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# 202110722-PERFORM-STUDENT-WORKBOOK-MATHEMATICS-G09-FY_Optimized

## Description: 202110722-PERFORM-STUDENT-WORKBOOK-MATHEMATICS-G09-FY_Optimized

practice workbook Mathematics Grade 9 Name: Roll No: Section: School Name:

This practice book is designed to support you in your journey of learning Mathematics for class 9. The contents and topics of this book are entirely in alignment with the NCERT syllabus. For each chapter, a concept map, expected objectives and practice sheets are made available. Questions in practice sheets address different skill buckets and different question types, practicing these sheets will help you gain mastery over the lesson. The practice sheets can be solved with the teacher’s assistance. There is a self-evaluation sheet at the end of every lesson, this will help you in assessing your learning gap.

TABLE OF CONTENTS • Assessment Pattern: 40 Marks • Assessment Pattern: 80 Marks • Syllabus & Timeline for Assessment Page 1: 1. Number Systems Page 13: 2. Polynomials Page 24: 3. Coordinate Geometry Page 32: 4. Linear Equations in Two Variables Page 41: 5. Introduction to Euclids Geometry Page 51: 6. Lines and Angles Page 65: 7. Triangles Page 78: 8. Quadrilaterals Page 91: 9. Area of Parallelograms and Triangles Page 102: 10. Circles Page 115: 11. Constructions Page 124: 12. Herons Formula Page 131: 13. Surface Areas and Volumes Page 141: 14. Statistics Page 155: 15. Probability

ASSESSMENT PATTERN Marks: 40 Grade 9 / Mathematics Max Internal PAPER: BEGINNER PAPER: PROFICIENT Mark Option Q.No Skill Level Difficulty Level Skill Level Difficulty Level Easy Medium Difficult Easy Medium Difficult Section A (Question Type: VSA) ·· ··· · ·· 11 Understanding ·· Understanding ·· Remembering 21 Remembering ····· · Applying ·· · Applying 31 Applying · Understanding · Understanding 41 Applying · Remembering ·· · Applying · 51 Understanding Remembering · Understanding · 61 Understanding · · · · ·7 1 Remembering · Applying · 81 91 Remembering ·· ·Section B (Question Type: SA) 10 1 Understanding ·11 2 Understanding Understanding Understanding 12 2 Understanding Understanding 13 2 Understanding ·Section C (Question Type: SA) Understanding 3 Understanding Understanding 15 3 Understanding Applying · ·16 3 Applying Applying Applying 17 3 Section D (Question Type: LA) ·18 4 Analysing Analysing Understanding 19 4 · Understanding 20 4 Analysing Analysing Beginner Paper: (Easy: 50%, Medium: 40%, Diﬀicult:10%) Proﬁcient Paper:(Easy: 40%, Medium: 40%, Diﬀicult: 20%) Easy Question: Remembering questions directly from the text or from the given exercises. (Mostly from content of book or end of chapter exercise). Medium Diﬀiculty Question: In-depth understanding of questions, not necessarily from the text. (Slightly modiﬁed concepts or end of chapter questions). Diﬀicult Question: Question involving creativity like story writing, analysis question like character analysis, justiﬁcation of title or extracts (mostly requires creative and thinking skills).

ASSESSMENT PATTERN Marks: 80 Grade 9 / Mathematics Max Internal PAPER: BEGINNER PAPER: PROFICIENT Mark Option Q.No Skill Level Difficulty Level Skill Level Difficulty Level Easy Medium Difficult Easy Medium Difficult Section A (Question Type: MCQ) 11 Remembering • Remembering • 21 Remembering • Remembering • 31 Understanding • Understanding • 41 Remembering • Remembering • 51 Remembering • Remembering • 61 Remembering • Remembering • 71 Understanding • Understanding • 81 Remembering • Remembering • 91 Understanding • Understanding • 10 1 Understanding • Understanding • Section B (Question Type: VSA) 11 1 • Applying • Applying • 12 1 Remembering • Remembering • 13 1 • Remembering • Remembering • 14 1 Applying • Applying • 15 1 Understanding • Understanding • 16 1 • Applying • Applying • 17 1 Remembering • Remembering • 18 1 Applying • Applying • 19 1 Understanding • Understanding • 20 1 • Understanding • Understanding • Section C (Question Type: SA) 21 2 Understanding • Understanding • 22 2 Understanding • Understanding • 23 2 Understanding • Understanding • 24 2 • Understanding • Understanding • 25 2 • Applying • Applying • 26 2 Remembering • Remembering • Section D (Question Type: SA) 27 3 • Applying • Applying • 28 3 • Remembering • Remembering • 29 3 • Remembering • Remembering • 30 3 Understanding • Understanding • 31 3 Remembering • Remembering • 32 3 Applying • Applying • 33 3 • Applying • Applying • 34 3 Understanding • Understanding • Section E (Question Type: LA) 35 4 • Understanding • Understanding • 36 4 • Understanding • Understanding • 37 4 Understanding • Understanding • 38 4 Analysing • Analysing • 39 4 Analysing • Analysing • 40 4 Applying • Applying •

SYLLABUS FOR ASSESSMENT Grade 9 / Mathematics CHAPTERS PT-1 TE-1 PT-2 MOCK 1. Number Systems ✓ ✓ ✓ ✓ 2. Polynomials ✓ ✓ ✓ ✓ 3. Coordinate Geometry ✓ ✓ ✓ ✓ 4. Linear Equations in Two Variables ✓ ✓ ✓ ✓ 5. Introduction to Euclid’s Geometry ✓ ✓ ✓ 6. Lines and Angles ✓ ✓ ✓ 7. Triangles ✓ ✓ ✓ 8. Quadrilaterals ✓ ✓ ✓ 9. Areas of Parallelograms and Triangles ✓ ✓ 10. Circles ✓ ✓ ✓ 11. Constructions ✓ ✓ 12. Heron’s Formula ✓ ✓ 13. Surface Area and Volumes ✓ 14. Statistics ✓ ✓ 15. Probability ✓ Assessment Timeline Periodic Test-1 1st July to 31st July Term Exam -1 23rd September to 21st October Periodic Test-2 16th December to 13th January Mock Test 17th February to 9th March

LESSON WISE PRACTICE SHEETS (This section has a set of practice questions grouped into different sheets based on different concepts. By solving these questions you will strengthen your subject knowledge. A self-evaluation sheet is provided at the end of every lesson.)

