DEVELOPMENT OF SUPPORT MATERIAL IN MATHEMATICS FOR CLASS XI GROUP LEADER. Sl. No. Name Designation TEAM MEMBERS

Transcription

1 DEVELOPMENT OF SUPPORT MATERIAL IN MATHEMATICS FOR CLASS XI GROUP LEADER Sl. No. Name Designation Dr. Vandita Kalra Vice Principal GGSSS, Kirti Nagar TEAM MEMBERS. Joginder Arora PGT Maths RPVV, Hari Nagar. Manoj Kumar PGT Maths RPVV, Kishan Ganj 3. Dr. Rajpal Singh PGT Maths RPVV Gandhi Nagar 4. Sanjeev Kumar PGT Maths RPVV, Raj Niwas Marg 5. Anita Saluja PGT Maths GGSSS, Kirti Nagar SCERT FACULTY Dr. Anil Kumar Teotia Sr. Lecturer DIET Dilshad Garden

2 REVIEW OF SUPPORT MATERIAL : 05-6 GROUP LEADER Sl. No. Name Designation SANJEEV KUMAR Vice Principal RPVV, Kishan Ganj TEAM MEMBERS. Pradeep Kumar Jain Lecturer Maths RPVV, Rajniwas Marg. Suman Arora Lecturer Maths RPVV, Paschim Vihar 3. Pawan Kumar Lecturer Maths RPVV, Kishan Ganj

3 CONTENTS S.No. Chapter Page. Sets 6. Relations and Functions 5 3. Trigonometric Functions 5 4. Principle of Mathematical Induction Complex Numbers and Quadratic Equations Linear Inequalities Permutations and Combinations Binomial Theorem Sequences and Series 6 0. Straight Lines 69. Conic Sections 76. Introduction to Three Dimensional Coordinate Geometry 8 3. Limits and Derivatives Mathematical Reasoning Statistics 0 6. Probability 08 Model Test Paper - I 6 Model Test Paper - II 33 Model Test Paper - III 38

4 COURSE STRUCTURE CLASS XI One Paper Three Hours Max. Marks. 00 Units Marks I. Sets and Functions 9 II. Algebra 37 III. Coordinate Geometry 3 IV. Calculus 06 V. Mathematical Reasoning 03 VI. Statistics and Probability 00 Unit-I : Sets and Functions. Sets : () Periods Sets and their representations. Empty set. Finite and Infinite sets. Equal sets. Subsets. Subsets of the set of real numbers especially intervals (with notations). Power set. Universal set. Venn diagrams. Union and Intersection of sets. Difference of sets. Complement of a set. Properties of Complement Sets.. Relations and Functions : (4) Periods Ordered pairs, Cartesian product of sets. Number of elements in the cartesian product of two finite sets. Cartesian product of the set of reals with itself (upto R R R). Definition of relation, pictorial diagrams, [XI Mathematics]

5 domain, codomain and range of a relation. Function as a special kind of relation from one set to another. Pictorial representation of a function, domain, co-domain and range of a function. Real valued functions, domain and range of these functions, constant, identity, polynomial, rational, modulus, signum exponential, logarithrnic functions and greatest integer functions, with their graphs. Sum, difference, product and quotients of functions. 3. Trigonometric Functions : (8) Periods Positive and negative angles. Measuring angles in radians and in degrees and conversion from one measure to another. Definition of trigonometric functions with the help of unit circle. Truth of the identity sin x + cos x =, for all x. Signs of trigonometric functions. Domain and range of trignometric functions and their graphs. Expressing sin (x ± y) and cos (x ± y) in terms of sin x, sin y, cosx and cos y. Deducing the identities like the following: tan x tan y cot x cot y tan x y, cot x y, tan x tan y cot y cot x sin x sin y x y x y sin cos, cos x cos y x y x y cos cos, x y x y sin x sin y cos sin, x y x y cos x cos y sin sin. Identities related to sinx, cosx, tan x, sin3x, cos3x and tan3x. General solution of trigonometric equations of the type sin = sin, cos = cos and tan = tan. Proof and simple applications of sine and cosine formulae. Unit-II : Algebra. Principle of Mathematical Induction : (06) Periods Process of the proof by induction, motivating the applications of the method by looking at natural numbers as the least inductive subset of real numbers. The principle of mathematical induction and simple applications. [XI Mathematics]

6 . Complex Numbers and Quadratic Equations : (0) Periods Need for complex numbers, especially, to be motivated by inability to solve some of the quardratic equations. Algebraic properties of complex numbers. Argand plane and polar representation of complex numbers. Statement of Fundamental Theorem of Algebra, solution of quadratic equations in the complex number system. Square root of a complex number. 3. Linear Inequalities : (0) Periods Linear inequalities. Algebraic solutions of linear inequalities in one variable and their representation on the number line. Graphical solution of linear inequalities in two variables. Graphical solution of system of linear inequalities in two variables. 4. Permutations and Combinations : () Periods Fundamental principle of counting. Factorial n (n!) Permutations and combinations, derivation of formulae and their connections, simple applications. 5. Binomial Theorem : (08) Periods History, statement and proof of the binomial theorem for positive integral indices. Pascal s triangle, General and middle term in binomial expansion, simple applications. 6. Sequence and Series : (0) Periods Sequence and Series. Arithmetic progression (A.P.) arithmetic mean (A.M.) Geometric progression (G.P.), general term of a G.P., sum of n terms of a G.P., Arithmetic and Geometric series, Infinite G.P. and its sum, geometric mean (G.M.), relation between A.M. and G.M. Sum to n terms of the special series n n n 3 k, k and k. k k k Unit-III : Coordinate Geometry. Straight Lines : (09) Periods Brief recall of two dimensional geometry from earlier classes. Shifting of origin. Slope of a line and angle between two lines. Various forms of equations of a line : parallel to axes, point-slope form, slope-intercept form, two-point form, intercept form and normal form. General equation of a line. Equation of family of lines passing through the point of intersection of two lines. Distance of a point from a line. 3 [XI Mathematics]

7 . Conic Sections : () Periods Sections of a cone : circles, ellipse, parabola, hyperbola, a point, a straight line and a pair of intersecting lines as a degenerated case of a conic section. Standard equations and simple properties of parabola, ellipse and hyperbola. Standard equation of a circle. 3. Introduction to Three-Dimensional Geometry (08) Periods Coordinate axes and coordinate planes in three dimensions. Coordinates of a point. Distance between two points and section formula. Unit-IV : Calculus. Limits and Derivatives : (8) Periods Limit of function introduced as rate of change of distance function and its geometric meaning. Limits of Polynomials and rational function, trigonometric, exponential and logarithmic functions. x o x o x log e x e lim, lim. Definition of derivative, relate it to slope x x of tangent of the curve, derivative of sum, difference, product and quotient of functions. Derivatives of polynomial and trigonometric functions. Unit-V : Mathematical Reasoning. Mathematical Reasoning : (08) Periods Mathematically acceptable statements. Connecting words/phrasesconsolidating the understanding of if and only if (necessary and sufficient) condition, implies, and/or, implied by, and, or, there exists and their use through variety of examples related to real life and Mathematics. Validating the statements involving the connecting words, difference between contradiction, converse and contrapositive. Unit-VI : Statistics and Probability. Statistics Measure of dispersion, range, mean deviation, variance and standard deviation of ungrouped/grouped data. Analysis of frequency distributions with equal mean but different variances. 4 [XI Mathematics]

