Maths Part 1 9th STD English Medium PDF

x +y = 4 2x + 3y =3 x= , y= 9 64.00  1 Sets Let’s study. Sets - Introduction Types of sets Venn diagrams Equal sets, subset Universal set Intersection and Union...

x +y = 4 2x + 3y =3 x= , y= 9 64.00  1 Sets Let’s study. Sets - Introduction Types of sets Venn diagrams Equal sets, subset Universal set Intersection and Union of sets Number of elements in a set Let’s recall. Some pictures are given below. It contains the group of things you know. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,... Flower Bunch of Flock of birds Pile of note- Collection of bouquet keys books numbers We use special word for each of the collection given above. In all the above examples we can clearly list the objects of that collection. We call the collection of such objects as ‘Set’. Now, observe the collection. ‘Happy children in the village’, ‘Brilliant students of the class’. In both the examples the words ‘Happy’ and ‘Brilliant’ are relative terms, because the exact meaning of these words ‘to be happy’ and ‘to be brilliant’ differ from person to person. Therefore, these collections are not sets. See the examples given below and decide whether it is a set or not. (1) Days of a week (2) Months in a year (3) Brave children in the class (4) First 10 counting numbers (5) Strong forts of Maharashtra (6) Planets in our solar system. 1  Let’s learn. Sets If we can definitely and clearly decide the objects of a given collection then that collection is called a set. Generally the name of the set is given using capital letters A, B, C,.....,Z The members or elements of the set are shown by using small letters a, b, c,... If a is an element of set A, then we write it as ‘a Î A’ and if a is not an element of set A then we write ‘a Ï A’. Now let us observe the set of numbers. N = { 1, 2, 3,...} is a set of natural numbers. W = {0, 1, 2, 3,...} is a set of whole numbers. I = {..., -3, -2, -1, 0, 1, 2,...} is a set of integers. Q is a set of rational numbers. R is a set of real numbers. Methods of writing sets There are two methods of writing set. (1) Listing method or roster method In this method, we write all the elements of a set in curly bracket. Each of the elements is written only once and separated by commas. The order of an element is not important but it is necessary to write all the elements of the set. e.g. the set of odd numbers between 1 and 10, can be written as as, A = {3, 5, 7, 9} or A = {7, 3, 5, 9} If an element comes more than once then it is customary to write that element only once. e.g. in the word ‘remember’ the letters ‘r, m, e’ are repeated more than once. So the set of letters of this word is written as A = {r, e, m, b} (2) Rule method or set builder form In this method, we do not write the list of elements but write the general element using variable followed by a vertical line or colon and write the property of the variable. e.g. A = {x ½ x Î N, 1 < x < 10 } and read as 'set A is the set of all ‘x’ such that ‘x’ is a natural number between 1 and 10'. 2  e.g. B = { x | x is a prime number between 1 and 10} set B contains all the prime numbers between 1 and 10. So by using listing method set B can be written as B = {2, 3, 5, 7} Q is the set of rational numbers which can be written in set builder form as p Q ={ q | p, q Î I, q ¹ 0} p and read as ‘Q’ is set of all numbers in the form q such that p and q are integers where q is a non-zero number.’ Illustrations : In the following examples each set is written in both the methods. Rule method or Set builder form Listing method or Roster method A = { x | x is a letter of the word ‘DIVISION’.} A = {D, I, V, S, O, N} B = { y | y is a number such that y2 = 9} B = { -3, 3} C = {z | z is a multiple of 5 and is less than 30} C = { 5, 10, 15, 20, 25} Ex. : Fill in the blanks given in the following table. Listing or Roster Method Rule Method A = { 2, 4, 6, 8, 10, 12, 14} A = {x | x is an even natural number less than 15}.................. B = { x | x is a perfect square number between 1 to 20} C = { a, e, i, o, u}.................................... D = { y | y is a colour in the rainbow}.................. P = {x | x is an integer and , -3 < x < 3} M = {1, 8, 27, 64, 125.......}.................. Practice set 1.1 (1) Write the following sets in roster form. (i) Set of even numbers (ii) Set of even prime numbers from 1 to 50 (iii) Set of negative integers (iv) Seven basic sounds of a sargam (sur) (2) Write the following symbolic statements in words. (i) 4 ÎQ (ii) -2 Ï N (iii) P = {p | p is an odd number} 3 3 (3) Write any two sets by listing method and by rule method. (4) Write the following sets using listing method. (i) All months in the indian solar year. (ii) Letters in the word ‘COMPLEMENT’. (iii) Set of human sensory organs. (iv) Set of prime numbers from 1 to 20. (v) Names of continents of the world. (5) Write the following sets using rule method. (i) A = { 1, 4, 9, 16, 25, 36, 49, 64, 81, 100} (ii) B = { 6, 12, 18, 24, 30, 36, 42, 48} (iii) C = {S, M, I, L, E} (iv) D = {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday} (v) X = {a, e, t} Let’s learn. Types of sets Name of set Definition Example A = {2} Singleton Set A set consisting of a single element is called a singleton set. A is the set of even prime numbers. If there is not a single element in B = {x | x is natural number Empty Set or the set which satisfies the given between 2 and 3.} Null Set condition then it is called a Null set or an empty set. Null set is \ B = { } or f represented by { } or a symbol f (phi) If a set is a null set or number of C = {p | p is a number from Finite Set elements are limited and countable 1 to 22 divisible by 4.} then it is called as ‘Finite set’. C = {4, 8, 12, 16, 20} If number of elements in a set is Infinite Set unlimited and uncountable then the N = {1, 2, 3,... } set is called ‘Infinite set’. 4 Ex. Write the following sets using listing method and classify into finite or infinite set. (i) A = {x | x Î N and x is an odd number} (ii) B = {x | x Î N and 3x -1 = 0} (iii) C = {x | x Î N, and x is divisible by 7 } (iv) D = {(a, b) | a, b Î W, a + b = 9} (v) E = {x | x Î I, x2 = 100} (vi) F = {(a, b) | a, b Î Q, a + b = 11} Solution : (i) A = {x | x Î N and x is an odd number.} A = {1, 3, 5, 7,......} This is an infinite set. (ii) B = {x | x Î N and 3x -1 = 0} 1 3x -1 = 0 \ 3x = 1 x= 3 1 But 3 Ï N \B = { } \ B is finite set. (iii) C = {x | x Î N and x is divisible by 7.} C = {7, 14, 21,... } This is an infinite set. (iv) D = {(a , b) | a, b Î W, a +b = 9} We have to find the pairs of a and b such that, a and b are whole numbers and a + b = 9. Let us first write the value of a and then the value of b. By keeping this order set D can be written as D = {(0, 9), (1, 8), (2, 7), (3, 6), (4, 5), (5, 4), (6, 3), (7, 2), (8, 1), (9, 0)}, In this set, number of pairs are finite and could be counted \ Set D is a finite set. (v) E = {x | x Î I, x2 = 100} E = {-10, 10}. \ E is a finite set (vi) F = {(a, b ) | a, b Î Q, a +b = 11 } F = {(6, 5), (3, 8), (3.5,7.5), (-15, 26),...} infinitely many such pairs can be written. \ F is an infinite set. Remember this ! N, W, I, Q, R all these sets are infinite sets. 5  Let’s learn. Equal sets Two sets A and B are said to be equal, if every element of set A is in set B and every element of set B is in set A. 'Set A and set B are equal sets', symbolically it is written as A = B. Ex (1) A = { x | x is a letter of the word ‘listen’.