1. Number Systems Learning Outcome • form, where p and q are integers and q ≠ 0. • Find a set of real numbers between two given By the end of this chapter, a student will be able to: • Explain the different types of number systems. numbers. • Find the decimal expansion real numbers. • Deduce the given real numbers into simpler • Determining the position of real numbers on forms. number line. • Convert rational numbers in decimal form to p q Concept Map Zero Natural Negative Terminating and Non- Numbers Numbers Terminating Recurring Whole Numbers Decimals Integers Rational Numbers Irrational Numbers Real Numbers Representation of Real Numbers Operations on Real numbers Finding Real Numbers (Rational and Commutative, Associative Irrational numbers) between two given and Distributive Laws for Addition numbers and Multiplication Law of Exponents Locating Real Numbers on the Number line Rationalizing the Denominator of Real Numbers Graphical Successive Magnification Method Method Decimal Expansion of Real Numbers (Rational and Irrational numbers ) Convert rational numbers (terminating, non terminating recurring) in decimal form to form, q ≠ 0 Key Points two given rational numbers ‘r’ and ‘s’, we have to add ‘r’ and ‘s’ and divide the sum by 2. The • Natural numbers include the numbers 1, 2, 3, 4... and are denoted by ‘N’. r+s number 2 lies between r and s. This step can • Natural numbers together with zero are called whole Numbers and denoted by ‘W’. be repeated to find the required number of rational numbers between two given rational • Positive and negative numbers with zero are called numbers. Example: Rational number between integers and denoted by ‘Z’. Negative numbers are called negative integers and positive numbers are 2 & 3 is (2 + 3) 5 called positive integers. = 22 • A number ‘r’ is called rational number if it can be written in the form of p , where p and q are integers rational number between 13 is q , and q ≠ 0. Rational numbers are denoted by Q. 1 3 10 24 • There are infinitely many rational numbers between + 8 10 5 any two given rational numbers. The methods to 2 4= == find the rational numbers between given numbers 2 2 16 8 are discussed below: – M ethod 1: To find a rational number between • – M ethod 2: To find n numbers between two given whole numbers ‘r’ and ‘s’; r is represented as 1

1. Number Systems represented as (n +1)×r and s is represented as (n +1)×s with a bar above the decimal digits that are repeating. ( ) ( )n + 1 n + 1 • Example: and the n number between r and s are – In the decimal expansion of 1 the remainder 2 (n +1)×r +1 (n +1)×r + 2 (n +1)×r + n becomes 0 and quotient is 0.5 Hence it is called ween r and s are , ,… . a terminating decimal. n+1 n+1 n+1 Example: 5 rational numbers between 2 and 3 is – In the decimal expansion of 2 the remainder 3 calculated as follows: never becomes zero and the quotient is 0.6666……. Hence it is a non-terminating (5 + 1) 12 (5 + 1) 18 recurring decimal and written as 0.6 2 = 2× (5+ 1) = 6 3 = 3× (5+ 1) = 6 • The decimal expansion of an irrational number is Rational numbers between 2 and 3 is same as non-terminating and non-recurring. rational numbers between 12 18 . • To convert a given terminating decimal number and 66 into p form, multiply and divide the decimal q The rational numbers are number with 10n, where n is the number of digits (5+1)×2+1 (5+1)×2+ 2 (5+1)×2+ 4 (5+1)×2+5 in the terminating decimal number. (5 + 1) , (5 + 1) ,…., (5 + 1) , (5 + 1) Example: 0.36 = 0.36 102 36 (n = 2 because × = 13 14 16 17 102 100 , ,....., , 66 6 6 there are two decimal digits) • A number‘s’ is called irrational, if it cannot be • To find ‘r’ number of rational number between two written in the form of p , where ‘p’ and ‘q’ are given rational numbers p1 and p2 : q q1 q2 integers and q ≠ 0. o Equalise the denominators of the given rational numbers by multiplying and dividing Examples of irrational numbers are ≠, 2, 12, 15, 17 , etc. numerator and denominator of the two given numbers with suitable integers. • Rational numbers together with irrational numbers are called real numbers. Every real number is o Let p and p whose denominators have been represented by a unique point on the number line. Also, every point on the number line represents a 3 4 unique real number. The number line is also called real number line. qq 34 • An irrational number of the form n where n is a positive integer, can be located on the number line rationalised, i.e., q3 = q4 ( )after n −1 has been located. By repeating this o Multiply the numerator and denominator with any integer value or 10n where n is any positive process any irrational can be located on the integer. number. • In the decimal expansion of real numbers, if the o Let X and X2 and the corresponding rational remainder becomes zero, then such numbers are 1 called terminating numbers and if the remainder never becomes zero but has a repeating block Y Y1 of digits in the quotient, then such numbers are called non-terminating recurring numbers. Non- 1 terminating and recurring numbers are denoted numbers whose value will be same as the given two numbers. Select the integer or n (in the previous step) such that there are ‘r’ whole numbers between X1 and X2. o If X2 > X1, then the rational numbers between p p :can be written as X1 + 2 ;…, 1 and X +1 , 2 1 qq Y1 Y1 12 X +r such that (X1 + r) < X2 and m is a integer. 1 Y1 2

1. Number Systems   Example: To find rational numbers between line. • Irrational numbers which do not have a terminating 11 and decimal can be marked on a number line to a certain degree (level) of accuracy depending on 36 the amount of magnification. (Number of decimals considered, more decimals considered means Equalizing the denominators of 1 and 1 we higher accuracy). 36 • To locate a real number or irrational number on the number line by successive magnification, get 1 = 1×2 = 2 then magnification is started from first digit of the 3 3×2 6 decimal point(digit next to the decimal point) and then the second digit and so on until the required Selecting n = 1 we get 1 = 2 10 = 20 accuracy is achieved in locating the real number. × 3 6 10 60 1 1 10 10 =× = 6 6 10 60 Here 20 > 10 , then rational numbers between 60 60 the given numbers are 11 , 12 , 13 ,⊃ ., 18 , 19 . 60 60 60 60 60 • To write a non-terminating and recurring rational number in p form: q o Assume the given number as x and then multiply the given number with 10n, where n is the number of repeating decimal digits. o Write the product in the form of A + x where A is constant value. o Solving for x would give the p form of the given q non-terminating recurring rational number. Example: Write 0.3 in p form . q Let x = x = 0.33333…. Here number of repeating decimals is 1 i.e., n = 1, Mulitply the above equation with 10n = 101 = 10 10x = 3.33333….. =3+x 9x = 3 31 x= = 93 • The process of visualisation of representation of numbers on the number line through a magnifying glass is called as the process of successive magnification. Using this method, it is possible to visualise the position of a real number with terminating decimal expansion on the number 3