8 . Probability Random experiments, outcomes; Sample spaces (set representation); Event; Occurence of events, "not', "and" and "or" events. exhaustive events, mutually exclusive events, Axiometic (set theoretic) probability connections with the theories of earlier classes. Probability of an event, probability of "not", "and" and "or" events. 5 [XI Mathematics]

9 CHAPTER - SETS KEY POINTS A set is a well-defined collection of objects. There are two methods of representing a set : (a) Roster or Tabular form e.g. :- (b) Types of sets : (i) (ii) (iii) (iv) natural numbers less than 5 = {,, 3, 4} Set-builder form or Rule method e.g. : Vowels in English alphabet = {x : x is a vowel in the English alphabet } Empty set or Null set or void set Finite set Infinite set Singleton set Subset : A set A is said to be a subset of set B if a A a B, a A. We write it as A B Equal sets : Two sets A and B are equal if they have exactly the same elements i.e A = B if A and B A Power set : The collection of all subsets of a set A is called power set of A, denoted by P(A) i.e. P(A) = { B : B A } If A is a set with n(a) = m then n [P(A)] = m. Equivalent sets : Two finite sets A and B are equivalent, if their cardinal numbers are same i.e., n(a) = n(b). Proper set and super set : If A B then A is called the proper subset of B and B is called the superset of A. 6 [XI Mathematics]

10 Types of Intervals Open Interval (a, b) = { x R : a < x < b } Closed Interval [a, b] = { x R : a x b } Semi open or Semi closed Interval, (a,b] = { x R : a < x b} [a,b) = { x R : a x < b} Union of two sets A and B is, A B = { x : x A or x B } U A B AUB Intersection of two sets A and B is, A B = { x : x A and x B} U A B A B Disjoint sets : Two sets A and B are said to be disjoint if A B = U A B 7 [XI Mathematics]

11 Difference of sets A and B is, A B = { x : x A and x B} U A B Difference of sets B and A is, A B B A = { x : x B and x A } U A B B A Complement of a set A, denoted by A' or A c is A' = A c = U A = { x : x U and x A} A Properties of complement sets :. Complement laws (i) A A' = U (ii) A A' = (iii) (A')' = A. De Morgan's Laws (i) (A B)' = A' B' (ii) (A B)' = A' B' 8 [XI Mathematics]

12 Note : This law can be extended to any number of sets. 3. ' = and ' = 4. If A B then B A Laws of Algebra of sets. (i) (ii) A = A A = A B = A B' = A (A B) Commutative Laws : (i) A B = B A (ii) A B = B A Associative Laws : (i) (A B) C = A (B C) (ii) (A B) C = A (B C) Distributive Laws : (i) A (B C) = (A B) (A C) (ii) A (B C) = (A B) (A C) If A B, then A B = A and A B = B When A and B are disjoint n(a B) = n(a) + n(b) When A and B are not disjoint n(a B) = n(a) + n(b) n(a B) n(a B C) = n(a) + n(b) + n(c) n(a B) n(b C) n(a C) + n(a B C) VERY SHORT ANSWER TYPE QUESTIONS ( MARK) Which of the following are sets? Justify your answer.. The collection of all the months of a year beginning with letter M. The collection of difficult topics in Mathematics. Let A = {,3,5,7,9}. Insert the appropriate symbol or in blank spaces : (Question- 3,4) 9 [XI Mathematics]

13 3. A 4. 5 A 5. Write the set A = { x : x is an integer, x < 4} in roster form 6. List all the elements of the set, A x : x Z, x 7. Write the set B = {3,9,7,8} in set-builder form. Which of the following are empty sets? Justify. (Question- 8,9) 8. A = { x : x N and 3 <x <4} 9. B = { x : x N and x = x} Which of the following sets are finite or Infinite? Justify. (Question-0,) 0. The set of all the points on the circumference of a circle.. B = { x : x N and x is an even prime number}. Are sets A = {,}, B = { x : x Z, x 4 = 0} equal? Why? 3. Write ( 5,9] in set-builder form 4. Write {x : x R, 3 x < 7} as interval. 5. If A = {,3,5}, how many elements has P(A)? 6. Write all the possible subsets of A = {5,6}. If A = {,3,4,5}, B = { 3,5,6,7} find (Question- 7,8) 7. A B 8. A B 9. If A = {,,3,6}, B = {,, 4, 8} find B A 0. If A = {p, q}, B = {p, q, r}, is B a superset of A? Why?. Are sets A = {,,3,4}, B = { x : x N and 5 x 7} disjoint? Why? 0 [XI Mathematics]

14 . If X and Y are two sets such that n(x) = 9, n(y) = 37 and n(x Y) =, find n(x Y). 3. Consider the following sets, A = {, 5}, B = {,, 3, 4}, C = {,, 3, 4, 5} Insert the correct symbol or between each pair of sets (i) B (ii) A B (iii) A C (iv) B C SHORT ANSWER TYPE QUESTIONS (4 MARKS) 4. If = {,,3,4,5,6,7,8,9}, A = {,3,5,7,9}, B = {,,4,6}, verify (i) (A ' = A' B' (ii) B A = B A' = B (A B) 5. Let A, B be any two sets. Using properties of sets prove that, (i) (ii) (A B) B = A B (A B) A = B A [ Hint : A B = A B' and use distributive law.] 6. In a group of 800 people, 500 can speak Hindi and 30 can speak English. Find (i) (ii) How many can speak both Hindi and English? How many can speak Hindi only? 7. A survey shows that 84% of the Indians like grapes, whereas 45% like pineapple. What percentage of Indians like both grapes and pineapple? 8. In a survey of 450 people, it was found that 0 play cricket, 60 play tennis and 70 play both cricket as well as tennis. How many play neither cricket nor tennis? 9. In a group of students, 5 students know French, 00 know Spanish and 45 know both. Each student knows either French or Spanish. How many students are there in the group? 30. If A = [ 3, 5), B = (0, 6] then find (i) A B, (ii) A B [XI Mathematics]