} \ A = { l, i, s, t, e, n} B = { y | y is a letter of the word ‘silent’.} \ B = { s, i, l, e, n, t} Though the elements of set A and B are not in the same order but all the elements are identical. \A=B Ex (2) A = {x | x = 2n, n Î N, 0 < x £ 10}, A = {2, 4, 6, 8, 10} B = { y | y is an even number, 1 £ y £ 10}, B = {2, 4, 6, 8, 10} \ A and B are equal sets. Now think of the following sets. C = {1, 3, 5, 7} D = { 2, 3, 5, 7} Are C and D equal sets ? Obviously ‘No’ Because 1 Î C, 1 Ï D, 2 Î D, 2 Ï C \ C and D are not equal sets. It is written as C ¹ D Ex (3) If A = {1, 2, 3} and B = { 1, 2, 3, 4}then A ¹ B verify it. Ex (4) A = {x | x is prime number and 10 < x < 20} and B = {11, 13, 17, 19} Here A = B. Verify, Practice set 1.2 (1) Decide which of the following are equal sets and which are not ? Justify your answer. A = { x | 3x -1 = 2} B = { x | x is a natural number but x is neither prime nor composite} C = {x | x Î N, x < 2} (2) Decide whether set A and B are equal sets. Give reason for your answer. A = Even prime numbers B = {x | 7x -1 = 13} (3) Which of the following are empty sets ? why ? (i) A = { a | a is a natural number smaller than zero.} (ii) B = {x | x2 = 0} (iii) C = { x | 5 x - 2 = 0, x Î N} 6 (4) Write with reasons, which of the following sets are finite or infinite. (i) A = { x | x < 10, x is a natural number} (v) Set of apparatus in laboratory (ii) B = {y | y < -1, y is an integer} (vi) Set of whole numbers (iii) C = Set of students of class 9 from your school. (vii) Set of rational number (iv) Set of people from your village. Let’s learn. Venn diagrams British logician John Venn was the first to use closed figures to represent sets. Such rep- resentations are called 'Venn diagrams'. Venn diagrams are very useful, in order to understand and illustrate the relationship among sets and to solve the examples based on the sets. Let us understand the use of Venn diagrams from the following example. e.g. A = { 1, 2, 3, 4, 5} John Venn is the first Set A is shown by Venn diagram. Mathematician who gave the Mathematical form to 1 2 A ‘logic’ and ‘probability’. 3 His famous book is ‘Logic 5 4 1834-1883 of chance’. B = {x | -10 £ x £ 0, x is an integer} 0 -1 -2 -3 Venn diagram given alongside represents the set B. -4 -5 -6 -7 B -8 -9 -10 Subset If A and B are two given sets and every element of set B is also an element of set A then B is a subset of A which is symbolically written as B Í A. It is read as 'B is a subset of A' or 'B subset A'. A Ex (1) A = { 1, 2, 3, 4, 5, 6, 7, 8} 1 B 3 B = {2, 4, 6, 8} 8 2 4 5 6 Every element of set B is also an element of set A. 7 \ B Í A. This can be represented by Venn diagram as shown above. 7  Activity : Set of students in a class and set of students in the same class who can swim, are shown by the Venn diagram. Students in a class Observe the diagram and draw Students Venn diagrams for the following subsets. who can (1) (i) set of students in a class swim (ii) set of students who can ride bicycles in the same class (2) A set of fruits is given as follows. {guava, orange, mango, jack fruit, chickoo, jamun, custard apple, papaya, plum} Show these subsets. (i) fruit with one seed (ii) fruit with more than one seed. Let’s see some more subsets. Ex (2) N = set of natural numbers. I = set of integers. Here N Í I. because all natural numbers are integers.. Ex (3) P = { x | x is square root of 25} S = { y | y Î I, -5 £ y £ 5} Let’s write set P as P = {-5, 5} Let’s write set S as S = {-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5} Here every element of set P is also an element of set S. \ PÍS Remember this ! (i) Every set is a subset of itself. i.e. A Í A (ii) Empty set is a subset of every set i.e. f Í A (iii) If A = B then A Í B and B Í A (iv) If A Í B and B Í A then A = B Ex. If A = { 1, 3, 4, 7, 8} then write all possible subsets of A. i.e. P = { 1, 3}, T = {4, 7, 8}, V = {1, 4, 8}, S = {1, 4, 7, 8} In this way many subsets can be written. Write five more subsets of set A. 8  Activity : Every student should take 9 triangular sheets of paper and one plate. Numbers from 1 to 9 should be written on each triangle. Everyone should keep some numbered triangles in the plate. Now the triangles in each plate form a subset of the set of numbers from 1 to 9. Sujata Hameed Mukta Nandini Joseph 1 1 3 2 4 1 4 2 3 4 2 3 5 7 5 6 7 5 6 8 9 7 9 8 9 Look at the plates of Sujata, Hameed, Mukta, Nandini, Joseph with the numbered triangles. Guess the thinking behind selecting these numbers. Hence write the subsets in set builder form. Let’s discuss. Ex.. Some sets are given below. A = {..., -4, -2, 0, 2, 4, 6,...} B = {1, 2, 3,...} C = {..., -12, -6, 0, 6, 12, 18.....} D = {..., -8, -4, 0, 4, 8,...} I = {..., -3, -2, -1, 0, 1, 2, 3, 4,.....} Discuss and decide which of the following statements are true. (i) A is a subset of sets B, C and D. (ii) B is a subset of all the sets which are given above. Let’s learn. Universal set Think of a bigger set which will accommodate all the given sets under consideration which in general is known as Universal set. So that the sets under consideration are the subsets of this Universal set. Ex (1) Suppose we want to study the students in class 9 who frequently remained absent. Then we have to think of all the students of class 9 who are in the school. So all the students in a school or the students of all the divisions of class 9 in the school is the Universal set. 9 Let us see the another example. Ex (2) A cricket team of 15 students is to be selected from a school. Here all the students from school who play cricket is the Universal set. A team of 15 cricket players is a subset of that Universal set. U All Students in school Generally, the universal set is denoted who play cricket by ‘U’ and in Venn diagram it is Team of cricket represented by a rectangle. players Complement of a set Suppose U is an universal set. If B Í U, then the set of all elements in U, which are not in set B is called the complement of B. It is denoted by B¢ or BC. B¢ is defined as follows. 3 5 1 A¢ U B¢ = {x | x Î U, and x Ï B} 2 A 7 Ex (1) U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10} 4 6 8 A = {2, 4, 6, 8, 10} 10 9 \ A¢ = {1, 3, 5, 7, 9} Ex (2) Suppose U = { 1, 3, 9, 11, 13, 18, 19} B¢ U B B = {3, 9, 11, 13} 1 3 18 9 \ B¢ = {1, 18, 19} 11 13 Find (B¢)¢ and draw the inference. 19 (B¢)¢ is the set of elements which are not in B¢ but in U. is (B ¢)¢ = B ? Understand this concept with the help of Venn diagram. Complement of a complement is the given set itself. Remember this ! Properties of complement of a set. (i) No elements are common in A and A¢. (ii) A Í U and A¢ Í U (iii) Complement of set U is empty set. U¢ = f (iv) Complement of empty set is U. f¢= U 10  Practice set 1.3 (1) If A = {a, b, c, d, e}, B = { c, d, e, f }, C = {b, d}, D = {a, e} then which of the following statements are true and which are false ? (i) C Í B (ii) A Í D (iii) D Í B (iv) D Í A (v) B Í A (vi) C Í A (2) Take the set of natural numbers from 1 to 20 as universal set and show set X and Y using Venn diagram. (i) X = { x | x Î N, and 7 < x < 15} (ii) Y = { y | y Î N, y is prime number from 1 to 20} (3) U = {1, 2, 3, 7, 8, 9, 10, 11, 12} P = {1, 3, 7, 10} then (i) show the sets U, P and P¢ by Venn diagram. (ii) Verify (P¢)¢ = P (4) A = {1, 3, 2, 7} then write any three subsets of A. (5) (i)Write the subset relation between the sets. P is the set of all residents in Pune. M is the set of all residents in Madhya Pradesh. I