1. Number Systems • Addition, subtraction, multiplication or division • If ‘a’ and ‘b’ are positive real numbers then (except) of any two rational numbers will result o  ab = a b in a rational number. Rational numbers satisfy commutative law, associative law and distributive o  a= a law. bb • If a, b, c are any three rational numbers than we ( )( ) o  a + b a − b = a − b have Commutative Addition Multiplication ( )( ) o  a + b a − b = a2 − b Law a+b=b+a a×b = b×a Associative Law a+(b+c) = (a+ b)+c a×(b×c) = (a×b)×c ( )( ) o  a + b c + d = ac + ad + bc + bd Distributive Law a×(b+c) = (a×b)+(a×c) ( ) o  2 • Irrational numbers also satisfy the commutative a+ law, associative law and distributive law for b = a+2 ab + b addition and multiplication. Also the sum, difference, quotients and products of irrational • For any two real numbers - a, b, we have numbers are not always irrational. ( a+b) ( a+b)=a2 -b2 • Properties of Irrational Numbers: o The sum or difference of rational number and • Let a > 0, m and n are integers such that m and n have no common factors other than 1 and n > 0, an irrational number is irrational. then o The product or quotient of a non-zero rational m n a m = n am number with an irrational number is irrational. o If we add, subtract, multiply or divide two an = ( ) irrationals, the result may be rational or irrational. • Laws of exponents for real numbers: Let, a, n and m • If ‘a’ is a real number and a > 0, then a = b means are rational numbers b2 = a and b>0. o am × an = am+n ( ) o  am n = amn • To find the value of x geometrically: o am = am-n ,m > n o  Draw a line segment AB of length x units. an o Draw a line segment BC of length 1 unit ( ) o ambm = ab m extending the line AB till C. o  Draw a semi-circle of diameter AC. o am  a m o Draw a line perpendicular to AC at B meeting bm =  b  the circle at D.  o The length of the line segment BD = x o a0 = 1 • Rationalizing the denominator means making the denominator into a rational number by multiplying and dividing the given number by suitable numbers. • If ‘a’ is real number and a > 0, n is a positive integer then n a = b ⇒ a = bn and b > 0. In the symbol n is called radical sign. 1 • In the language of exponents, n a = an 4

1. Number Systems Work Plan COVERAGE DETAILS PRACTICE SHEET PS – 1 CONCEPT COVERAGE • Types of Number systems Pre-requisites • Finding required quantity of numbers PS – 2 PS – 3 Number Systems between two given numbers. PS - 4 • Decimal numbers to p form q • Commutative Law, Associative Law, Distributive law for addition and multiplication. • Laws of Exponents • Concept of Irrational Numbers • Locating Irrational Numbers on Number Line using graphical method. • Real Numbers and their Decimal Expansion • Representing Real Numbers on Number Line using successive magnification method. • Operations on Real Numbers • Determine x geometrically where x has a terminating decimal. • Laws of exponents on real numbers Worksheet for “Number Systems” PS - 5 Evaluation with Self Check or Peer ---- Self Evaluation Sheet Check* 5

PRACTICE SHEET - 1 (PS-1) 1. Justify the following statements. i) 0.5 is a rational number ii) -1 is not a whole number 2. Find 3 rational numbers between -1 and 1. 3. Show the following numbers on a number line. i)-1 2 34 ii) - iii) iv) 5 42 4. Express the following numbers in p form and find 4 rational numbers between the given set of q rational numbers. i) 0.256, 0.259 ii)15.01, 15.011 5. Name the property or law that is used in the following equations. (a, b and c are rational numbers.) i) a + b = b + a ii) a x (b x c) = (a x b) x c iii) a x (b + c) = ab+ ac 6. Determine the result of the following operations: i) 3 × 5 iii) 27 ii) 2 + 4 2 6 iv) 5 5 + 250 6

PRACTICE SHEET - 2 (PS-2) 1. Name the type of numbers listed below. i) 0     ii) – 0.11   iii) 9.35 iv) 4    v) 5     vi) −1 5 2. Locate 5 following irrational numbers on number line. 3. Find the decimal expansion of the following numbers and mention the quotient, divisor and remainder. 22 4 1 i) ii) iii) 50 7 11 4. Express the following numbers in p form. q i) 0.2856 ii) 0.001001 iii) 1.356 iv) 0.88888 5. Express the following numbers in p form. q i) 0.6 ii) 1.101 6. Express the following numbers in the form of p where q ≠ 0. q i)5.3386 ii)102.2335 7. Find two rational numbers and two irrational numbers between 0.5 and 0.6. 7

PRACTICE SHEET - 3 (PS-3) 1. Locate 5.055 on a number line. 2. Locate 35.91 upto four decimals on a number line. 3. Tom and Jerry are running on a track of 100 m and Jerry wins the race. Tom was behind Jerry by 0.434 m. Locate the position of Tom on the track when Jerry was at winning line by measuring from the starting line. 8

PRACTICE SHEET - 4 (PS-4) p 1. If 2 andπ are irrational numbers then find whether 2 + 2 and 2 are rational numbers or irrational numbers. 2. Give examples to show that i) Addition of two irrational numbers is a rational number. ii) Addition of two irrational numbers is a irrational number iii) Multiplication of two irrational numbers is a rational number iv) Multiplication of two irrational numbers is a irrational number 3. Find the value of 7.6 geometrically. 4. Rationalise the denominator and simplify the given number � 5 2 +3 2 −3  65 5 iv) 6.44 iii)  210  3.22 ii) 55 5. Find the value of ( )i) 22 ×23 6. Simplify the given number and rationalie the denominator if needed. 32 + 3− 2 2− 3 7. Simplify the given number. 1 1 = 20 − 2 + 8 − 5 + ( )( )8 − 5 20 − 2 8 − 5 20 − 2 9