15 Hots (4 Marks) 3. Show that n{p {P (P()}} = 4 LONG ANSWER TYPE QUESTIONS (6 MARKS) 3. In a town of 0,000 families it was found that 40% families buy newspaper A, 0% families buy newspaper B and 0% families by newspaper C. 5% families buy A and B, 3%, buy B and C and 4% buy A and C. If % families buy all the three newspapers, find the no of families which buy () A only () B only (3) none of A, B and C (4) exactly two newspapers (5) exactly one newspaper (6) A and C but not B (7) atleast one of A, B, C. 33. In a group of 84 persons, each plays atleast one game out of three viz. tennis, badminton and cricket. 8 of them play cricket, 40 play tennis and 48 play badminton. If 6 play both cricket and badminton and 4 play tennis and badminton and no one plays all the three games, find the number of persons who play cricket but not tennis. 34. In a class, 8 students took Physics, 3 students took Chemistry and 4 students took Mathematics of these 3 took both Chemistry and Mathematics, took both Physics and Chemistry and took both Physics and Mathematics. If 6 students offered all the three subjects, find : () The total number of students. () How many took Maths but not Chemistry. (3) How many took exactly one of the three subjects. 35. Using properties of sets and their complements prove that () (A / B) (B / A) = (A B) / (A B) () (A B) (A C) = A (B C) (3) A (B C) = (A B) (A C) (4) A (B C) = (A B) (A C) (5) (A B) (A C) = A (B C) where A, B, C are any three sets. [XI Mathematics]

16 ANSWERS. Set. Not a set 3. A = {, 0,,, 3} 6. A = { 0,,,3,4,5} 7. B = { x : x = 3 n, n N and n 4} 8. Empty set because no natural number is lying between 3 and 4 9. Non-empty set because B = {} 0. Infinite set because circle is a collection of infinite points whose distances from the centre is constant called radius.. Finite set because B = {}. Yes because x 4 = 0 ; x =, both are integers 3. { x : x R, 5 < x 9} 4. [ 3,7) 5. 3 = 8 6., { 5}, {6}, {5,6} 7. A B = {,3,4,5,6,7} 8. A B = {3, 5} 9. B A = {4,8} 0. Yes, because A is a subset of B. Yes, because A B =. n(x Y) = (i) ; (ii) ; (iii) ; (iv) 6. (i) 0 people can speak both Hindi and English (ii) 480 people can speak Hindi only 7. 9% of the Indians like both grapes and pineapple. 8. Hint : set of people surveyed A set of people who play cricket B set of people who play tennis 3 [XI Mathematics]

17 Number of people who play neither cricket nor tennis = n[(a B)'] = n(u) n(a B) = = There are 80 students in the group. 30. (i) [ 3, 0]; (ii) [ 3, 6] 3. (i) 3300 (ii) 400 (iii) 4000 (iv) 800 (v) 4800 (vi) 400 (vii) (i) 35 (ii) (iii) 4 [XI Mathematics]

18 CHAPTER RELATIONS AND FUNCTIONS KEY POINTS Cartesian Product of two non-empty sets A and B is given by, A B = { (a,b) : a A, b B} If (a,b) = (x, y), then a = x and b = y Relation R from a non-empty set A to a non-empty set B is a subset of A B. Domain of R = {a : (a,b) R} Range of R = { b : (a,b) R} Co-domain of R = Set B Range Co-domain If n(a) = p, n(b) = q then n(a B) = pq and number of relations = pq Image : If the element x of A corresponds to y B under the function f, then we say that y is image of x under f f (x) = y If f (x) = y, then x is preimage of y. A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B. D f = {x : f(x) is defined} R f = {f(x) : x D f } Let A and B be two non-empty finite sets such that n(a) = p and n(b) = q then number of functions from A to B = q p. Identity function, f : R R; f(x) = x x R where R is the set of real numbers. D f = R R f = R 5 [XI Mathematics]

19 X Y O f(x) = x X Constant function, f : R R; f(x) = c x R where c is a constant Y D f = R R f = {c} X O Y f(x) = c }c X Y Modulus function, f : R R; f(x) = x x R D f = R R f = R + {0} = { x : x R: x 0} Y X O X Y Signum function, f : R R ; D f = R R f = {,0,}, If x 0 f x 0, if x 0, if x 0 6 [XI Mathematics]

20 X Y O y = Y y = Greatest Integer function, f : R R; f(x) = [x], x R assumes the value of the greatest integer, less than or equal to x X D f = R R f = Z f : R R, f(x) = x X 3 O Y 3 4 Y X D f = R R f = [0, Y X O X f : R R, f(x) = x 3 Y D f = R R f = R Y X O X Y 7 [XI Mathematics]

21 Exponential function, f : R R ; f(x) = a x, a > 0, a D f = R R f = (0, ) Y Y X O (0, ) (0, ) X X O X Y Y 0 < a < a > Natural exponential function, f(x) = e x e =..., e 3!! 3! Logarithmic functions, f : (0, ) R ; f(x) log a x, a > 0, a D f = (0, ) R f = R Y Y X O (0, ) X X O (0, ) X Y Y Natural logarithrnic function f(x) log e x or log x. Let f : X R and g : X R be any two real functions where x R then (f ± g) (x) = f(x) ± g(x) x X (fg) (x) = f(x) g(x) x X f g f x x x X provided g x 0 g x 8 [XI Mathematics]

22 VERY SHORT ANSWER TYPE QUESTIONS ( MARK). Find a and b if (a, b + 5) = (, 3) If A = {,3,5}, B = {,3} find : (Question-, 3). A B 3. B A Let A = {,}, B = {,3,4}, C = {4,5}, find (Question- 4,5) 4. A (B C) 5. A (B C) 6. If P = {,3}, Q = {,3,5}, find the number of relations from A to B 7. If A = {,,3,5} and B = {4,6,9}, R = {(x, y) : x y is odd, x A, y B} Write R in roster form Which of the following relations are functions. Give reason. (Questions 8 to 0) 8. R = { (,), (,), (3,3), (4,4), (4,5)} 9. R = { (,), (,), (,3), (,4)} 0. R = { (,), (,5), (3,8), (4,0), (5,), (6,)} Which of the following arrow diagrams represent a function? Why? (Question-,).. X a b c d X Y Y [XI Mathematics]

23 Let f and g be two real valued functions, defined by, f(x) = x, g(x) = 3x +, find : (Question 3 to 6) 3. (f + g)( ) 4. (f g)() 5. (fg)( ) f g If f(x) = x 3, find the value of, f 5 f 5 8. Find the domain of the real function, f x x 4 9. Find the domain of the function, f x x x 3 x 5x 6 Find the range of the following functions, (Question- 0,) f x 4 x 0.. f(x) = x +. Find the domain of the relation, R = { (x, y) : x, y Z, xy = 4} Find the range of the following relations : (Question-3, 4) 3. R = {(a,b) : a, b N and a + b = 0} 4. R x, : x z, 0 x 6 x 0 [XI Mathematics]

24 SHORT ANSWER TYPE QUESTIONS (4 MARKS) 5. Let A = {,,3,4}, B = {,4,9,6,5} and R be a relation defined from A to B as, (a) R = {(x, y) : x A, y B and y = x } Depict this relation using arrow diagram. (b) Find domain of R. (c) Find range of R. (d) Write co-domain of R. 6. Let R = { (x, y) : x, y N and y = x} be a relation on N. Find : (i) (ii) (iii) Domain Codomain Range Is this relation a function from N to N? 7. Let f x x, when 0 x. x, when x 5 g x x, when 0 x 3. x, when 3 x 5 Show that f is a function while g is not a function. 8. Find the domain and range of, f(x) = x Draw the graph of the Greatest Integer function 30. Draw the graph of the Constant function, f : R R; f(x) = x R. Also find its domain and range. 3. Draw the graph of the function x Find t h e d o main an d ran g e o f t h e f o llowing real f u n ct ions (Question 33 to 38) 3. (x) f x 4 [XI Mathematics]