is the set of all residents in Indore. B is the set of all residents in India. H is the set of all residents in Maharashtra. (ii) Which set can be the universal set for above sets ? (6*) Which set of numbers could be the universal set for the sets given below? (i) A = set of multiples of 5, B = set of multiples of 7. C = set of multiples of 12 (ii) P = set of integers which are multiples of 4. T = set of all even square numbers. (7) Let all the students of a class is an Universal set. Let set A be the students who secure 50% or more marks in Maths. Then write the complement of set A. Let’s learn. Operations on sets Intersection of two sets Suppose A and B are two sets. The set of all common elements of A and B is called the intersection of set A and B. It is denoted as A Ç B and read as A intersection B. \ A Ç B = {x | x Î A and x Î B} 11 Ex (1) A = { 1, 3, 5, 7} B = { 2, 3, 6, 8} B Let us draw Venn diagram. A 2 1 The element 3 is common in set A and B. 6 5 3 \ A Ç B = {3} 7 8 Ex (2) A = {1, 3, 9, 11, 13} B = {1, 9, 11} The elements 1, 9, 11 are common in set A and B. A B9 \ A Ç B = {1, 9, 11} But B = {1, 9, 11} B 3 1 11 \ AÇB=B Here set B is the subset of A. 13 \ If B Í A then A Ç B = B, similarly, if B Ç A = B, then B Í A Remember this ! Properties of Intersection of sets (1) A Ç B = B Ç A (2) If A Í B then A Ç B = A (3) If A Ç B = B then B Í A (4) A Ç B Í A and A Ç B Í B (5) A Ç A¢ = f (6) A Ç A = A (7) A Ç f = f Activity : Take different examples of sets and verify the above mentioned properties. Let’s learn. Disjoint sets A B Let, A = { 1, 3, 5, 9} 1 3 2 4 and B = {2, 4, 8} are given. 5 9 8 Confirm that not a single element is common in set A and B. These sets are completely different from each other. So the set A and B are disjoint sets. Observe its Venn diagram. Activity I : Observe the set A, B, C given by A B 1 2 3 8 9 Venn diagrams and write which of these are 4 7 6 10 C disjoint sets. 5 11 12 12  Activity II : Let the set of English alphabets be E S N the Universal set. Q T I J The letters of the word 'LAUGH' is one set L A U G H C RY and the letter of the word 'CRY' is another set. K B P We can say that these are two disjoint sets. X M O V W F Observe that intersection of these two sets is empty. Z D Union of two sets Let A and B be two given sets. Then the set of all elements of set A and B is called the Union of two sets. It is written as A È B and read as 'A union B'. A È B = {x | x Î A or x Î B} A B Ex (1) A = {-1, -3, -5, 0} -3 3 B = {0, 3, 5} 0 A È B = {-3, -5, 0, -1, 3, 5} -5 5 Note that, A È B = B È A -1 Ex (2) U Observe the Venn diagram and write the following sets A B using listing method. 6 2 8 1 (i) U (ii) A (iii) B (iv) A È B (v) A Ç B 4 10 3 7 (vi) A¢ (vii) B¢ (viii)(A È B)¢ (ix) (A Ç B)¢ 5 12 11 9 Solution : U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} A = {2, 4, 6, 8, 10}, B = {1, 3, 5, 7, 8, 10} A È B ={1, 2, 3, 4, 5, 6, 7, 8, 10} A Ç B = {8, 10} A¢ = {1, 3, 5, 7, 9, 11, 12} B¢ = {2, 4, 6, 9, 11, 12} (A È B)¢ ={9, 11, 12} (A Ç B)¢ = {1, 2, 3, 4, 5, 6, 7, 9, 11, 12} Ex (3) A = {1, 2, 3, 4, 5} B = {2, 3} Let us draw its Venn diagram. A 1 2 A È B = {1, 2, 3, 4, 5} 4 B Observe that set A and A È B have the same 3 elements. 5 Hence, if B Í A then A È B = A 13  Remember this ! Properties of Union of sets (1) A È B = B È A (2) If A Í B then A È B = B (3) A Í A È B, B Í A È B (4) A È A¢= U (5) A È A = A (6) A È f = A Let’s learn. Number of elements in a set Let A = {3, 6, 9, 12,15} is a given set with 5 elements. Number of elements in set A is denoted as n (A). \ n (A) = 5 Let B = { 6, 12, 18, 24, 30, 36} \ n (B) = 6 Number of elements in Union and Intersection of sets. Let us consider the set A and set B as given above, n (A) + n (B) = 5 + 6 = 11 ----(I) A È B = {3, 6, 9, 12, 15, 18, 24, 30, 36} \ n (A È B) = 9--------(II) To find A Ç B means to find common elements of set A and set B. A Ç B = {6, 12} \ n (A Ç B) = 2--------(III) In n (A) and n (B) elements in A Ç B are counted twice. \ n (A) + n (B) -n (A Ç B ) = 5 + 6 -2 = 9 and n (A È B ) = 9 From equations (I), (II) and (III), we can write it as follows \ n (A È B ) = n (A) + n (B) -n (A Ç B) A 3 B Verify the above rule for the given Venn diagram. 9 6 18 15 12 24 n (A) = , n (B) = 30 36 n (A È B )= , n (A Ç B)= \ n (A È B ) = n (A) + n (B) - n (A Ç B) Remember this ! n (A È B ) = n (A) + n (B) -n (A Ç B) means n (A) + n (B) = n (A È B ) + n (A Ç B) Ex. Let A = {1, 2, 3, 5, 7, 9, 11, 13} B = {1, 2, 4, 6, 8, 12, 13} Verify the above rule for the given set A and set B. 14  Let’s learn. Word problems based on sets Ex. In a class of 70 students, 45 students like to play Cricket. 52 students like to play Kho-kho. All the students like to play atleast one of the two games. How many students like to play Cricket or Kho-kho ? Solution : We will solve this example in wo ways. Method I : Total number of students = 70 Let A be the set of students who likes to play Cricket. Let B be the set of students who likes to play Kho-kho. Hence the number of students who likes to play Cricket or Kho-kho is n (A È B ) \ n (A È B ) = 70 Number of students who likes to play both Cricket and Kho-kho = n (A Ç B) n (A) = 45, n (B) = 52 We know, n (A È B ) = n (A) + n (B) - n (A Ç B). \ n (A Ç B) = n (A) + n (B) - n (A È B ) = 45 + 52 - 70 = 27 \ Number of students who likes to play both the games are 27, Number of students who likes to play Kho-kho are 45. \ Number of students who likes to play only Cricket = 45 -27 = 18 \ A Ç B is the set of students who play both the games. \ n (A Ç B)= 27 Method II : Given information can be shown by Venn diagrams as follows. Let n (A Ç B) = x, n (A) = 45, n (B) = 52, We know that, n (A È B ) = 70 A B \ n (A Ç B) = x = n (A) + n (B) - n (A Ç B) (45-x) x (52-x) = 52 + 45 - 70 = 27 Students who like to play only cricket = 45 - 27 =18 15  Practice set 1.4 (1) If n (A) = 15, n (A È B ) = 29, n (A Ç B) = 7 then n (B) = ? (2) In a hostel there are 125 students, out of which 80 drink tea, 60 drink coffee and 20 drink tea and coffee both. Find the number of students who do not drink tea or coffee. (3) In a competitive exam 50 students passed in English. 60 students passed in Mathematics. 40 students passed in both the subjects. None of them fail in both the subjects. Find the number of students who passed at least in one of the subjects ? (4*) A survey was conducted to know the hobby of 220 students of class IX. Out of which 130 students informed about their hobby as rock climbing and 180 students informed about their hobby as sky watching. There are 110 students who follow both the hobbies. Then how many students do not have any of the two hobbies ? How many of them follow the hobby of rock climbing only ? How many students follow the hobby of sky watching only ? (5) Observe the given Venn diagram and write the following sets. U A B (i) A (ii) B (iii) A È B (iv) U x m p y q (v) A¢ (vi) B¢ (vii) (A È B)¢ n z r s t Problem set 1 (1) Choose the correct alternative answer for each of the following questions. (i) If M = {1, 3, 5}, N = {2, 4, 6}, then M Ç N = ? (A) {1, 2, 3, 4, 5, 6} (B) {1, 3, 5} (C) f (D) {2, 4, 6} (ii) P = {x | x is an odd natural number, 1 < x £ 5} How to write this set in roster form? (A) {1, 3, 5} (B) {1, 2, 3, 4, 5} (C) {1, 3} (D) {3, 5} (iii) P = {1, 2,........., 10}, What type of set P is ? (A) Null set (B) Infinite set (C) Finite set (D) None of these (iv) M È N = {1, 2, 3, 4, 5, 6} and M = {1, 2, 4} then which of the following represent set N ? (A) {1, 2, 3} (B) {3, 4, 5, 6} (C) {2, 5, 6} (D) {4, 5, 6} 16  (v) If P Í M, then Which of the following set represent PÇ(P È M) ? (A) P (B) M (C) PÈM (D) P¢ÇM (vi) Which of the following sets are empty sets ? (A) set of intersecting points of parallel lines (B) set of even prime numbers. (C) Month of an english calendar having less than 30 days. (D) P = {x | x Î I, -1< x < 1} (2) Find the correct option for the given question. (i) Which of the following collections is a set ? (A) Colours of the rainbow (B) Tall trees in the school campus. (C) Rich people in the village (D) Easy examples in the book (ii) Which of the following set represent N Ç W? (A) {1, 2, 3,.....