PRACTICE SHEET - 5 (PS-5) I. Choose the correct option. 1. Set of natural numbers is a subset of (A) Set of even numbers (B) Set of real numbers (C) Set of odd numbers (D) Set of composite numbers 2. Between two integers, there exist ______________ number of integers. (A) Only one (B) Finite (C) Two (D) Infinite 3. Rational number between 5 and 7 is: (A) 5 + 7 (B) 5× 7 (C) 5.4 (D) 5.8 2 2 4. The rationalizing factor of 6 x2y3z4 is: (A) 6 x2y5z3 (B) 3 x2y5z3 (C) (D) 3 x4 y3z2 6 x4y3z2 5. 1 is not equal to: ( 3 - 2) (A) ( 3 - 2) 2 (C) 3 + 2 3 (5 - 2 6) (B) (D) ( 6 - 2) (9 - 6) 6. 2 6 = 2+ 3+ 5 (A) 1( 2+ 5- 3) (B) 2 + 3 - 5 (C) 4 - 2 - 3 (D) 2 + 3 + 6 - 5 2 7. If m = cab , then b equals: a-b (A) m(a - b) (B) 1 (C) ma (D) mab ca 1+ c m + ca 8. If a ≥ 0, then a a a = (A) 8 a7 (B) a 8 a (C) 8 a (D) a a 9. Which of the following number has the terminating decimal representation? (A) 17 (B) 3 (C) 1 (D) 1 3 5 3 7 10. Which of the following expression is same as 1 ? ( 3 2 -1) (A) 3 4 + 33 2 + 1 (B) 3 5 + 2 (C) 3 4 + 3 2 + 1 (D) 3 2 + 1 10

PRACTICE SHEET - 5 (PS-5) II. Short answer questions. 1. Simplify the following expression. (a) ( 3 + 5)2 (b) ( 5 - 2)2 2. If x= 1 a+ 1  x2 -1 = a -1   , then show that x - x2 -1 2 2  a 3. Find the value of 1 + 2 1 + 2 1 + 2 1 + ...... III. Long answer questions. 1. Express the following recurring decimal expansions in the form p , where p and q are integers and q ≠ 0. q (a) 0.7 (b) 0.257 2. Find the value of 1 + 1 + 1 + 1 + 1 . 4+ 5 5+ 6 6+ 7 7+ 8 8+ 9 11

Self-Evaluation Sheet Marks: 15 Time: 30 Mins 1. Which of the following statements are true? 5. Find the value of  1  1   (1 Mark) Explain the reasons. 3 − 2 3 + 2  i) All whole numbers are integers. ii) All rational numbers are not integers.(1 Mark) 6. Rationalize the denominator and simplify: 2. Locate 6 on a number line. (3 Marks) 7 − 2 (2 Marks) 3− 5 3. Lpolaccaetse.14.61 on a number line up to 4 decimal (4 Marks) 7. Simplify: 4 (2 Marks)  2 4 1  7 3 1 i) 73 + 53  ×73 ii) 64  ×   3 4. Find the value of 2.3 geometrically. (2 Marks)    5 5 64    12

2. Polynomials Learning Outcome By the end of this chapter, a student will be able to: • Compute of division of polynomials • Identify elements and classify the given algebraic • Factorise the given polynomials. • Apply principles of algebraic identities to expression. • Determine the value of polynomial at a given value simplify polynomials of the variable. Concept Map Addition Subtraction Multiplication Term Division Coe icient Variable Constant Polynomials Naming of Finding the value of Zero of Algebraic Polynomial polynomial for given value Polynomial Identities of variable. Calculate Remainder Finding factors of polynomials Long Division Factor Theorem Remainder Theorem Factorisation by Splitting the Middle term Using Identities Key Points polynomial. Also in a polynomial, the exponent of the variables should only be whole numbers. • Expressions are formed from variables and • Among a given setof terms, if thecoefficientis different constants. The value of an expression changes and the variable is same, then such terms are called with the value of the variables. Terms are added Like terms. to form expressions. Terms themselves can be Example: 3x, −5x, 10x, 0.5x are all like terms. formed as product of factors. The numerical factor • Addition and subtraction of polynomials is done by of a term is called its coefficient or numerical adding or subtracting the coefficients of like terms. coefficient. Subtraction is same as addition inverse. Thus Example: In 3x + 4y − 3xy + y2 + 11: subtracting ‘−a’ is similar to adding ‘a’. 3x, 4y, −3xy, y2, 11 are the terms and the numbers in Example: 5x + 2x = x(5+2) = 7x each term is called coefficient. +11 is the constant. • In multiplication of polynomials, the individual terms are multiplied and then the like terms are • Expressions that contain only one term is called combined using identities. Monomial. Example: 5x, 3y, −2t, etc. • An equality which will be true for every value of the Expression containing two terms is called a variable in it is called an identity. An equation is binomial. true for only certain values of the variable in it. An Example: 3x + 4y, 2x + 5, y −1, etc. equation is not true for all values of the variables. Expression having three terms is called a trinomial. Examples: 2x − 3y−12, 3x2 + 3y + 3x, etc. In general, an expression containing one or more terms with non-zero coefficients is called 13