25 f 33. (x) f 34. (x) x x x x 35. f (x) x 9 x 3 f 36. (x) 4 x x f (x) = [x 3] ANSWERS. a = 3, b =. A B = {(,), (,3), (3,), (3,3), (5,), (5,3)} 3. B A = { (,), (,3), (,5), (3,), (3,3), (3,5)} 4. {(,4), (,4)} 5. {(,), (,3), (,4), (,5), (,), (,3), (,4), (,5)} 6. 6 = R = { (,4), (,6), (,9), (3,4), (3,6), (5,4), (5,6)} 8. Not a function because 4 has two images. 9. Not a function because does not have a unique image. 0. Function because every element in the domain has its unique image.. Function because every element in the domain has its unique image.. Not a function because is having more than one image 0 and [XI Mathematics]

26 (, ] [, ) 9. R {,3} 0. (, 0) [, 4 ). [,). { 4,,,,,4} 3. {,4,6,8} (a) A 3 4 B ,,,, (b) {,,3,4} (c) {,4,9,6} (d) {,4,9,6,5} 6. (i) N (ii) N (iii) Set of even natural numbers yes, R is a function from N to N. 8. Domain is R Range is [ 3, ) 30. Domain = R Range = {} 3. Y (0, ) y = x X O (, 0) Y y = x X 3 [XI Mathematics]

27 3. Domain = R Range = {, } 33. Domain = R {} Range = R {} 34. Domain = R Range = {, } 35. Domain = R {3} Range = R {6} 36. Domain = R 4} Range = { } 37. Domain = R Range = {, } 4 [XI Mathematics]

28 CHAPTER - 3 TRIGONOMETRIC FUNCTIONS KEY POINTS A radian is an angle subtended at the centre of a circle by an arc whose length is equal to the radius of the circle. We denote radian by c. radian = 80 degree radian = 80 degree degree = radian 80 If an arc of length l makes an angle radian at the centre of a circle of radius r, we have l r Quadrant I II III IV t- functions which All sin x tan x cos x are positive cosec x cot x sec x Function x x x x + x x + x sin sin x cos x cos x sin x sin x sin x sin x cos cos x sin x sin x cos x cos x cos x cos x tan tan x cot x cot x tan x tan x tan x tan x cosec cosec x sec x sec x cosec x cosec x cosec x cosec x sec sec x cosec x cosec x sec x sec x sec x sec x cot cot x tan x tan x cot x cot x cot x cot x 5 [XI Mathematics]

29 Function Domain Range sin x R [,] cos x R [,] tan x R (n ) ; n z R Cosec x R {n; n z} R (,) Sec x R (n ) ; n z R (,) cot x R {n, n z} R Some Standard Results sin (x + y) = sinx cosy + cosx siny cos (x + y) = cosx cosy sinx siny tan x tan y tan(x y) tan x. tan y cot(x y) cot x. cot y cot y cot x sin (x y) = sinx cosy cosx siny cos (x y) = cosx cosy + sinx siny tan x tan y tan(x y) tan x.tany cot(x y) cot x. cot y cot y cot x tan(x y z) tan x tan y tan z tan x tan y tan z tan x tan y tan y. tan z tan z tan x sinx cosy = sin(x + y) + sin(x y) cosx siny = sin(x + y) sin(x y) cosx cosy = cos(x + y) + cos(x y) sinx siny = cos(x y) cos(x + y) 6 [XI Mathematics]

30 x y x y sin x sin y sin cos x y x y sin x sin y cos sin x y x y cos x cos y cos cos x y x y cos x cos y sin sin sin x sin x cos x tan x tan x cos x = cos x sin x = cos x = sin x = tan x tan x tan x tan x tan x sin 3x = 3 sinx 4 sin 3 x cos 3x = 4 cos 3 x 3 cos x tan 3x = 3 3 tan x tan x 3 tan x sin(x + y) sin(x y) = sin x sin y = cos y cos x cos(x + y) cos(x y) = cos x sin y = cos y sin x Principal solutions The solutions of a trigonometric equation for which 0 x < are called its principal solutions. General solution A solution of a trigonometric equation, generalised by means of periodicity, is known as the general solution. 7 [XI Mathematics]

31 General solutions of trigonometric equations : sin = 0 = n n z cos = 0 = (n n z tan = 0 = n n z sin = sin = n ( ) n n z cos = cos = n n z tan = tan = n n z Law of sines or sine formula The lengths of sides of a triangle are proportional to the sines of the angles opposite to them i.e., A a b c sin A sin B sin C Law of cosines or cosine formula In any ABC B C cos A cos B cos C b c a bc c a b ca a b c ab VERY SHORT ANSWER TYPE QUESTIONS ( MARK). Find the radian measure corresponding to (i) 5 37' 30''; (ii) Find the degree measure corresponding to (i) 6 c ; (ii) 4c 8 [XI Mathematics]

32 3. (i) Find the length of an arc of a circle of radius 5 cm subtending a central angle measuring 5. (ii) Find each interior angle of a regular decagon in radian Find the value of tan 3 5. Find the value of (i) sin( 5 );(ii) cos ( 070 ) 6. Find the value of (i) tan 5 ; (ii) sin If sin A = 3 5 and < A <, find cos A 8. If tan A = a a and tan B = a then find the value of A + B. 9. Express sin + sin 4 as the product of sines and cosines. 0. Express cos4x sinx as an algebraic sum of sines or cosines.. Write the range of cos. What is domain of sec? 3. Find the principal solutions of (i) cosec x = ; (ii) cos x =. 4. Write the general solution of sin x 0 5. If sinx = 6. If cosx = 5 3 and 0 < x < find the value of cos x 3 7. Evaluate : sin 75 sin 5 8. If and x lies in quadrant III, find the value of sin x tan, then find cos 7 9. Evaluate : sin ( + x) sin ( x) cosec x x x 0. What is sign of cos sin when (i) 0 < x < / (ii) 0 < x < 9 [XI Mathematics]

33 . What is maximum value of 3 7cos 5x?. Find the range of f (x) = sin x A. In any ABC, if a =, b = 3 and sin A =, 3 find B. B. Find-the angle between hour hand and minute hand of a clock at quarter to five. SHORT ANSWER TYPE QUESTIONS (4 MARKS) 3. A horse is tied to a post by a rope. If the horse moves along a circular path, always keeping the rope tight and describes 88 metres when it traces 7 at the centre, find the length of the rope. 4. If the angles of a triangle are in the ratio 3:4:5, find the smallest angle in degrees and the greatest angle in radians. 5. If sinx = and x lies in the second quadrant, show that secx + tanx = If sec x = and 3 tan x cosec x x find the value of cot x cosec x. 7. Prove that tan 50 = tan 40 + tan If cot, sec where 3 the value of tan ( ) 9. tan 3x = tan 4x + tan 9x + tan 4x tan 9x tan 3x Prove the following Identities 3 and, find tan 5 tan 3 4 cos cos 4 tan 5 tan 3 cos x sin x cos x sin x tan x cos x sin x cos x sin x cos 4x sin 3x cos x sin x tan x sin 4x sin x cos 6x cos x sin cos tan sin cos 30 [XI Mathematics]