} (B) {0, 1, 2, 3,....} (C) {0} (D) { } (iii) P = {x | x is a letter of the word ' indian'} then which one of the following is set P in listing form ? (A) {i, n, d} (B) {i, n, d, a} (C) {i, n, d, i, a} (D) {n, d, a} (iv) If T = {1, 2, 3, 4, 5} and M = {3, 4, 7, 8} then T È M = ? (A) {1, 2, 3, 4, 5, 7} (B) {1, 2, 3, 7, 8} (C) {1, 2, 3, 4, 5, 7, 8} (D) {3, 4} (3) Out of 100 persons in a group, 72 persons speak English and 43 persons speak French. Each one out of 100 persons speak at least one language. Then how many speak only English ? How many speak only French ? How many of them speak English and French both ? (4) 70 trees were planted by Parth and 90 trees were planted by Pradnya on the occasion of Tree Plantation Week. Out of these; 25 trees were planted by both of them together. How many trees were planted by Parth or Pradnya ? (5) If n (A) = 20, n (B) = 28 and n (A È B) = 36 then n (A Ç B) = ? (6) In a class, 8 students out of 28 have a dog as their pet animal at home, 6 students have a cat as their pet animal. 10 students have dog and cat both, then how many students do not have a dog or cat as their pet animal at home ? (7) Represent the union of two sets by Venn diagram for each of the following. (i) A ={3, 4, 5, 7} B ={1, 4, 8} (ii) P = {a, b, c, e, f} Q ={l, m, n , e, b} 17  (iii) X = {x | x is a prime number between 80 and 100} Y = {y | y is an odd number between 90 and 100 } (8) Write the subset relations between the following sets.. X = set of all quadrilaterals. Y = set of all rhombuses. S = set of all squares. T = set of all parallelograms. V = set of all rectangles. (9) If M is any set, then write M È f and M Ç f. (10*) U 4 A B Observe the Venn diagram and write the 2 3 1 10 7 5 8 given sets U, A, B, A È B and A Ç B. 9 11 13 (11) If n (A) = 7, n (B) = 13, n (A Ç B) = 4, then n (A È B} = ? Activity I : Fill in the blanks with elements of that set. U = {1, 3, 5, 8, 9, 10, 11, 12, 13, 15} A = {1, 11, 13} B = {8, 5, 10, 11, 15} A¢ = {........} B¢ = {........} A Ç B = {............} A¢ Ç B¢ = {...............} A È B = {............} A¢ È B¢ = {............} (A Ç B)¢ = {............} (A È B)¢ = {............} Verify : (A Ç B)¢ = A¢ È B¢, (A È B)¢ = A¢ Ç B¢ Activity II : Collect the following information from 20 families nearby your house (i) Number of families subscribing for Marathi Newspaper. (ii) Number of families subscribing for English Newspaper. (iii) Number of families subscribing for both English as well as Marathi Newspaper. Show the collected information using Venn diagram. 18  2 Real Numbers Let’s study. Properties of rational numbers Properties of irrational numbers Surds Comparison of quadratic surds Operations on quadratic surds Rationalization of quadratic surds. Let’s recall. In previous classes we have learnt about Natural numbers, Integers and Real numbers. N = Set of Natural numbers = {1, 2, 3, 4,...} W = Set of Whole numbers = {0, 1, 2, 3, 4,...} I = Set of Integers = {..., -3, -2, -1, 0, 1, 2, 3,...} p Q = Set of Rational numbers = { | p, q Î I, q ¹ 0} q R = Set of Real numbers. N Í W Í I Í Q Í R. Order relation on rational numbers : p r q and s are rational numbers where q > 0, s > 0 p p (i) If p ´ s = q ´ r then q = rs (ii) If p ´ s > q ´ r then q > rs p (iii) If p ´ s < q ´ r then q < rs Let’s learn. Properties of rational numbers If a, b, c are rational numbers then Property Addition Multiplication 1. Commutative a+b=b+a a´b=b´a 2. Associative (a + b) + c = a + (b + c) a ´(b ´ c) = (a ´ b) ´ c 3. Identity a+0=0+a=a a´1=1´a=a 1 4. Inverse a + (-a) = 0 a´ =1 (a ¹ 0) a 19  Let’s recall. Decimal form of any rational number is either terminating or non-terminating recurring type. Terminating type Non terminating recurring type 2 17 (1) = 0.4 (1) = 0.472222... = 0.472 5 36 7 33 (2) - 64 = - 0.109375 (2) = 1.2692307692307... = 1.2 692307 26 101 56 (3) = 12.625 (3) = 1.513513513... = 1. 513 8 37 Let’s learn. p To express the recurring decimal in q form. p Ex. (1) Express the recurring decimal 0.777.... in q form. Solution : Let x = 0.777... = 0.7 \ 10 x = 7.777... = 7.7 \ 10x -x = 7.7 - 0.7 \ 9x = 7 7 \ x= 9 \ 0.777... = 79 p Ex. (2) Express the recurring decimal 7.529529529... in q form.. Solution : Let x = 7.529529... = 7. 529 \ 1000 x = 7529.529529... =7529. 529 Use your brain power! \ 1000 x - x = 7529. 529 - 7. 529 How to convert 7522 p \ 999 x = 7522.0 \ x = 2.43 in form ? 999 q 7522 \ 7. 529 = 999 20  Remember this ! (1) Note the number of recurring digits after decimal point in the given rational number. Accordingly multiply it by 10, 100, 1000 e.g. In 2.3, digit 3 is the only recurring digit after decimal point, hence to convert 2.3 p in q form multiply 2.3 by 10. In 1. 24 digits 2 and 4 both are recurring therefore multiply 1. 24 by 100. In 1. 513 digits 5, 1 and 3 are recurring so multiply 1. 513 by1000. (2) Notice the prime factors of the denominator of a rational number. If the prime factors are 2 or 5 only then its decimal expansion is terminating. If the prime factors are other than 2 or 5 also then its decimal expansion is non terminating and recurring. Practice set 2.1 1. Classify the decimal form of the given rational numbers into terminating and non-terminating recurring type. 2 11 (i) 13 (ii) (iii) 29 (iv) 17 (v) 5 11 16 125 6 2. Write the following rational numbers in decimal form. 127 4 17 (i) (ii) 25 (iii) 23 (iv) (v) 8 200 99 7 5 p 3. Write the following rational numbers in q form. (i) 0.6 (ii) 0. 37 (iii) 3. 17 (iv) 15. 89 (v)2. 514 Let’s recall. The numbers 2 and 3 shown on a number line are not rational numbers means they are irrational numbers. D C B O A P - 3 - 2 -1 0 1 2 3 On a number line OA = 1 unit. Point B which is left to the point O is at a distance of 1 unit. Co-ordinate of point B is -1. Co-ordinate of point P is 2 and its opposite number - 2 is shown by point C. The co-ordinate of point C is - 2. Similarly, opposite of 3 is - 3 which is the co-ordinate of point D. 21  Let’s learn. Irrational and real numbers 2 is irrational number. This can be proved using indirect proof. p Let us assume that 2 is rational. So 2 can be expressed in q form. p q is the reduced form of rational number hence p and q have no common factor other than 1. p P2 2 = q \ 2 = q2 (Squaring both the sides) \ 2q2 = p2 \ p2 is even. \ p is also even means 2 is a factor of p.....(I) \ p = 2t \ p2 = 4t2 tÎI \ 2q2 = 4t2 ( \ q2 = 2t2 \ q2 is even. \ q is even. \ p2 = 2q2) \ 2 is a factor of q..... (II) From the statement (I) and (II), 2 is a common factor of p and q both. p This is contradictory because in ; we have assumed that p and q have no common q factor except 1. \ Our assumption that 2 is rational is wrong. \ 2 is irrational number. Similarly, one can prove that 3 , 5 are irrational numbers. Numbers 2 , 3 , 5 can be shown on a number line. The numbers which are represented by points on a number line are real numbers. In a nutshell, Every point on a number line is associated with a unique a 'Real number' and every real number is associated with a unique point on the number line. We know that every rational number is a real number. But 2 , 3 , - 2 , p, 3 + 2 etc. are not rational numbers. It means that Every real number may not be a rational number. 