2. Polynomials Some of the standard identities are: A non zero constant polynomial has no zero and every real number is a zero of the zero polynomial. ( )Identity I: a + b 2 = a2 + 2ab + b2 Example: Zero of p(x) = 5 does not exist ( )Identity II: a − b 2 = a2 − 2ab + b2 Zero of p(x) = 0 is any real number • A Linear polynomial has only one zero. Identity III: (a+b)(a-b)=a2-b2 If p(x) = ax + b, a ≠, then finding zero of the Identity IV: (x + a)(x + b) = x2 + (a + b) x + ab polynomial means finding the value of x such that p(x) = 0 • If an algebraic expression has only one variable then it can be called polynomial in one variable. A ⇒ ax + b = 0 polynomial in one variable can have any number of ⇒ x = −b terms in it. If the polynomial has no variables in it then it a is called constant polynomial and the constant Example: Zero of 5x + 3 is obtained by equating the polynomial ‘0’ is called zero polynomial. polynomial to 0 • In a polynomial, the highest power of the variable 5x + 3 = 0 ⇒ x = −3 is called Degree of polynomial. The degree of a non-zero constant polynomial is zero. The degree 5 of a Zero polynomial is not defined. A polynomial of degree one is called linear • If p(x) and g(x) are two polynomials such that degree polynomial. Ex: 3x − 5 of p(x) ≥ degree of g(x), then we find polynomials A polynomial of degree two is called quadratic q(x) and r(x) such that polynomial. Ex: 4 y2 − 2 y − 1 p(x) = g(x) × q(x) + r(x). A polynomial of degree three is called cubic Dividend = Divisor ×Quotient + Remainder polynomial. Ex: x3 + 5x − 2 In the above equation r(x) = 0 or • If the variable in a polynomial is ‘x’, it can be degree of r(x) < degree of g(x). denoted as p(x). If there is one variable then it is We say that p(x) is divided by g(x), q(x) is quotient called polynomial in one variable. A polynomial and r(x) is remainder. can be of more than one variable. If the remainder r(x) is zero, then g(x), q(x) are considered as factors of p(x) and the zero of g(x) or • A polynomial in one variable ‘x’ and degree n is an q(x) will be zeros of p(x). expression of the form anxn + an−1xn−1 + a2xn−2 + …+ a1x + a0 Example: x2 + 3x + 6 = (x + 1)(x + 2) + 3 Here an ,an−1 ,an−2 ,…..,a1 ,a0 are constants and an ↑ 0 Here x2 + 3x + 6 is dividend, . The standard form of writing a polynomial is If (x + 1) is divisor then (x + 2) is quotient. arranging the terms in the descending order of If (x + 2) is divisor then (x + 1) is quotient. their degrees. 3 is the remainder of the division. Example: Standard form of x3 − 5x + x2 is • To divide one polynomial with another: The process is similar to division of numbers. x3 + x2 − 5x o Write the dividend in standard form i.e., in • Inpolynomialp(x),xiscalledvariable.Ifxsubstituted descending order of the degrees of the terms of by a real number, say ‘m’ then polynomial is then the polynomial. represented as p(m). o Multiply the divisor by the first term of the Example: p(x) = x2 + 5 quotient and subtract this product from the p(1) = 12 + 5 = 6 dividend. Select the quotient such the first term of the dividend is eliminated during subtraction. • A real number, ‘c’ is called the zero of a polynomial o The result of the subtraction will become the p(x) if p(c) = 0. new dividend but the divisor remains the same. Example: p(x) = x − 5 o Repeat this process until the remainder is ‘0’ or Zero of p(x) is 5 because p(5) = 5 − 5 = 0 the degree of the new dividend is less than the degree of divisor. o The sum of the quotients of the individual steps will be the whole quotient. • Remainder Theorem: Let p(x) be any polynomial 14

2. Polynomials with degree greater than or equal to 1 and let ‘a’ be any real number. If p(x) is divided by a linear polynomial (x − a), then the remainder is p(a). • Factor Theorem: If p(x) is a polynomial of degree n ≥ 1 and ‘a’ is any real number then i) (x − a) is a factor of p(x) if p(a) = 0, and ii) p(a) = 0 if (x − a) is a factor of p(x). • Factorizing a polynomial by splitting the middle term x2 + lx + m can be factorised as (x + a)(x + b) where l = a + b and m = ab • Algebraic Identities: ( )Identity V: x + y + z 2 = x2 + y2 + z2 + 2xy + 2 yz + 2zx ( ) ( )Identity VI: x + y 3 = x3 + y3 + 3xy x + y ( ) ( )Identity VII: x − y 3 = x3 − y3 − 3xy x − y Identity VIII: ( )( )x3 + y3 + z3 − 3xyz = x + y + z x2 + y2 + z2 − xy − yz − zx Work Plan CONCEPT COVERAGE COVERAGE DETAILS PRACTICE SHEET Pre-requisites • Terms, Variables, Constants and coefficients PS – 1 • Monomial, Binomial and Polynomials • Addition, Subtraction, Multiplication of polynomials. • Identities and Equations Polynomials • Degree of Polynomial, Naming of PS – 2 Algebraic expression. • Zeroes of Linear polynomial • Value of a polynomial. • Division of polynomials PS – 3 • Remainder Theorem • Factor Theorem PS − 4 • Factorizing by Splitting the middle term • Algebraic Identities Worksheet for \"Polynomials\" PS − 5 Evaluation with Self Check Self Evaluation Sheet or Peer Check* 15

PRACTICE SHEET - 1 (PS-1) 1. Mention term, coefficient and constants of the polynomial. 3xy + 4 y2 − 3x2 + 2 2. Write examples of a monomial, binomial and trinomial. 3. Subtract 3x3 − 4xy2 + x2 y − xy − 20 from 5x3 + x2 y − 5x ( )( )4. Simplify the expression 3x + 4 y −12 x2 + y + 2 5. Simplify the expression (5x − 2y + 3)(3x + y − 5) . 6. Find the value of (3x − 5) (3x + 5). 7. Find the value of (x − 1) (x − 2). 8. Find the value of (x + 1) (x − 2) (x + 3). 9. Find the value of: i) (2x + 3)2 ii) ( x − 3)2 − ( x + 3)2 ( ) ( )2 2 iii) x2 − 2 iv) x2 − 2x + 3 10. Using Identities, determine the value of: i) 992   ii) 1032   iii) 102 x 98   iv) 105 x 107 16

PRACTICE SHEET - 2 (PS-2) 1. Explain the various elements of the algebraic expression and name it. 5x3 − 25 x − 35. 3 2. State whether the expression x2 + x − x is a polynomial? 3. Mention the terms, coefficient of terms, constant and degree of the following polynomials. 3 i) 3x3-5x2 +2 ii) x2+x4 iii) 5x3 iv) y2+3y+ 5 4. Is 3 + 3 a polynomial? If yes, mention the terms, coefficients, constants and the degree of the polynomial. 5. Give two examples for the following algebraic expressions in one variable. i) Linear Monomial ii) Quadratic Monomial iii) Cubic binomial iv) Quadratic trinomial 6. Given p(x) = 3x + 5x2 − 9 . Find p(0), p(−1), p(1), p(2) 7. Find the zeroes of the following polynomials: 42 1 i) 3t − 2.4 ii) t + iii) 6 − 2x iv) + 1 34 x 17