34 34. tan tan tan(60 + ) = tan Show that (i) cos0 cos40 cos80 = 8 (ii) sin0 sin 40 sin60 sin80 = Show that cos 4 cos 37. cos x x Pr ove that tan sin x cos 0 + cos 0 + cos 30 = x y x x y x y x x y sin sin sin cos cos cos tan x x 3x 40. sin x sin x sin 4x sin 5x 4 cos cos sin 3x. 4. sec 8x tan 8 x. sec 4x tan x 4. Evaluate (i) cos 36 (ii) 8 (iii) 3 tan 43. Draw the graph of cosx in [0, Find the general solution of the following equations (Q.No. 44 to Q. No. 49) 44. If sin sin x y a b x y a b then prove that tan tan x y a b [Hint : By applying Companando and Dividendo A C A B C D B D A B C D 45. (i) sin 7x = sin 3x. (ii) cos 3x sin x = 0 3 [XI Mathematics]

35 46. 3 cos x sin x tanx + cotx = 5 cosec x. 48. tan x + tan x + 3 tan x tan x = tan x + sec x = In any triangle ABC, prove that a(sin B sin C) + b(sinc sina) + c(sina sinb) = In any triangle ABC, prove that (Q. 5 to Q. 55) c c a b cos a b sin c. 5. (b c ) cot A + (c a ) cot B + (a b ) cot c = (i) a = b cosc + c cosb (ii) c tan A/ tan B/ a b tan A/ tan B/ 54. (i) A B a b cos. (ii) c C sin a b tan (A B) cos C a c cos (A C) cos B 55. (i) If cos A sin B sin C then prove that the triangle is isoscles. (ii) If a cos A = b cos B then prove that the triangle is either isosceles or right angled LONG ANSWER TYPE QUESTIONS (6 MARKS) 56. Prove that sin 6A cosa cosa cos4a cos8a = 6 sin A 57. Prove that sin0 sin30 sin50 sin70 = Find the general solution of (i) sinx + sin4x + sin6x = 0 (ii) 4sinx sinx sin4x = sin3x 59. Find the general solution of cos cos cos3 = 4 3 [XI Mathematics]

36 Draw the graph of tanx in, 6. In any triangle ABC, prove that a b b c c a sin A sin B sin C 0 a b c 6. Prove that 4 sin sin sin sin *. Prove that sin x sin x sin sin 3 3 x 3 x In any ABC prove that (c + b a ) tan A = (a + c b ) tan B = (a + b c ) tan C 65. In any ABC prove that 3 if c 60 a c b c a b c 66. If and are distinct roots of cos + b sin = c prove that sin ( + ) = ab a b ANSWERS. (i) 3 c ; (ii) 5 4 c. (i) 39 '30''; (ii) (i) 5 cm (ii) (i) 6 ; (ii) 0 6. (i) 3 ; (ii) [XI Mathematics]

37 9. sin8 cos4 sin 6x sinx R n ; n z. [,]. 3. (i) 3 ; (ii) 0, 4. n (i) Positive; (ii) Negative. 0. [, ] A. / B m , radians 4. (i) 5 4 (ii) (iii) (i) n (n ),, n z (ii) n, n, n z n, n z 47. n, n z n n (i) n, n, n z (ii) n, m, m, n z (n ), n, n z [XI Mathematics]

38 CHAPTER - 4 PRINCIPLE OF MATHEMATICAL INDUCTION KEY POINTS Induction and deduction are two basic processes of reasoning. Deduction is the application of a general case to a particular case. In contrast to deduction, induction is process of reasoning from particular to general. Principle of Mathematical Induction : Let P(n) be any statement involving natural number n such that (i) (ii) P() is true, and If P(k) is true implies that P(k +) is also true for some natural number k then P(n) is true n N SHORT ANSWER TYPE QUESTIONS (4 MARKS) Using the principle of mathematical induction prove the following for all n N :. If P(n) : (3n ) = 3 n n Verify P(n) for n =,, 0. Given P(n) : n < 8 n Verify P(n) for n =, 3. P(n) " 3 n + 8n 9 is a multiple of 64 Verify P(n) for n = and n (3n + 3) = 3n(n + )(n + ) n n 35 [XI Mathematics]

39 6. n + n is an even natural number. 7. 3n is divisible by n when divided by 8 leaves the remainder n + 5n is divisible by 9 0. n 3 + (n + ) 3 + (n + ) 3 is a multiple of 9.. x n is divisible by x, x. 3 n > n 3. If x and y are any two distinct integers then x n y n is divisible by (x y) 4. n < n n a n d 5. a + (a + d) + (a + d) [a +(n )d] = 3 n n x 6. 3x + 6x + 9x +... to n terms 7. n+ + n+ is divisible by Using induction prove that n nx sin x sin sin x sinx sin3 x... sinnx x sin 9. Using PMI, Prove n 0 n N to n terms n 0. Using PMI prove that sin n + sin 3x sin (n ) x =. Using PMI, prove that sin nx, n N sin x cos. cos. cos 4... cos( n ) = n sin, n N n sin. n, n N prove that n n 36 [XI Mathematics]

40 CHAPTER - 5 COMPLEX NUMBERS AND QUADRATIC EQUATIONS KEY POINTS The imaginary number = i, is called iota For any integer k, i 4k =, i 4k+ = i, i 4k+ =, i 4k+3 = i a b ab if both a and b are negative real numbers A number of the form z = a + ib, where a, b R is called a complex number. a is called the real part of z, denoted by Re(z) and b is called the imaginary part of z, denoted by Im(z) a + ib = c + id if a = c, and b = d z = a + ib, z = c + id. In general, we cannot compare and say that z > z or z < z but if b, d = 0 and a > c then z > z i.e. we can compare two complex numbers only if they are purely real. 0 + i 0 is additive identity of a complex number. z = a + i( b) is called the Additive Inverse or negative of z = a + ib + i 0 is multiplicative identity of complex number. z = a ib is called the conjugate of z = a + ib 37 [XI Mathematics]

41 z = a ib z z a b z z = a + ib (a 0, b 0) is called the multiplicative Inverse of The coordinate plane that represents the complex numbers is called the complex plane or the Argand plane Polar form of z = a + ib is, z = r (cos + i sin) where r = of z, a b = z is called the modulus is called the argument or amplitude of z. The value of such that, < is called the principle argument of z. z + z z + z z z = z. z z z z z, z z, z z z z, z z z z + z z + z z z z z If z = r (cos + i sin ) n n z = r (cos + i sin ) then z z = r r [cos ( + ) + i sin ( + )] z z r cos r i sin For the quadratic equation ax + bx + c = 0, a, b, c R, a 0, if b 4ac < 0 then it will have complex roots given by, b i 4ac b x a 38 [XI Mathematics]