22  Decimal form of irrational numbers Find the square root of 2 and 3 using devision method. Square root of 2 Square root of 3 1.41421... 1.732.... 1 2. 00 00 00 00.... 1 3. 00 00 00 00.... +1 -1 +1 -1 24 100 27 200 +4 -96 +7 -189 281 400 343 1100 + 1 -281 + 3 -1029 2824 11900 3462 007100 + 4 -11296 + 2 -6924 28282 60400 3464 0176 + 2 - 56564 28284 1 0383600 \ 2 = 1.41421... \ 3 = 1.732... Note that in the above division, numbers after decimal point are unending, means it is non-terminating. Even no group of numbers or a single number is repeating in its quotient. So decimal expansion of such numbers is non terminating, non-recurring. 2 , 3 are irrational numbers hence 1.4142... and 1.732... are irrational numbers too. Moreover, a number whose decimal expansion is non-terminating, non-recurring is irrational. Number p Activity I Draw three or four circles of different radii on a card board. Cut these circles. Take a thread and measure the length of circumference and diameter of each of the circles. Note down the readings in the given table. No. radius diameter circumference c c From table the ratio is Ratio = d d (r) (d) (c) nearly 3.1 which is con- 1 7 cm stant. This ratio is denoted 2 8 cm by p (pi). 3 5.5 cm 23  Activity II To find the approximate value of p, take the wire of length 11 cm, 22 cm and 33 cm each. Make a circle from the wire. Measure the diameter and complete the following table.. Circle Circumference Diameter Ratio of No. (c) (d) (c) to (d) Verify ratio of circumference 1 11 cm to the diameter of a circle is 2 22 cm 22 approximately. 3 33 cm 7 Ratio of the circumference of a circle to its diameter is constant number which is irrational. This constant number is represented by the symbol p. So the approximate value of p is 22 or 3.14. 7 The great Indian mathematician Aryabhat in 499 CE has calculated the value of p as 62832 = 3.1416. 20000 We know that, 3 is an irrational number because its decimal expansion is non-terminating, non-recurring. Now let us see whether 2 + 3 is irrational or not ? Let us assume that, 2 + 3 is not an irrational number. p p If 2 + 3 is rational then let 2 + 3 = q. \ We get 3 = q - 2. In this equation left side is an irrational number and right side rational number, which is contradictory, so 2 + 3 is not a rational but it is an irrational number. Similarly it can be proved that 2 3 is irrational. The sum of two irrational numbers can be rational or irrational. Let us verify it for different numbers. i.e., 2 + 3 +(- 3 ) = 2, 4 5 ¸ 5 = 4, (3 + 5 ) - ( 5 ) = 3, 2 3 ´ 3 = 6 2 ´ 5 = 10 , 2 5- 5 = 5 Remember this ! Properties of irrational numbers (1) Addition or subtraction of a rational number with irrational number is an irrational number. (2) Multiplication or division of non zero rational number with irrational number is also an irrational number. (3) Addition, subtraction, multiplication and division of two irrational numbers can be either a rational or irrational number. 24  Let’s learn. Properties of order relation on Real numbers 1. If a and b are two real numbers then only one of the relations holds good. i.e. a = b or a < b or a > b 2. If a < b and b < c then a < c 3. If a < b then a + c < b + c 4. If a < b and c > 0 then ac < bc and If c < 0 then ac > bc Verify the above properties using rational and irrational numbers. Square root of a Negative number We know that, if a = b then b2 = a. Hence if 5 = x then x2 = 5. Similarly we know that square of any real number is always non-negative. It means that square of any real number is never negative. But ( -5 )2 = -5 \ -5 is not a real number. Hence square root of a negative real number is not a real number. Practice set 2.2 (1) Show that 4 2 is an irrational number. (2) Prove that 3 + 5 is an irrational number. (3) Represent the numbers 5 and 10 on a number line. (4) Write any three rational numbers between the two numbers given below. (i) 0.3 and -0.5 (ii) -2.3 and -2.33 (iii) 5.2 and 5.3 (iv) -4.5 and -4.6 Let’s learn. Root of positive rational number We know that, if x2 = 2 then x = 2 or x = - 2 , where. 2 and - 2 are irrational numbers. This we know, 3 7, 4 8 , these numbers are irrational numbers too. If n is a positive integer and xn = a, then x is the nth root of a. x = 5 2. This root may be rational or irrational. For example, 25 = 32 \ 2 is the 5th root of 32, but if x5 = 2 then x = 5 2 , which is an irrational number. 25  Surds We know that 5 is a rational number but 5 is not rational. Square root or cube root of any real number is either rational or irrational number. Similarly nth root of any number is either rational or irrational. If n is an integer greater than 1 and if a is a positive real number and nth root of a is x then it is written as xn = a or n a =x If a is a positive rational number and nth root of a is x and if x is an irrational number then x is called a surd. (surd is an irrational root) n In a surd a the symbol is called radical sign, n is the Order of the surd and a is called radicand. (1) Let a = 7, n = 3, then 3 7 is a surd because 3 7 is an irrational number. (2) Let a = 27, n = 3 then 3 27 is not a surd because 3 27 = 3 is not an irrational number.. (3) 3 8 is a surd or not ? Let 3 8 = p p3 = 8. Cube of which number is 8 ? We know 2 is cube-root of 8 or cube of 2 is 8. \ 3 8 is not a surd. 4 (4) Whether 8 is surd or not ? Here a = 8, Order of surd n = 4; but 4th root of 8 is not a rational number. \ 4 8 is an irrational number. \ 4 8 is a surd. This year we are going to study surds of order 2 only, means 3 , 7 , 42 etc. The surds of order 2 is called Quadratic surd. Simplest form of a surd (i) 48 = 16 ´ 3 = 16 ´ 3 =4 3 (ii) 98 = 49 ´ 2 = 49 ´ 2 =7 2 2, 3, 5,.... these type of surds are in the simplest form which cannot be simplified further. Similar or like surds 4 2 , -3 2 , 5 2 are some like surds. If p and q are rational numbers then p a , q a are called like surds. Two surds are said to be like surds if their order is equal and radicands are equal. 26  45 and 80 are the surds of order 2, so their order is equal but radicands are not same, Are these like surds? Let us simplify it as follows : 45 = 9´5 = 9 ´ 5 = 3 5 and 80 = 16 ´ 5 = 16 ´ 5 =4 5 \ 3 5 and 4 5 are now similar or like surds means 45 and 80 are similar surds. Remember this ! In the simplest form of the surds if order of the surds and redicand are equal then the surds are similar or like surds. Let’s learn. Comparison of surds Let a and b are two positive real numbers and If a < b then a × a < b × a If a 2< ab...(I) Similarly ab < b2...(II) \ a 2< b2...[from (I) and (II)] But if a > b then a 2> b2 and if a = b then a 2 = b2 hence if a < b then a2< b2 Here a and b both are real numbers so they may be rational numbers or surds. By using above properties, let us compare the surds. (i) 6 2 , 5 5 (ii) 8 3 , 192 (iii) 7 2 , 5 3 ? ? 36 ´ 2 ? 25 ´ 5 64 ´ 3 192 49 ´ 2 25 ´ 3 72 ? 125 192 ? 192 98 ? 