PRACTICE SHEET - 32 (PS-–32)) 1. Divide 5x4 − x + x3 by i) x- 2 ii) x2 + 1 . State the quotient and remainder. 2. Divide x3 + 1 − x2 with x and find the quotient and remainder. 3. Find the remainder when 3x2 − 5x − 24 is divided by x + 1, 2x − 1. 4. Which among the following polynomials are factors of 7x2 − 18x + 11 i) x + 2 ii) x − 1 iii) x 5. The census of deer in a forest is done every year starting from 2000. It was found that the population of deer is given as x3 − x2 + 10x − 10 where x is the number of years after 2000. The number of guards in the forest is given by x − 1. Calculate an algebraic expression for the ratio of number of deer to the number of guards. 6. Check whether x−3 is a factor of x2 + 9 − 6x. Are there any other factors for the given expression. 7. Determine the remainder when (x + 1) divides the following polynomials. i)15x2-25x+12 ii) x3-8x2+24x-9 iii) x2 iv) x State whether x+1 is a factor of the polynomials. 8. Ravi plans his birthday party after school for ‘x’ number of his friends and orders 5x3 + 36x2 sweets from a hotel. How many sweets Ravi wanted to give to each of his friend. 18

PRACTICE SHEET - 24 (PS-42) 1. If (x − 1) is a factor of the following polynomials, then find the value of k. i) 5x-kx2 ii) x3- x2 +5 k 2. Factorise the given polynomials by splitting the middle term. i) x2+3x+2 ii) 3x2-2x-1 3. Factorise x3 + 5x2 − 150. 4. Factorise x4 − 4x2 + 4 5. Find the factors of 3x2 + 9x + 6 by splitting the middle term. State the factors of x2 + 3x + 2 using the factors of 3x2 + 9x + 6 6. Find the value of ( )( ) 3 3 i) 2x+4 ii) 3x2-x 7. Find the value of i) 983 ii) 1043 iii) 103 × 98 8. Factorize using appropriate identities 125x3 -8y3 -150x2y+60xy 2 9. If the area of a square is given by x2 + 2x + 1 what is the side of the square? 10. Factorise 4x2 + 9 + 12x using factor theorem. 11. If xy = −yz, find the value of x3+y3+z3 . 12. Factorise x3 + 216y3 + z3 − 9xyz 8 19

PRACTICE SHEET - 5 (PS-5) I. Choose the correct option. 1. If a + b + c = 0, then a3 + b3 + c3 = (A) 3abc (B) -3abc (C) 2abc (D) abc 2. Degree of the polynomial p(x) = 2x5 + 5x2 + 9 is: (A) 5 (B) 4 (C) 3 (D) 1 3. A real number ‘a’ is a root of polynomial equation p(x)=0 if: (A) p(a) = 1 (B) p(a) = 0 (C) p(a) = -1 (D) p(a) = 2 4. Which of the following algebraic expressions is not a polynomial? (A) 6x2 + 9x 2/ 3 - 8 (B) 17 x2 + x - 3 (C) 0 (D) 3 2 5. Factorisation of x2 + 3 2x + 4 iiss: (A) (x + 2 2)(x - 2) (B) (x - 2 2)(x - 2) (C) (x - 2 2)(x + 2) (D) (x + 2 2)(x + 2) 6. For the polynomial p(x) = x5 + 4x3 – 5x2 + x – 1, one of the factor is: (A) (x+1) (B) x (C) (x-1) (D) (x+2) 7. One of the dimensions of the cuboid whose volume is 16x2 – 26x + 10 is: (A) 2 (B) (8x-5) (C) (x-1) (D) All of these 8. If both x =2, and x = 1 are factors of px2 + 5x + r , then p = (A) 2r 2 (B) 3r (C) 4r (D) r 9. In the method of factorization of an algebraic expression, which of the following statement is false? (A) Taking out a common factor from two or more terms. (B) Taking out a common factor from a group of terms. (C) Using remainder theorem. (D) Using standard identities. 10. The following are the steps involved in finding the value of a4 + 1 , when a + 1 =1. Arrange them in sequential order from the first to the last. a4 a (A) a2 + 1 + 2 =1 ⇒ a2 + 1 = -1 (B) (a2 )2 +  1  + 2 =1 a2 a2  a2  (C)  a + 1 2 =12 (D)  a2 + 1 2 = (-1)2 (E) a4 + 1 =-1  a   a2  a4 (A)CADBE (B) CDBAE (C) CBADE (D) CEDAB II. Short answer questions. 1. If (2a+b) = 12 and ab = 15, then find the value of 8a3 + b3. 2. If the perimeter of a rectangle is 24 units and the length exceeds the breadth by 4 units, then find the area of a rectangle. 3. Without actually calculating the cubes, evaluate the expression (30)3 + (-18)3 + (-12)3. 20

PRACTICE SHEET - 5 (PS-5) III. Long answer questions. a3 + b3 + c3 - 3abc 1. Value of ab + bc + ca - a2 - b2 - c2 , When a = - 5, b = -6, c = 10 is 2. The polynomial ax3+ 3x2 – 3 and 2x3 – 5x + a when divided by x-4 leaves the remainder R1 and R2 respectively. Find the value of a, if 2R1 – R2 = 0 21

Self-Evaluation Sheet Marks: 20 Time: 30 Mins 1. If s(t) = 32t2 + 24t + 350 , find p(2) and p(10) 4. Divide 2x3 − 11x2 + 24x − 22 with (x − 5) and find quotient and remainder. (3 Marks) where t is the time in seconds. Find p(2) and p(10). (2 Marks) 2. The population of tigers is reducing continuously and the number of tigers in any 5. Find the remainder when 5x3 − 7x2 + 20 is given year 2000-5y where y is the number of years after 2010. Determine in what year there divided by (x − 2). (2 Marks) will be no tigers left. (2 Marks) 6. Find k if x + 2 is a factor of 3x2+4x-k .(2 Marks) 3. Find the zero of the following linear expressions. i) 35t − 5 ii) t − 7  (2 Marks) 58 22

Self-Evaluation Sheet Marks: 20 Time: 30 Mins 7. Factorise 7x2 + 15x + 8 by splitting the middle 3 10. If x + y = xy , find the value of x+y-z (1 Mark) ( )term. (1 Mark) 8. Factorise 2x3 + 3x2 − 32x + 15� using factor t heorem. (4 Marks) 9. Find the value of 1013. (1 Mark) 23

