Inequdlity sign remdins the sume number Multiplying or dividing both sides by a negative Inequality sign is reversed number >2 6. LINEAR INEQUALITIES IN 5. GRAPHICAL REPRESENTATION ON NUMBER LINE TWO VARIABLES Open circle (o) for or General form: Closed circle for < or > by < c 5 X by > c shows direction of solution ax Arrow by < c X by > c aX SOLUTION OF LINEAR INEQUALITY IN 8. GRAPHICAL RULES TWO VARIABLES Dotted line for < or Steps: Solid line for < or 2 Replace inequality with equality: ax by = @ Shaded region represents solution set Draw the boundary line Choose a test point (usually origin) Check which region satisfies the inequality Shade the required region | 10. IMPORTANT POINTS 9. SYSTEM OF LINEAR INEQUALITIES Inequalities have infinitely many solutions The common shaded region satisfying all Graphical method is compulsory for two] inequdlities is the solution variables Boundary line may or may not be included) - ShareChat")
![maths - Complete Formulas Functions] 1, FUNCTION २. DOMAIN CODOMAIN & RANGE DEFINITION Domain > Sef of all possible inpufs A funcfion f from sef A to set Bl Set of all possible outputs rule that assigns each element is Codomain > exacfly one elemenf of 8. Sef of acfual oufpufs of A Range _ Range = Codomain Writren as: f:A-8 If x ٤ A Then f(x) & B TYPES OF FUNCTIONS 4 (a) One-One Function (Injective): f(x1) = f(x2) => X1 = X2 ३. REPRESENTATION OF FUNCTIONS Set of ordered pairs b) Many-One Funcfion: Two or more Arrow diagram] elements of domain have the same Graph image. Algebraic expression] (c) Onto Function (Surjective): 5. ALGEBRAIC FUNCTIONS Range Codomain] d) Info Funcfion : Range C Codomain (a) Constant Function: f(x) = c (e) Bijective Function: One-one and onto (b) Constant Function: f(x) = c function (b) Identity Function: f(x) = X (c) Polynomial Function: 6. MODULUS FUNCTION f(x) + ٠٠٠ + 00 anxn ৭n-1X"-1 f(x) = Ixll "(X) (d) Rationdl Function: F(x) = Ixl = X X>0 q(x)+0 Ixl =-xl X < 0 7. GREATEST INTEGER FUNCTION 8 COMPOSITION OF FUNCTIONS f(x) = [x] If f : A -> B ond g : B -> C greatest integer < x [x] (g ৭ f)(x) g(f(x)) Example: [2.7] = 2, [-1.3] =-2 10. GRAPH OF FUNCTIONS 9. INVERSE FUNCTION A function f has an inverse if and Straight line] Linear funcfion ) f is bijecfive only if Quadraric function Parabola If f(x) then f-1(y) = * Horizonfql line' ٧ Constant function Property: f(f-1(x)) = X , f-1(f(x)) = X Complete Formulas Functions] 1, FUNCTION २. DOMAIN CODOMAIN & RANGE DEFINITION Domain > Sef of all possible inpufs A funcfion f from sef A to set Bl Set of all possible outputs rule that assigns each element is Codomain > exacfly one elemenf of 8. Sef of acfual oufpufs of A Range _ Range = Codomain Writren as: f:A-8 If x ٤ A Then f(x) & B TYPES OF FUNCTIONS 4 (a) One-One Function (Injective): f(x1) = f(x2) => X1 = X2 ३. REPRESENTATION OF FUNCTIONS Set of ordered pairs b) Many-One Funcfion: Two or more Arrow diagram] elements of domain have the same Graph image. Algebraic expression] (c) Onto Function (Surjective): 5. ALGEBRAIC FUNCTIONS Range Codomain] d) Info Funcfion : Range C Codomain (a) Constant Function: f(x) = c (e) Bijective Function: One-one and onto (b) Constant Function: f(x) = c function (b) Identity Function: f(x) = X (c) Polynomial Function: 6. MODULUS FUNCTION f(x) + ٠٠٠ + 00 anxn ৭n-1X"-1 f(x) = Ixll "(X) (d) Rationdl Function: F(x) = Ixl = X X>0 q(x)+0 Ixl =-xl X < 0 7. GREATEST INTEGER FUNCTION 8 COMPOSITION