75 < = > But 72 125 But 192 192 But 98 75 < = > \6 2 5 5 \ 192 192 \7 2 5 3 = \ 8 3 192 Or Or (6(5 5 )2, (7 2 )2 2) 2 (5 3 )2, 72 < 125 98 > 75 < > \6 2 5 5 \7 2 5 3 27  Operations on like surds Mathematical operations like addition, subtraction, multiplication and division can be done on like surds. Ex (1) Simplify : 7 3 + 29 3 Use your brain power ! Solution : 7 3 + 29 3 = (7 + 29) 3 = 36 3 9 + 16 ? = 9 + 16 Ex (2) Simplify : 7 3 - 29 3 100 + 36 ? = 100 + 36 Solution : 7 3 - 29 3 = (7 - 29) 3 = -22 3 1 Ex (3) Simplify : 13 8 + 2 8 - 5 8 1 1 26 + 1 − 10 ) 8 Solution : 13 8 + 8 - 5 8 = (13 + -5) 8 =( 2 2 2 17 17 = 8= 4´ 2 2 2 17 = ´ 2 2 =17 2 2 Ex (4) Simplify : Solution : = = = (8 + 2 - 5) 5 = 5 5 Ex. (5) Multiply the surds 7 ´ 42 Solution : 7 ´ 42 = 7 ´ 42 = 7´7´6 =7 6 ( 7 6 is an irrational number.) Ex. (6) Divide the surds : 125 ¸ 5 125 125 Solution : 5 = 5 = 25 = 5 (5 is a rational number.) Ex. (7) 50 ´ 18 = 25 ´ 2 ´ 9 ´ 2 = 5 2 ´3 2 = 15 ´ 2 = 30 Note that product and quotient of two surds may be rational numbers which can be observed in the above Ex. 6 and Ex. 7. 28  Rationalization of surd If the product of two surds is a rational number, each surd is called a rationalizing factor of the other surd. Ex. (1) If surd 2 is multiplied by 2 we get 2´ 2 = 4. 4 = 2 is a rational number. \ 2 is rationalizing factor of 2. Ex. (2) Multiply 2 ´ 8 2 ´ 8 = 16 = 4 is a rational number. \ 2 is the rationalizing factor of 8. Similarly 8 2 is also a rationalizing factor of 2. because 2 ´8 2 = 8 2 ´ 2 = 8 ´ 2 = 16. 6 , 16 50 are the rationalizing factors of 2. Remember this ! Rationalizing factor of a given surd is not unique. If a surd is a rationalizing factor of a given surd then a surd obtained by multiplying it with any non zero rational number is also a rationalizing factor of the given surd. Ex. (3) Find the rationalizing factor of 27 Solution : 27 = 9#3 =3 3 \ 3 3 ´ 3 = 3 ´ 3 = 9 is a rational number. \ 3 is the rationalizing factor of 27. Note that, 27 = 3 3 means 3 3 ´ 3 3 = 9 ´ 3 = 27. Hence 3 3 is also a rationalizing factor of 27. In the same way 4 3 , 7 3 ,... are also the rationalizing factors of 27. Out of all these 3 is the simplest rationalizing factor. Ex. (4) Rationalize the denominator of 1 5 1 1 5 5 Solution : = ´ = 5....(multiply numerator and denominator by 5.) 5 5 5 3 Ex. (5) Rationalize the denominator of. 2 7 3 3 7 3 7 3 7 Solution : 2 7 = 2 7 ´ = 2´ 7 = 14 7...(multiply 2 7 by 7 is sufficient to rationalize.) 29  Remember this ! We can make use of rationalizing factor to rationalize the denominator. It is easy to use the numbers with rational denominator, that is why we rationalize it. Practice set 2.3 (1) State the order of the surds given below. 3 (i) 7 (ii) 5 12 (iii) 4 10 (iv) 39 (v) 3 18 (2) State which of the following are surds. Justify. 3 4 5 22 (i) 51 (ii) 16 (iii) 81 (iv) 256 (v) 3 64 (vi) 7 (3) Classify the given pair of surds into like surds and unlike surds. (i) 52 , 5 13 (ii) 68 , 5 3 (iii) 4 18 , 7 2 (iv) 19 12 , 6 3 (v) 5 22 , 7 33 (vi) 5 5 , 75 (4) Simplify the following surds. (i) 27 (ii) 50 (iii) 250 (iv) 112 (v) 168 (5) Compare the following pair of surds. (i) 7 2 , 5 3 (ii) 247 , 274 (iii) 2 7 , 28 (iv) 5 5 , 7 2 (v) 4 42 , 9 2 (vi) 5 3 , 9 (vii) 7, 2 5 (6) Simplify. (i) 5 3 + 8 3 (ii) 9 5 - 4 5 + 125 3 (iii) 7 48 - 27 - 3 (iv) 7 - 7 +2 7 5 (7) Multiply and write the answer in the simplest form. (i) 3 12 ´ 18 (ii) 3 12 ´7 15 (iii) 3 8 ´ 5 (iv) 5 8 ´2 8 (8) Divide, and write the answer in simplest form. (i) 98 ¸ 2 (ii) 125 ¸ 50 (iii) 54 ¸ 27 (iv) 310 ¸ 5 (9) Rationalize the denominator. 1 5 6 11 (i) 3 (ii) (iii) (iv) (v) 5 14 7 9 3 3 30  Let’s recall. We know that, If a > 0, b > 0 then ab ab=== aa×× bb ( a) 2 (a + b)(a − b) = a 2 − b 2 ; = a ; a2 = a Multiply. Ex. (1) 2 ( 8 + 18 ) Ex. (2) ( 3 - 2 )(2 3 -3 2 ) == 2 × 8 + 2 ×18 == ( ) 3 2 3 −3 2 − 2 2 3 −3 2 ( ) = = 16 + 36 == 3×2 3 − 3×3 2 − 2 × 2 3 + 2 ×3 2 == 2 × 3 − 3 6 − 2 6 + 3 × 2 = 4+6 == 6 − 5 6 + 6 = 10 == 12 − 5 6 Let’s learn. Binomial quadratic surd yy 5 + 3 ; 3 + 5 are the binomial quadratic surds form. 5 - 3; 3- 5 are also 4 4 binomial quadratic surds. Study the following examples. y y ( a + b ) ( a - b ) = ( a )2 - ( b )2 = a -b yy ( 5 + 3 )( 5 - 3 ) = ( 5 )2 - ( 3 )2 = 5 - 3 = 2 y y ( 3 + 7 )( 3 - 7 ) = ( 3 )2 - ( 7 )2 = 3 - 7 = -4 3 3 3 9 9 - 20 11 yy ( 2 + 5 )( 2 - 5 ) = ( 2 )2 - ( 5 )2 = - 5 = =- 4 4 4 The product of these two binomial surds ( 5 + 3 ) and ( 5 - 3 ) is a rational number, hence these are the conjugate pairs of each other. Each binomial surds in the conjugate pair is the rationalizing factor for other. Note that for 5 + 3 , the conjugate pair of binomial surd is 5 - 3 or 3 - 5. Similarly for 7 + 3 , the conjugate pair is 7 - 3 or 3 - 7. 31  Remember this ! The product of conjugate pair of binomial surds is always a rational number. Let’s learn. Rationalization of the denominator The product of conjugate binomial surds is always a rational number - by using this property, the rationalization of the denominator in the form of binomial surd can be done. 1 Ex..(1) Rationalize the denominator. 5- 3 Solution : The conjugate pair of 5 - 3 is 5 + 3. 1 1 5+ 3 5+ 3 5+ 3 5+ 3 = ´ 5+ 3 = ( 5 ) 2 -( 3 ) 2 = 5−3 = 2 5- 3 5- 3 8 Ex. (2) Rationalize the denominator. 3 2+ 5 Solution : The conjugate pair of 3 2 + 5 is 3 2 - 5 8 = 8 ´ 3 2- 5 3 2+ 5 3 2+ 5 3 2- 5 = ( 8 3 2 − ) 5 (3 2 ) − ( 5 ) 2 2 = 8 ´ 3 2 - 8 5 = 24 2 - 8 5 = 24 2 - 8 5 9´ 2 - 5 18 - 5 13 Practice set 2.4 (1) Multiply (i) 3( 7 - 3 ) (ii) ( 5 - 7 ) 2 (iii) (3 2 - 3 )(4 3 - 2 ) (2) Rationalize the denominator. 1 3 4 5- 3 (i) (ii) (iii) (iv) 7+ 2 2 5 -3 2 7 +4 3 5+ 3 32  Let’s learn. Absolute value If x is a real number then absolute value of x is its distance from zero on the number line which is written as ½x½, and ½x½ is read as Absolute Value of x or modulus of x. If x > 0 then ½x½ = x If x is positive then absolute value of x is x. If x = 0 then ½x½ = 0 If x is zero then absolute value of x is zero. If x < 0 then ½x½ = -x If x is negative then its absolute value is opposite of x. Ex. (1) ½3½ = 3, ½-3½ = -(-3) = 3, ½0½ =0 The absolute value of any real number is never negative. Ex. (2) Find the value. (i) ½9-5½= ½4½ = 4 (ii) ½8-13½= ½-5½= 5 (iii) ½8½-½-3½= 5 (iv) ½8½´½4½= 8 ´ 4 = 32 Ex. (3) Solve ½x - 5½= 2. Solution : ½x - 5½= 2 \ x - 5 = +2 or x - 5 = -2 \x=2+5 or x = -2+5 \ x = 7 or x=3 Practice set 2.5 (1) Find the value. i) ½15 - 2½ (ii) ½4 - 9½ (iii) ½7½´½-4½ (2) Solve. 8- x x (i) ½3x-5½= 1 (ii) ½7-2x½= 5 (iii) ½ 2 ½= 5 (iv) ½5+ 4 ½= 5 33  3 Activity (I) : There are some real numbers 2......... -4 -9 -5 12 5 written on a card sheet. Use these numbers and construct two examples each of -11 9 2 3 11 5 7 addition, subtraction, multiplication and division. Solve these examples. -3 2 Activity (II) : Start ¯ 3 ´2 ´ 8 4 6 2 3 +10 6 ¸ 3 ´ 2 +3 ´3 5 5 ´ 465 31 ¸5 End Problem set 2 (1) Choose the correct alternative answer for the questions given below. (i) Which one of the following is an irrational number ? 16 3 (A) 25 (B) 5 (C) 9 (D) 196 (ii) Which of the following is an irrational number? (A) 0.17 (B) 1.513 (C) 0.2746 (D) 0.101001000..... (iii) Decimal expansion of which of the following is non-terminating recurring ? 2 3 3 137 (A) 5 (B) (C) 11 (D) 25 16 (iv) Every point on the number line represent, which of the following numbers? (A) Natural numbers (B) Irrational numbers (C) Rational numbers (D) Real numbers. p (v) The number 0.4 in q form is..... 4 40 3.6 36 (A) (B) 9 (C) 9 (D) 9 9 34  (vi) What is n , if n is not a perfect square number ? (A) Natural number (B) Rational number (C) Irrational number (D) Options A, B, C all are correct. (vii) Which of the following is not a surd ? 