3. Coordinate Geometry Learning Outcome • Mark the points based on their coordinate values. By the end of this chapter, a student will be able to: • Determine the coordinates of the point located in a plane. Concept Map Key Points • The horizontal line is called X-axis and the vertical line is called Y-axis and the point where x-axis (X’X) • The point from which the distances are marked is and y-axis (Y’Y) meet is called origin and denoted called origin. A point in the positive direction at a by O. OX and OY represent the positive direction distance ‘r’ from the origin represents the number and OX’ and OY’ represent the negative direction of r. A point in the negative direction at a distance ‘r’ x-axis and y-axis respectively. from the origin represents the number (–r). Example: • The two axes divide the plane into four parts and each part is called quadrant. Quadrants are numbered I, II, III and IV anticlockwise from OX. The plane is called Cartesian plane or coordinate plane or the xy-plane and the axes are called coordinate axis. • Two number lines are combined in such a way that they cross each other at their zeroes or origins. • The x-coordinate of a point is its perpendicular distance from the y-axis measured along the x-axis (positive along positive direction of x-axis and 24

3. Coordinate Geometry negative along the negative direction of x-axis). of y axis. The x-coordinate is also called abscissa. o B will be the point with coordinates (x,y) • The y-coordinate of a point is its perpendicular • If x ≠ y, then the position of the point (x,y) in the distance from x-axis measured along y-axis Cartesian plane is different from the position of (positive along the positive direction of y-axis and the point (y,x). Hence the order of the x and y is negative along the negative direction of y-axis). important. Therefore (x,y) is called an ordered pair. The y-coordinate is also called ordinate. Also (x,y) = (y,x), if x = y. • The coordinates of a point are denoted as (a,b). • It is possible to locate very large values or small The first number ‘a’ is x-coordinate and the second values on a Cartesian plane using scaling. Here a number ‘b’ is the y-coordinate. The coordinates large number N is represented as a smaller value n. describe a point uniquely in the plane. The labels are marked accordingly to represent the • The origin has zero distance from x-axis and y-axis values that are being represented. and hence the coordinates of origin are (0,0). • Example: • A point on the x axis and y axis does not belong to - 100 km distance can be marked as 1 unit. The any quadrant. • To plot the given point (x,y) in Cartesian coordinate distance between two divisions on the axis would system: be considered as 100 km. The first division will be o From the origin, count x units along x axis 100 km, the second division will be 200 km, etc. - 1 year time can be marked as a 1 unit. The and mark it as A. If x is positive then count the distance between two divisions on the axis distance in the positive direction of the x axis, would be considered as 1 year. The first division and if x is negative the distance should be will be 1 year, the second division will be 2 year, counted in the negative direction of x axis. etc. o From A, count y units along y axis and mark it - 1 second time can be marked as 1 unit. The as B. If y is positive count the distance along the distance between two divisions on the axis positive direction of y axis and if y is negative would be considered as 1 second. The first count the distance along the negative direction division will be 1 second, the second division will be 2 second, etc. Work Plan CONCEPT COVERAGE COVERAGE DETAILS PRACTICE SHEET PS – 1 Pre-requisites • Locating elements on number line, gird. PS – 2 PS – 3 Co-ordinate Geometry • Obtain the coordinates of a point in a PS - 4 Cartesian plane. Self Evaluation Sheet • Locate points in a Cartesian plane. Worksheet for \"Coordinate Geometry\" Evaluation with Self ---- Check or Peer Check* 25

PRACTICE SHEET - 1 (PS-1) 1. Locate the following numbers on a number line. i) 0  ii) 3   iii) -2 2. From the figure below, find the positions of the following: i) A ii) B iii) c iv) D v) E vi) F Write the row and column of the following as (x, y) where x represents the column number and y represents the row number. 26

PRACTICE SHEET - 2 (PS-2) 1. Determine the coordinates of the points marked from A – K, quadrant wise. 2. What will be coordinates of the point that lie: i) On x-axis ii) On y-axis iii) On origin 3. What is the abscissa and ordinate of P(3,5)? 4. Determine the coordinates of the marked points A – F and mention which of the points have positive abscissa and negative abscissa. 5. Determine the quadrants of the following points based on their coordinates: i) A(4,4) ii) B(4,-4) iii) C(0,4) iv) D(4,0) v) E(0,-4) vi) F(-4,0) vii) G(-4,4) viii) H(0,-4) 27

PRACTICE SHEET - 3 (PS-3) 1. Mark the following points on a Cartesian plane. A (1,1) B (3,3) C (0,3) D (-3,0) E (-2,2) F (-2,-2) G (0,-2) H (2.-3) 2. Mark the following points on the Cartesian plane using appropriate units of distance. A(100,100) B(-150,250) C(50,-200) D(-100,-150) and the values of x and y of all points are in meters. 3. Mark the points for the coordinates given in the table. 3 4 x12 5 3 y 0 1.5 4. A class teacher records the number of absentees for 5 days in a week. Day Monday Tuesday Wednesday Thursday Friday No. of 30 5 12 Absentees Explain the axes details and plot the information in a Cartesian plane. 28

PRACTICE SHEET - 4 (PS-4) III. Long answer questions. 1. (a) Define abscissa and ordinate. (b) What is the use of two perpendicular lines in coordinate geometry? What are those perpendicular lines named as? 2. From the given graph, find the area of ABCD. 30

Self-Evaluation Sheet Marks: 15 Time: 30 Mins 1. State whether the following statements are true 4. If North, South, East and West represent +ve y or false. axis, -ve y axis, +ve x axis and –ve x axis, then i) If a point has positive value of x coordinate, state which quadrant represents North West, the point lies in second quadrant. North East, South West and South East?(2 Marks) ii) The ordinate of a point which lies on x axis is always same. (2 Marks) 5. Mark the points on a Cartesian plane. (2 Marks) A(4,5), B(-3,4), C( -3,-3), D(0,-5). 2. Determine the coordinates of the points A- H. Mention the quadrants of all points. (4 Marks) 6. Give the coordinates of at least 4 points which are equidistant from x and y axis in third quadrant. (2 Marks) 7. Mark the following points on a Cartesian plane. A(-1,-1), B(-2,-2), C(-3,-3), D(-4,-4) (2 Marks) 3. Points P(a,b) and Q(c,d) are present in Quadrant I and represent the same point. What is the relation between the coordinates of P and Q? (1 Mark) 31