42 a ib is called square root of z = a + ib a ib x iy squaring both sides we get a + ib = x y + i(xy) x y = a, xy = b. Solving these we get x and y. VERY SHORT ANSWER TYPE QUESTIONS ( MARK). Evaluate : (i) (ii) i 6 i 5 49 i 49 4 (iii) 3 (iv) i 35 i (v) i 5 + i 0 + i 5 (vi) ( 3 ) ( 3 i) 35 (vii) (3 5i) (3 5i) ( 3 i) ( 3 i). Find x and y if 3x + (y 3) i is equal to + 4i 3. Find additive inverse of 6 i i Find multiplicative inverse of + i 5. Find modulus of If z 4 i, z 3 5i then find (i) R e (z z ) (ii) I m (z z ) (iii) R e (z z ) (iv) I m (z + z ) 7. Express i in the form of a + ib. 8. If modulus of z is and argument of z is 5 6 a + ib. then write z in form of 39 [XI Mathematics]

43 9. If z = (cos 30 + i sin 30 ), z = 3(cos 60 + i sin 30 ) Find R e (z z ) 0. Find the value of 4 n n is any positive integer.. Find conjugate of i 7. Find the solution of equation x + 3 = 0 in complex numbers. 3. Represent sin i cos in polar form Represent in (a + ib) 5. If z = cos 30 + i sin 30 z = cos 60 + i sin cos i sin then find (i) z z ; (ii) zz SHORT ANSWER TYPE QUESTIONS (4 MARKS) 6. For Complex numbers z = + i, z = 3 i show that, Im (z z ) = Re (z ) Im(z ) + Im (z ) Re (z ) i 7. If x + iy i, prove that x + y = 8. Find real value of such that, i cos is a real number i cos z 5i 9. If, show that z is a real number. z 5i 0. Find the value of x and y if (i) x 7x + 9yi = y i + 0i (ii) (iii) x y i 3 i 3 i (x 4 + xi) (3x + iy) = (3 5i) + ( + iy) 40 [XI Mathematics]

44 . Express in the form of a + ib i 3 4 3i. Convert the following in polar form : (i) 3 3 i (ii) (iii) i ( + i ) (iv) 3. If x iy 3 a ib, prove that, 3 3 i 5 i 3i x y a b = 4(a b ) 4. For complex numbers z = 6 + 3i, z = 3 i find z z 5. If i i n,find the least positive integral value of n. 6. Solve (i) ix + 4x 4i = 0 (ii) x ( + i )x ( 7i ) = 0 (iii) x + 3ix + = 0 (iv) x 3 x 0 7. Find square root of the following complex number (i) 7 + 4i (ii) 3 4i (iii) 5 8i (iv) 7 30 z 8. If z, z are complex numbers such that 3z then find z z z z. is purely imaginary number 9. Prove that x = (x + + i) (x + i) (x + i) (x i). 30. If z = x + iy and w = i z i show that w = z is purely real. 4 [XI Mathematics]

45 3. If z = 3i show that z 4z + 3 = 0, hence find the value of 4z 3 3z Find all non-zero complex number z satisfying z = iz. 33. If iz 3 + z z + i = 0 then show that z =. LONG ANSWER TYPE QUESTIONS (6 MARKS) z 34. If z, z are complex numbers such that, 3z 3 z z then find z 35. If x = + i then find the value of x 4 + 4x 3 + 4x + and z *36. If z = x + iy and w iz z i show that if w = then z is purely real. *37. Prove by Principle of Mathematical Induction (cos + i sin ) n = cos n + i sin n if n is any natural number. 38. Solve x (7 i) x + 8 i = Solve x (3 + 7i) x (3 9i) = 0 ANSWERS. (i) 0; (ii) 9; (iii) i ; (iv) 0; (v) ; (vi) 0 98i (viii) 7 i. x 7, y i 3 4. i (i) 6; (ii) ; (iii) ; (iv) 4 [XI Mathematics]

46 7. i 8. 3 i i. 3 i 3. cos sin 6 6 i 4. 3i 5. (i) ; (ii) ; 8. n 0. (i) x = 4, 3; y = 5, 4 (ii) x = 4, y = 6.. (i) 4. (iii) x = and y = 3 or x = and y = / i (iii) z z cos i sin 4 4 ; (ii) 5 5 cos i sin ; 3 3 cos i sin 4 4 ; (iv) cos i sin i 5. n = 4 6. (i) i, i (ii) 3 i, + i (iii), i i (iv) i 7. (i) ± (3 + 4i ) (ii) ±( i ) (iii) ± ( 4i ) (iv) ± (5 3i ) z 0, i, i, i 34. z = i and 3 + i i and 3i 43 [XI Mathematics]

47 CHAPTER - 6 LINEAR INEQUALITIES KEY POINTS The inequality containing < or > is called strict inequality. The inequality containing or is called slack inequality. Two real numbers or two algebraic expressions related by the symbol '<', '>', '' or '' form an inequality. The inequalities of the form ax + b > 0, ax + b < 0, ax + b 0, ax + b 0 ; a 0 are called linear inequalities in one variable x The inequalities of the form ax + by + c > 0, ax + by + c < 0, ax + by + c 0, ax + by + c 0, a 0, b 0 are called linear inequalities in two variables x and y Rules for solving inequalities : (i) (ii) a b then a ± k b ± k where k is any real number. but if a b then ka is not always kb. If k > 0 (i.e. positive) then a b ka kb If k < 0 (i.e. negative) then a b ka kb Solution Set : A solution of an inequality is a number which when substituted for the variable, makes the inequality true. The set of all solutions of an inequality is called the solution set of the inequality. The graph of the inequality ax + by > c is one of the half planes and is called the solution region When the inequality involves the sign or then the points on the line are included in the solution region but if it has the sign < or > then the points on the line are not included in the solution region and it has to be drawn as a dotted line. 44 [XI Mathematics]

48 VERY SHORT ANSWER TYPE QUESTIONS ( MARK). Solve 5x < 4 when x N. Solve 3x < when x Z 3. Solve 3 x < 9 when x R 4. Show the graph of the solution of x 3 > x 5 on number line. 5. Solve 5x 8 8 graphically 6. Solve x 0 7. Solve x 0 3 Write the solution in the form of intervals for x R. for Questions 8 to 0 8. x x + < x > 4 3x. Draw the graph of the solution set of x + y 4.. Draw the graph of the solution set of x y SHORT ANSWER TYPE QUESTIONS (4 MARKS) Solve the inequalities for real x x 3 4x x 3 x x [XI Mathematics]