3 3 (A) 7 (B) 17 (C) 64 (D) 193 3 (viii) What is the order of the surd 5 ? (A) 3 (B) 2 (C) 6 (D) 5 (ix) Which one is the conjugate pair of 2 5 + 3 ? (A) -2 5 + 3 (B) -2 5 - 3 (C) 2 3 - 5 (D) 3 + 2 5 (x) The value of ½12 - (13+7) ´ 4½is............... (A) -68 (B) 68 (C) -32 (D) 32. p (2) Write the following numbers in q form. (i) 0.555 (ii) 29. 568 (iii) 9.315 315... (iv) 357.417417... (v)30. 219 (3) Write the following numbers in its decimal form.. (i) -5 (ii) 9 (iii) 5 (iv) 121 (v) 29 7 11 13 8 (4) Show that 5 + 7 is an irrational number. (5) Write the following surds in simplest form. 5 (i) 3 8 (ii) - 45 4 9 (6) Write the simplest form of rationalising factor for the given surds. 3 (i) 32 (ii) 50 (iii) 27 (iv) 10 (v) 3 72 (vi) 4 11 5 (7) Simplify. 4 3 1 1 3 (i) 147 + 192 - 75 (ii) 5 3 +2 27 + (iii) 216 - 5 6 + 294 - 7 8 5 3 6 (iv) 4 12 - 75 - 7 48 (v*) 2 48 - 75 - 1 3 (8) Rationalize the denominator. 1 2 1 1 12 (i) 5 (ii) 3 7 (iii) 3- 2 (iv) (v) 3 5 +2 2 4 3- 2 35  3 Polynomials Let’s study. Introduction to Polynomials Operations on polynomials Degree of the polynomial Synthetic Division Value of the polynomial Remainder theorem Let’s discuss. p3 - 12 p2 + p ; m2 + 2n3 - 3 m5; 6 are all algebraic expressions. 1 Teacher : Dear Students, consider each term of the expressions p3 - 2 p2 + p, m2 + 2n3 - 3 m5, 6 and state the power of each variable. 1 Madhuri : In the expressions p3 - 2 p2 + p powers of p are 3, 2, 1 respectively. Vivek : Sir, in the expression m2 + 2n3 - 3 m5 the powers of the variable are 2, 3, 5 respectively. Rahul : Sir, apparently there is no variable in the expression 6. But 6 = 6 × 1 = 6 × x0. Therefore, the power of the variable is 0. Teacher : In all algebraic expressions given above the powers of the variable are positive integers or zero. i.e. whole numbers. In an algebraic expression, if the powers of the variables are whole numbers then that algebraic expression is known as polynomial. 6 is also a polynomial. 1 6, - 7, 2 , 0, 3 etc. are constant numbers can be called as Constant polynomial. 0 is also a constant polynomial. 1 Are y + 5 and - 3 polynomials? y 1 Sara : Sir, y + 5 is not a polynomial, because y + 5 = y 2 + 5, here power of y is 12 which is not a whole number. 1 1 John : Sir, y - 3 is also not a polynomial because y - 3 = y-1 - 3, here power of y is - 1 which is not a whole number. Teacher : Write any five algebraic expressions which are not polynomials. Explain why these expressions are not polynomials ? Justify your answer. Is every algebraic expression a polynomial ? Is every polynomial an algebraic expression ? 36  Types of polynomials (based on number of terms) 1 2 2x , 5 x4 + x , m2 - 3m 2 y - 2y + 5, x3 - 3 x2 + 5x Only one term Two terms Three terms in the polynomial in the polynomial in the polynomial Monomial Binomial Trinomial Polynomials are written as p(x), q(m), r(y) according to the variable used. 1 For example, p(x) = x3 + 2x2 + 5x - 3, q(m) = m2 + 2 m - 7, r(y) = y2 + 5 Let’s learn. Degree of a polynomial in one variable Teacher : In the polynomial 2x7 - 5x + 9 which is the highest power of the variable ? Jija : Sir, the highest power is 7. Teacher : In case of a polynomial in one variable, the highest power of the variable is called the Degree of the polynomial. Now tell me, what is the degree of the given polynomial? Ashok : Sir, the degree of the given polynomial 2x7 - 5x + 9 is 7. Teacher : What is the degree of the polynomial 10 ? Radha : 10 = 10 × 1 = 10 × x0 therefore the degree of the polynomial 10 is 0. Teacher : Just like 10, degree of any non zero constant polynomial is 0. Degree of zero polynomial is not defined. Degree of a polynomial in more than one variable The highest sum of the powers of variables in each term of the polynomial is the degree of the polynomial. Ex. 3m3n6 + 7m2n3 - mn is a polynomial in two variables m and n. Degree of the polynomial is 9. (as sum of the powers 3 + 6 = 9, 2 + 3 = 5, 1 + 1 = 2) 37 Activity I : Write an example of a monomial, a binomial and a trinomial having variable x and degree 5. Monomial Binomial Trinomial Activity II : Give example of a binomial in two variables having degree 5. Types of polynomial (based on degree) Polynomial Polynomial in one in one variable variable 3x -3x1,- 7y 1, 7y 2y2 2y + 2y++y 1,+ -1,3x-2 3x2 x3 +x3x+2 +x22x ++2x +3 , 3m, -mm-3 m3 1 Degree Degree 1 Degree Degree 2 2 3 Degree Degree 3 Linear Linear polynomial polynomial Quadratic Quadratic polynomial polynomial Cubic Cubic polynomial polynomial Standard Standard formform Standard Standard formform Standard Standard formform ax + + b axb ax2 ax 2 + bx ++ + c ax3 ax bxc + bx+ bx 3 2 + cx+ + 2 cxd+ d herehere a and a and b are herehere b are a, cb,are a, b, c are herehere a, c,b,dc,are a, b, d are coefficients coefficients andand a ≠a0≠ 0 coefficients coefficients andand coefficients a ≠a0≠ 0 coefficients andand a ≠a0≠ 0 Polynomial : anxn + an-1xn - 1 +... + a2x2 + a1x + a0 is a polynomial in x with degree n an, an-1,....., a2, a1, a0 are the coefficients and an ≠0 Standard form, coefficient form and index form of a polynomial p(x) = x - 3x2 + 5 + x4 is a polynomial in x, which can be written in descending powers of its variable as x4 - 3x2 + x + 5. This is called the standard form of the polynomial. But in this polynomial there is no term having power 3 of the variable we can write it as 0x3. It can be added to the polynomial and it can be rewritten as x4 + 0x3 - 3x2 + x + 5. This form of the polynomial is called Index form of the polynomial. 38  One can write the coefficients of the variables by considering all the missing terms in the standard form of the polynomial. For example : x3 - 3x2 + 0x - 8 can be written as (1, -3, 0, -8). This form of the polynomial is called Coefficient form. Polynomial (4, 0, -5, 0, 1) can be written by using variable y as 4y4 + 0y3 - 5y2 + 0y + 1. This form is called Index form of the polynomial. Ex. p(m) = 3m5 - 7m + 5m3 + 2 Write the polynomial in standard form 3m5 + 5m3 - 7m + 2 Write it in the index form by considering all 3m5 + 0m4 + 5m3 + 0m2 - 7m + 2 the missing terms with coefficient zero. Write it in a coefficient form (3, 0, 5, 0, - 7, 2) Degree of the polynomial 5 Ex (1) Write the polynomial x3 + 3x - 5 in coefficient form. Solution : x3 + 3x - 5 = x3 + 0x2 + 3x - 5 ∴ given polynomial in coefficient form is (1, 0, 3, - 5) Ex (2) (2, -1, 0, 5, 6) is the coefficient form of the polynomial. Represent it in index form. Solution : Coefficient form of the polynomial is (2, - 1, 0, 5, 6) ∴ index form of the polynomial is 2x4 - x3 + 0x2 + 5x + 6 i.e. 2x4 - x3 + 5x + 6 Practice set 3.1 1. State whether the given algebraic expressions are polynomials ? Justify. 1 (i) y + (ii) 2 - 5 x (iii) x2 + 7x + 9 y (iv) 2m-2 + 7m - 5 (v) 10 2. Write the coefficient of m3 in each of the given polynomial. -3 -2 m3 - (i) m3 (ii) 2 + m- 3m 3 (iii) 3 5m2 + 7m - 1 3. Write the polynomial in x using the given information. (i) Monomial with degree 7 (ii) Binomial with degree 35 (iii) Trinomial with degree 8 39 4. Write the degree of the given polynomials. (i) 5 (ii) x° (iii) x2 (iv) 2 m10 - 7 (v) 2p - 7 (vi) 7y - y3 + y5 (vii) xyz + xy - z (viii) m3n7 - 3m5n + mn 5. Classify the following polynomials as linear, quadratic and cubic polynomial. 