OF FUNCTIONS f(x) = [x] If f : A -> B ond g : B -> C greatest integer < x [x] (g ৭ f)(x) g(f(x)) Example: [2.7] = 2, [-1.3] =-2 10. GRAPH OF FUNCTIONS 9. INVERSE FUNCTION A function f has an inverse if and Straight line] Linear funcfion ) f is bijecfive only if Quadraric function Parabola If f(x) then f-1(y) = * Horizonfql line' ٧ Constant function Property: f(f-1(x)) = X , f-1(f(x)) = X - ShareChat](https://cdn4.sharechat.com/33d5318_1c8/3174ddb4_1772289770826_sc.jpeg?tenant=sc&referrer=user-profile-service%2FrequestType50&f=26_sc.jpeg "maths - Complete Formulas Functions] 1, FUNCTION २. DOMAIN CODOMAIN & RANGE DEFINITION Domain > Sef of all possible inpufs A funcfion f from sef A to set Bl Set of all possible outputs rule that assigns each element is Codomain > exacfly one elemenf of 8. Sef of acfual oufpufs of A Range _ Range = Codomain Writren as: f:A-8 If x ٤ A Then f(x) & B TYPES OF FUNCTIONS 4 (a) One-One Function (Injective): f(x1) = f(x2) => X1 = X2 ३. REPRESENTATION OF FUNCTIONS Set of ordered pairs b) Many-One Funcfion: Two or more Arrow diagram] elements of domain have the same Graph image. Algebraic expression] (c) Onto Function (Surjective): 5. ALGEBRAIC FUNCTIONS Range Codomain] d) Info Funcfion : Range C Codomain (a) Constant Function: f(x) = c (e) Bijective Function: One-one and onto (b) Constant Function: f(x) = c function (b) Identity Function: f(x) = X (c) Polynomial Function: 6. MODULUS FUNCTION f(x) + ٠٠٠ + 00 anxn ৭n-1X\"-1 f(x) = Ixll \"(X) (d) Rationdl Function: F(x) = Ixl = X X>0 q(x)+0 Ixl =-xl X < 0 7. GREATEST INTEGER FUNCTION 8 COMPOSITION OF FUNCTIONS f(x) = [x] If f : A -> B ond g : B -> C greatest integer < x [x] (g ৭ f)(x) g(f(x)) Example: [2.7] = 2, [-1.3] =-2 10. GRAPH OF FUNCTIONS 9. INVERSE FUNCTION A function f has an inverse if and Straight line] Linear funcfion ) f is bijecfive only if Quadraric function Parabola If f(x) then f-1(y) = * Horizonfql line' ٧ Constant function Property: f(f-1(x)) = X , f-1(f(x)) = X Complete Formulas Functions] 1, FUNCTION २. DOMAIN CODOMAIN & RANGE DEFINITION Domain > Sef of all possible inpufs A funcfion f from sef A to set Bl Set of all possible outputs rule that assigns each element is Codomain > exacfly one elemenf of 8. Sef of acfual oufpufs of A Range _ Range = Codomain Writren as: f:A-8 If x ٤ A Then f(x) & B TYPES OF FUNCTIONS 4 (a) One-One Function (Injective): f(x1) = f(x2) => X1 = X2 ३. REPRESENTATION OF FUNCTIONS Set of ordered pairs b) Many-One Funcfion: Two or more Arrow diagram] elements of domain have the same Graph image. Algebraic expression] (c) Onto Function (Surjective): 5. ALGEBRAIC FUNCTIONS Range Codomain] d) Info Funcfion : Range C Codomain (a) Constant Function: f(x) = c (e) Bijective Function: One-one and onto (b) Constant Function: f(x) = c function (b) Identity Function: f(x) = X (c) Polynomial Function: 6. MODULUS FUNCTION f(x) + ٠٠٠ + 00 anxn ৭n-1X\"-1 f(x) = Ixll \"(X) (d) Rationdl Function: F(x) = Ixl = X X>0 q(x)+0 Ixl =-xl X < 0 7. GREATEST INTEGER FUNCTION 8 COMPOSITION OF FUNCTIONS f(x) = [x] If f : A -> B ond g : B -> C greatest integer < x [x] (g ৭ f)(x) g(f(x)) Example: [2.7] = 2, [-1.3] =-2 10. GRAPH OF FUNCTIONS 9. INVERSE FUNCTION A function f has an inverse if and Straight line] Linear funcfion ) f is bijecfive only if Quadraric function Parabola If f(x) then f-1(y) = * Horizonfql line' ٧ Constant function Property: f(f-1(x)) = X , f-1(f(x)) = X - ShareChat")