49 6. 3x 7. 4 x + < x x x 5 0. x 4 x 5. Solve x 0, x x R, x. 3x 7 > (x 6), 6 x > x 3. The water acidity in a pool is considered normal when the average PH reading of three daily measurements is between 7. and 7.8. If the first two PH readings are 7.48 and 7.85, find the range of PH value for the third reading that will result in the acidity level being normal. 4. While drilling a hole in the earth, it was found that the temperature (T C) at x km below the surface of the earth was given by T = (x 3), when 3 x 5. Between which depths will the temperature be between 00 C and 300 C? 5. Solve for real x, x + + x > 3 Solve the following systems of inequalities graphically : (Questions 6, 7) 6. x + y > 6, x y > x + 4y 60, x + 3y 30, x 0, y 0 LONG ANSWER TYPE QUESTIONS (6 MARKS) Solve the system of inequalities for real x 8. 5x 3x 39 and [XI Mathematics]

50 x x 3x 3 4 Solve the following system of inequalities graphically (Questions 8 to 30) 9. 3x + y 4, x + y 6, x + y 0, x 0, y x + y 4, x + y 3, x 3y 6 3. x + y 000, x + y 500, y 600, x 0, y y x < 4, x + 3y > 3 and x + y 5 ANSWERS. {,,3,4}. {...,,, 0,,, 3} 3. x > 3 6. x < 7. 3 < x < 0 8. (, 3) 9. 5, , 5 Y Y X' O Y 63, 0 34, 3 3 x + y 4 X. X' Y x y Y ' 3, 5, 6 O (, ) (, ) 7. (, 6) 8. (, ) (5, ) 9. (, 5) (5, ) 0. (, ) (7, ) X 47 [XI Mathematics]

51 . [, ] (, ) (, ). (5, ) 3. Between 6.7 and Between 9.8 m and 3.8 m 5. ( -, ) (, ) 8. (3, ) 48 [XI Mathematics]

52 CHAPTER - 7 PERMUTATIONS AND COMBINATIONS KEY POINTS Multiplication Principle (Fundamental Principle of Counting : If an event can occur in m different ways, following which another event can occur in n different ways, then the total no. of different ways of occurence of the two events in order is m n. Fundamental Principle of Addition : If there are two events such that they can occur independently in m and n ways respectively, then either of the two events can occur in (m + n) ways. Factorial : Factorial of a natural number n, denoted by n! or n is the continued product of first n natural numbers. n! = n (n ) (n )... 3 Permutation : A permutation is an arrangement in a definite order of a number of objects taken some or all at a time. The number of permutation of n different objects taken r at a time where 0 r n and the objects do not repeat is denoted by n P n or P(n, r) n Pr = n! (n r)! The number of permutations of n objects, taken r at a time, when repetition of objects is allowed is n r. The number of permutations of n objects of which p, are of one kind, p are of second kind,.. p k are of k th kind and the rest if any, are of different kind, is n! p! p!... p!. k Combination : Each of the different selections made by choosing some 49 [XI Mathematics]

53 or all of a number of objects, without considering their order is called a combination. The number of combination of n object taken r at a time where 0 r n, is denoted by n C r or C(n, r) or Some important result : 0! = n C 0 = n C n = n r n ( ) where C n C r = n C n r where 0 r n, and r are positive integer n P r = r! n C r where 0 r n, n and r are positive integers. n C r + n C r = n+ C r where r n, n, r N r n!. r! (n r)! n C a = n C b if either a + b = n or a = b VERY SHORT ANSWER TYPE QUESTIONS ( MARK). Using the digits,, 3, 4, 5 how many 3 digit numbers (without repeating the digits) can be made?. In how many ways 7 pictures can be hanged on 9 pegs? 3. Ten buses are plying between two places A and B. In how many ways a person can travel from A to B and come back? 4. There are 0 points on a circle. By joining them how many chords can be drawn? 5. There are 0 non collinear points in a plane. By joining them how many triangles can be made? 6. x If find x 6! 8! 9! 7. If n P 4 : n P =, find n. 8. How many different words (with or without meaning) can be made using all the vowels at a time? 50 [XI Mathematics]

54 9. Using,, 3, 4, 5 how many numbers greater than 0000 can be made? (Repetition not allowed) 0. If n C = n C 3 then find the value of 5 C n.. In how many ways 4 boys can be choosen from 7 boys to make a committee?. How many different words can be formed by using all the letters of word SCHOOL? 3. In how many ways can the letters of the word PENCIL be arranged so that I is always next to L. 4. In an examination there are there multiple choice questions and each question has 4 choices. Find the number of ways in which a student can fail to get all answer correct. 5. A gentleman has 6 friends to invite. In how many ways can he send invitation cards to them if he has three servants to carry the cards? 6. If there are persons in a party, and if each two of them Shake hands with each other, how many handshakes happen in the party? SHORT ANSWER TYPE QUESTIONS (4 MARKS) 7. In how many ways boys can be seated on 0 chairs in a row so that two particular boys always take seat? 8. In how many ways 7 positive and 5 negative signs can be arranged in a row so that no two negative signs occur together? 9. From a group of 7 boys and 5 girls, a team consisting of 4 boys and girls is to be made. In how many different ways it can be done? 0. In how many ways can one select a cricket team of eleven players from 7 players in which only 6 players can bowl and exactly 5 bowlers are to be included in the team?. A student has to answer 0 questions, choosing atleast 4 from each of part A and B. If there are 6 questions in part A and 7 in part B. In how many ways can the student choose 0 questions? 5 [XI Mathematics]

55 . Using the digits 0,,,, 3 how many numbers greater than 0000 can be made? 3. If the letters of the word PRANAV are arranged as in dictionary in all possible ways, then what will be 8 nd word. 4. From a class of 5 students, 0 are to choosen for a picnic. There are two students who decide that either both will join or none of them will join. In how many ways can the picnic be organized? 5. Using the letters of the word, ARRANGEMENT how many different words (using all letters at a time) can be made such that both A, both E, both R and both N occur together. 6. A polygon has 35 diagonal. Find the number of its sides. [Hint : Number of diagonal of n sided polygon is given by n C n] 7. How many different products can be obtained by multiplying two or more of the numbers, 5, 6, 7, 9? 8. Determine the number of 5 cards combinations out of a pack of 5 cards if atleast 3 out of 5 cards are ace cards? 9. How many words can be formed from the letters of the word ORDINATE so that vowels occupy odd places? 30. Find the number of all possible arrangements of the letters of the word MATHEMATICS taken four at a time. 3. Prove that 33! in divisible by 5 what is the largest integer n such that 33! is divisible by n? 3. A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected if a team has (i) no girl (ii) at least 3 girls (iii) at least one girl and one boy? 33. Find n if n n C8 P A boy has 3 library tickets and 8 books of his interest in the library of these 8, he does not want to borrow Mathematics Part II. Unless 5 [XI Mathematics]