1 (i) 2x2 + 3x + 1 (ii) 5p (iii) 2 y- 2 5 (iv) m3 + 7m2 + m- 7 (v) a2 (vi) 3r3 2 6. Write the following polynomials in standard form. 1 (i) m3 + 3 + 5m (ii) - 7y + y5 + 3y3 - 2 + 2y4 - y2 7. Write the following polynomials in coefficient form. 2 (i) x3 - 2 (ii) 5y (iii) 2m4 - 3m2 + 7 (iv) - 3 8. Write the polynomials in index form. (i) (1, 2, 3) (ii) (5, 0, 0, 0, - 1) (iii) (- 2, 2, - 2, 2) 9. Write the appropriate polynomials in the boxes. Quadratic polynomial x + 7, Binomial............................. x2,............................. x3 + x2 + x + 5, Cubic polynomial 2x2 + 5x + 10, Trinomial............................. x3+ 9,............................. 3x2 + 5x Linear polynomial Monomial.......................................................... Let’s recall. (1) Coefficients are added or subtracted while adding or subtracting like algebraic terms, e.g. 5m3 - 7m3 = (5 - 7)m3 = -2m3 (2) While multiplying or dividing two algebraic terms, we multiply or divide their coefficients. We also use laws of indices. -4y3 ´ 2y2z = -8y5z ; 12a2b ¸ 3ab2 = 4ba 40  Let’s learn. Operations on polynomials The methods of addition, subtraction, multiplication and division of polynomials is similar to the operation of algebraic expressions. Ex (1) Subtract : 5a2 - 2a from 7a2 + 5a + 6. Solution : (7a2 + 5a + 6) - (5a2 - 2a) = 7a2 + 5a + 6 - 5a2 + 2a = 7a2 - 5a2 + 5a + 2a + 6 = 2a2 + 7a + 6 Ex (2) Multiply : - 2a × 5a2 Solution : - 2a × 5a2 = -10a3 Ex (3) Multiply :(m2 - 5) × (m3 + 2m - 2) × (m3 + 2m - 2) } Solution : (m2 - 5) (Each term of second polynomials is multiplied by = m2 (m3 + 2m - 2) - 5 (m3 + 2m - 2) each term of first polynomial) = m5 + 2m3 - 2m2 - 5m3 - 10m + 10 = m5 + 2m3 - 5m3 - 2m2 - 10m + 10 (Like terms taken together.) = m5 - 3m3 - 2m2 - 10m + 10 Here the degree of the product is 5. Ex (4) Add : 3m2n + 5mn2 - 7mn and 2m2n - mn2 + mn. Solution : (3m2n + 5mn2 - 7mn) + (2m2n - mn2 + mn) = 3m2n + 5mn2 - 7mn + 2m2n - mn2 + mn = 3m2n + 2m2n + 5mn2 - mn2 - 7mn + mn (Like terms are arranged.) = 5m2n + 4mn2 - 6mn (Like terms are added.) 41  Let’s recall. Degree of one polynomial is 3 and the degree of other polynomials is 5. Then what is the degree of their product ? What is the relation between degree of multiplicand and degree of a multiplier with degree of their product ? ÷ (x + 2) and write the answer in the given form Ex (5) Divide (2 + 2x2) Dividend = Divisor × Quotient + Remainder Solution : Let us write the polynomial in standard form. p(x) = 2 + 2x2 2 + 2x2 = 2x2 + 0x + 2 2x - 4 Dividend = divisor × quotient + remainder x + 2) 2x2 + 0x + 2 Method I : - 2x2 + 4x 2 + 2x2 = (x + 2) × (2x - 4) + 10 - - q(x), divisor = (x + 2) - 4x + 2 - - 4x - 8 s(x), quotient = 2x - 4 and +  + r(x), remainder = 10 10 ∴ p(x) = q(x) × s(x) + r(x). Method II : Linear method of division : Divide (2x2 + 2) ÷ (x + 2) To get the term 2x2 multiply (x + 2) by 2x and subtract 4x. 2x(x+2) - 4x = 2x2 ∴ Dividend = 2x2 + 2 = 2x(x+2) - 4x + 2...(I) To get the term -4x multiply (x+2) by -4 and add 8. -4 (x+2) + 8 = -4x ∴ (2x2 + 2) = 2x(x+2) - 4(x+2) + 8 + 2...from (I) ∴ (2x2 + 2) = (x + 2) (2x - 4) + 10 Dividend = divisor ´ quotient + remainder. 42  Remember this ! Euclid's division lemma If s(x) and p(x) are two polynomials such that degree of s(x) is greater than or equal to the degree of p(x) and after dividing s(x) by p(x) the quotient is q(x) then s(x) = p(x) ´ q(x) + r(x), where r(x) = 0 or degree of r(x) < 0. Practice set 3.2 (1) Use the given letters to write the answer. (i) There are ‘a’ trees in the village Lat. If the number of trees increases every year by ‘b’, then how many trees will there be after ‘x’ years? (ii) For the parade there are y students in each row and x such row are formed. Then, how many students are there for the parade in all ? (iii) The tens and units place of a two digit number is m and n respectively. Write the polynomial which represents the two digit number. (2) Add the given polynomials. (i) x3 - 2x2 - 9 ; 5x3 + 2x + 9 (ii) - 7m4 + 5m3 + 2 ; 5m4 - 3m3 + 2m2 + 3m - 6 (iii) 2y2 + 7y + 5 ; 3y + 9 ; 3y2 - 4y - 3 (3) Subtract the second polynomial from the first. (i) x2 - 9x + 3 ; - 19x + 3 + 7x2 (ii) 2ab2 + 3a2b - 4ab ; 3ab - 8ab2 + 2a2b (4) Multiply the given polynomials. (i) 2x ; x2- 2x -1 (ii) x5-1 ; x3+ 2x2 +2 (iii) 2y +1; y2- 2y3 + 3y (5) Divide first polynomial by second polynomial and write the answer in the form ‘Dividend = Divisor ´ Quotient + Remainder’. (i) x3- 64; x - 4 (ii) 5x5 + 4x4-3x3 + 2x2 + 2; x2 - x (6*) Write down the information in the form of algebraic expression and simplify. There is a rectangular farm with length (2a2 + 3b2) metre and breadth (a2 + b2) metre. The farmer used a square shaped plot of the farm to build a house. The side of the plot was (a2 - b2) metre. What is the area of the remaining part of the farm ? 43  Activity : Read the following passage, write the appropriate amount in the boxes and discuss. Govind, who is a dry land farmer from Shiralas has a 5 acre field. His family includes his wife, two children and his old mother. He borrowed one lakh twenty five thousand rupees from the bank for one year as agricultural loan at 10 p.c.p.a. He cultivated soyabean in x acres and cotton and tur in y acres. The expenditure he incurred was as follows : He spent Rs. 10,000 on seeds. The expenses for fertilizers and pesticides for the soyabean crop was 2000 x rupees and 4000 x2 rupees were spent on wages and cultivation of land. He spent 8000 y rupees on fertilizers and pesticides and rupees 9000 y2 for wages and cultivation of land for the cotton and tur crops. Let us write the total expenditure on all the crops by using variables x and y. + 2000 x + 4000 x2 + 8000 y + rupees He harvested 5 x2 quintals soyabean and sold it at Rs. 2800 per quintal. The cotton 5 2 crop yield was y quintals which fetched Rs. 5000 per quintal. The tur crop yield was 3 4y quintals and was sold at Rs. 4000 per quintal. Let us write the total income in rupees that was obtained by selling the entire farm produce, with the help of an expression using variables x and y. + + rupees Let’s learn. Synthetic division We know, how to divide one polynomial by other polynomial. Now we will learn an easy method for division of polynomials when divisor is of the form x + a or x - a. Ex (1) Divide the polynomial (3x3 + 2x2 - 1) by (x + 2). Solution : Let us write the dividend polynomial in the coefficient form. Index form of the dividend polynomial is 3x3 + 2x2 - 1 = 3x3 + 2x2 + 0 x - 1 ∴ coefficient form of the given polynomial = (3, 2, 0, - 1) Divisor polynomial = x + 2 44 Let us use the following steps for synthetic First row division. Second row (1) Draw one horizontal and one vertical line Third row as shown alongside. (2) Divisor is x + 2. Hence take opposite number of 2 which is -2 - 2 3 2 0 - 1 First row Write -2 to the left of the vertical line as shown. Write the coefficient form of 3 Third row the dividend polynomial in the first row. (3) Write the first coefficient as it is in the third row. (4) The product of 3 in the third row with - 2 3 2 0 -1 divisor -2 is -6. Write this -6 in the -6 8 -16 second row below the coefficient 2. 3 -4 8 - 17 Remainder Addition of 2 and -6 which is -4, is to be written in the third row. Similarly by multiplying and adding, last addition is the remainder, which is (- 17) and coefficient form of the Quotient is (3, - 4, 8). ∴ Quotient = 3x2 - 4x + 8 and Remainder = - 17 ∴ 3x3 + 2x2 - 1 = (x + 2)(3x2 - 4x + 8) - 17 This method is called the method of synthetic division. The same division can be done by linear meth