56 Mathematics Part I is also borrowed. In how many ways can he choose the these books to be borrowed? 35. Three married couples are to be seated in a row having six seats in a cinema hall. If spouces are to be seated next to each other, in how many ways can they be seated? Find also the number of ways of their seating if all the ladies sit together. LONG ANSWER TYPE QUESTIONS (6 MARKS) 36. Using the digits 0,,, 3, 4, 5, 6 how many 4 digit even numbers can be made, no digit being repeated? 37. There are 5 points in a plane out of which only 6 are in a straight line, then (a) (b) How many different straight lines can be made? How many triangles can be made? 38. If there are 7 boys and 5 girls in a class, then in how many ways they can be seated in a row such that (i) (ii) No two girls sit together? All the girls never sit together? 39. Using the letters of the word 'EDUCATION' how many words using 6 letters can be made so that every word contains atleast 4 vowels? 40. What is the number of ways of choosing 4 cards from a deck of 5 cards? In how many of these, (a) (b) (c) (d) 3 are red and is black. All 4 cards are from different suits. Atleast 3 are face cards. All 4 cards are of the same colour. 4. How many 3 letter words can be formed using the letters of the word INEFFECTIVE? 53 [XI Mathematics]

57 4. How many 5 letter words containing 3 vowels and consonants can be formed using the letters of the word EQUATION so that 3 vowels always occur together? 43. If all letters of word MOTHER are written in all possible orders and the word so formed are arranged in a dictionary order, then find the rank of word MOTHER? ANSWERS !! n = = P , PAANVR 4. 3 C C (i) ; (ii) 9; (iii) , [XI Mathematics]

58 (a) 9 (b) (i) 7! 8 P 5 (ii)! 8! 5! C 4 (a) 6 C 6 C 3 (b) (3) 4 (c) 995 (Hint : Face cards : 4J + 4K + 4Q) (d) 6 C (Hint : make 3 cases i.e. (i) All 3 letters are different (ii) are identical different (iii) All are identical, then form the words.) [XI Mathematics]

59 CHAPTER - 8 BINOMIAL THEOREM n KEY POINTS n n n n a b n a n a b n a b n b C 0 C C C n n r 0 C r nr r n a b, n N T r + = General term = C r nr r n a b 0 r n Total number of terms in (a + b) n is (n + ) If n is even, then in the expansion of (a + b) n, middle term is term i.e. n th term. n If n is odd, then in the expansion of (a + b) n, middle terms are th n n 3 and th terms In (a + b) n, r th term from the end is same as ( n r + ) th term from the beginning. r th term from the end in (a + b) n = r th term from the beginning in (b + a) n In ( + x) n, coefficient of x r is n C r Some particular cases : th 56 [XI Mathematics]

60 ( + x) n = ( x) = n n C r r x r 0 n r 0 r h ( ) C x r r Some properties of Binomial coefficients : n C 0 + n C + n C n C n = n n C 0 n C + n C n C ( ) n n C n = 0 n C 0 n C + n C = n C + n C 3 + C n = n VERY SHORT ANSWER TYPE QUESTIONS ( MARK). Compute (98), using binomial theorem.. Expand x x 3 using binomial theorem. 3. Find number of terms in the expansionof the following : (i) 7 3x y 8 (ii) ( + x + x ) 7 4. Write value of n n n C 5 C 6 C 7 [Hint : Use n C r + n C r = n + C r ] 5. In the expansion, ( + x) 4, write the coefficient of x 6. Find the sum of the coefficients in (x + y) 8 [Hint : Put x =, y = ] 7. If n C n 3 = 70, find n. [Hint : Express 70 as the product of 3 consecutive positive integers] 8. Find middle term in expansion of ( + x) n. 57 [XI Mathematics]

61 9. Find (i) 3rd term in expansion of 3x x 8 (ii) 4th term in expansion of (x y) (iii) 4th term from end in the expansion of 3 x x 9 SHORT ANSWER TYPE QUESTIONS (4 MARKS) 0. If the first three terms in the expansion of (a + b) n are 7, 54 and 36 respectively, then find a, b and n.. In 3x x 8, which term contains x?. In x 3 3 x 0, find the term independent of x Evaluate : using binomial theorem. 4. Evaluate (0.9) 4 using binomial theorem. 5. In the expansion of ( + x ) 8, find the difference between the coefficients of x 6 and x In 8 3 x x, find 7th term from end. 7. In x 3 x, find the coefficient of x. 8. Find the coefficient of x 4 in ( x) ( + x) 5 using binomial theorem. 58 [XI Mathematics]

62 9. Using binomial theorem, show that 3 n + 8n 9 is divisible by 8. [Hint : 3 n + = 9 3 n = 9 ( + 8) n, Now use binomial theorem.] 0. Prove that, 0 0 0r 0 r r r 0 C t t. Find the middle term(s) in x x. If the coefficients of three consecutive terms in the expansion of ( + x) n are in the ratio :3:5, then show that n = Show that the coefficient of middle term in the expansion of ( + x) 0 is equal to the sum of the coefficients of two middle terms in the expansion of ( + x) 9 LONG ANSWER TYPE QUESTIONS (6 MARKS) 4. Show that the coefficient of x 5 in the expansion of product ( + x) 6 ( x) 7 is If the 3 rd, 4 th and 5 th terms in the expansion of (x + a) n are 84, 80 and 560 respectively then find the values of a, x and n 6. In the expansion of ( x) n, find the sum of coefficients of x r and xn r 7. If the coefficients of x 7 in equal, then show that ab =. 8 ax bx and x 7 in ax bx 8. If three successive coefficients in expansion of ( + x) n are 0, 495 and 79 then find n. 9. In the expansion of 3 3 to the 7th term from the end is : 6 find n. 3 n are the ratio of 7th term from the beginning 59 [XI Mathematics]

63 30. If a, a, a 3 and a 4 are the coefficients of any four consecutive terms in the expansion of ( + x) n a a3 a prove that a a a a a a Find n if the coefficients of 5th, 6th and 7th terms in the expansion of ( + x) n are in A.P. 3. Find the ratio of the coefficients of x 5 to the term independent of x in the expansion 5 x. x ANSWERS x 3 3x 3 3 x x 3. (i) 9; (ii) 5; 4. n C n = 0 8. n n c (i) 3 6 x 0 ; (ii) 760x 9 y 3 ; (iii) 3 x n x 0. a = 3, b =, n = 3. 9 th term. T 3 = x a =, x =, n = or or : 3 60 [XI Mathematics]

64 CHAPTER - 9 SEQUENCES AND SERIES KEY POINTS A sequence is a function whose domain is the set N of natural numbers or some subset of it. A sequence whose range is a subset of R is called a real sequence. A sequence is said to be a progression if the term of the sequence can be expressed by some formula A sequence is called an arithmetic progression if the difference of a term and previous term is always same, i.e., a n + a n = constant (=d) for all n N. The term series is associated with the sequence in following way : Let a, a, a 3... be a sequence. Then, the expression a + a + a is called series associated with given sequence. A series is finite or infinite according as the given sequence is finite or infinite. General A.P. is, a, a + d, a + d,... a n = a + (n )d = n th term of A.P. = l S n = Sum of first n terms of A.P. n a l where l = last term. = n a n d = If a, b, c are in A.P. then a ± k, b ± k, c ± k are in A.P., 6 [XI Mathematics]