4. Linear Equations In Two Variables Learning Outcome • Draw the graph of the linear equations. By the end of this chapter, a student will be able to: • Derive a linear equation in two variables for a problem. • Determine the solutions of Linear equations in two variables by solving the equation. Concept Map Key Points • A linear equation in one variable is of the form ax • The solution of a linear equation is not affected + b = 0, where a, b are called constants and x is when: called the variable. o The same number is added to (or subtracted Example: 5x + 3 = 0, 2x – 4 = 2, −x + 5 = 2 , etc. from) both sides of the equation. 5 o Multiplying or dividing both sides of the equation by the same non-zero number. • The value of x which satisfies the equation is called Example: Root of 5x + 3 = 0 is −3 the root of the equation or solution of the linear 5 equation. Example: To find the roots of linear equations Adding 5 on both Subtracting 4 on Multiply 2 on both sides i) 5x+3=0 ii) 2x-4=2 -x sides both sides 2 (5x + 3) = 2 (0) iii) + 5= 2 5x + 3 + 5 = 0 5x + 3 – 4 = 0 – 4 10x + 6 = 0 ⇒ 5x = −3 ⇒ 2x = 2 + 4 + 5 5x – 1 = – 4 10x = –6 ⇒ x = −3 6 5 5x = – 3 x = −6 = −3 5x + 8 = 5 5 ⇒x = =3 ⇒ −x = 2 − 5 10 5 2 5 5x = −3 x = −3 x = −3 5 Root of 10x + 6 = ⇒ x = −5( 2 − 5) 0 is −3 -3 iiss rroooottoof f 3 is root of 5 −5( 2 − 5) is root 5 5 2x − 4 = 2 Root of 5x + 8=5 Root of 5x – 1 = of 5x+3=0 5x + 3 = 0 is −3 – 4 is −3 5 5 32

4. Linear Equations In Two Variables • A linear equation in one variable has one root and the root can be represented on a number line. • Any equation which can be put in the form: ax+by+c=0 , where a, b and c are real numbers and a, b are both non zero is called is a linear • Two points are needed to draw a straight line equation in two variables. x and y are the two and hence the graph of a linear equation in two variables in the equation. It is customary to variables can be drawn with two solutions (points). denote the variables by x and y but other letters Locate any two solutions of the given equation may also be used. on the Cartesian plane and connect them with a Example: 5x – 3y = 24, 2t – 3x = 25, 8p – 5q = –3 , straight line. The straight line will be the graph of the given linear equation. etc • A solution to a linear equation in two variables • Every point on the x axis is for the from (x,0) and every point on the y axis is of the form (0,y). means a pair of values, one for x and one for y which satisfy the given equation. The solution is • x = constant; represents an equation of a line written as an ordered pair. A linear equation in two parallel to y axis variables has infinitely many solutions. y = constant; represents an equation of a line A solution to the linear equation in two variables parallel to x axis. is obtained by taking a value for one variable and then solving the equation for the second variable. Example: For the equation x + y = 1, the possible solutions are as follows. Let x = 0 Let x = 4 Let x = 2 Let y = –1 ⇒0+ y =1 ⇒4+ y =1 ⇒ x-1=1 ⇒ y=1 ⇒ y = 1 − 4 = −3 ⇒ 2 + y = 1 ⇒ x = 1 + 1 = 2 (0,1) is ⇒ y =1− 2 solution (4,-3) is ( 2 ,1- 2 ) (2,-1) is solution is solution solution • The graphical representation of ax + by + c = 0 will be a straight line in Cartesian plane. A linear equation in two variables is represented geometrically by a line whose points make up the collection of solutions of the equation. This geometrical representation is called graph of the linear equation. Example. Graph of x – y = 0, Select a value for x and then find the value of y. Mark the points in a Cartesian plane and then join the points with a straight line. x2 -3 y2 -3 33

4. Linear Equations In Two Variables Work Plan CONCEPT COVERAGE COVERAGE DETAILS PRACTICE SHEET PS – 1 Pre-requisites • Roots of linear equations • Representing numbers on number line PS – 2 • Locating a point in a Cartesian plane PS – 3 Linear Equations in Two • S tandard form of linear equation in two PS - 4 Variables variables Self Evaluation Sheet • Express a given problem statement as a linear equation. • F inding solution to Linear equations by selecting value for one variable and solving the second variable. • Graph of linear equations in two variables • E quations of lines parallel to x axis and y axis. Worksheet for \"Linear Equations in Two Variables\" Evaluation with Self ---- Check or Peer Check* 34

PRACTICE SHEET - 1 (PS-1) 1. Find the roots of the following equations and represent them on the number line. i) 2x- 12 = 32 ii) 4x = 4 iii) 7x = 12 + 3x 2. Locate the roots of 7x + 1 = 3. 3. Locate the following points on a Cartesian plane i) (3.5, 4) ii) (-2.5, –2) iii) (3, –4) iv) (-2, 4.5) 35

PRACTICE SHEET - 2 (PS-2) 1. The following are the linear equations in two variables. Convert them to the form ax + by + c = 0 and find the constants (a,b,c) and variables (x,y). i) 3x – 5y = 24 ii) 12t – 24s = 2t – 5 iii) 8x – 5y = 12x + 3y – 7 2. Express the given equation in the form: ax + by + c = 0 and then determine the constants and variables. 2p−5 1 2p+ − 2 3y 3y− =3 + 43 3. Rita and Reena go to a book shop to buy a book. The cost of the book was Rs. 20 and the bought it using all the money they had. Write an equation to represent this and explain the terms. 4. A book costs Rs. 10 and a pen costs Rs. 3. Ravi was given Rs. 200 and was asked to buy books and pens. Write an equation to represent the purchase. 5. Find two solutions for the equation: 2x – 3y = 20. 6. Find two solutions to i) 5x + 2y – 20 = 0 ii) 4x + 3y – 180 = 0 7. Find two solutions such that x and y are non zero for the following equations: i) 3x – 5y + 16=0 ii) 2x + 3y = 2 6 8. A shop keeper was asked to pack some apples and oranges in a bag and was given Rs. 100. The cost of one apple was Rs. 20 and cost of one orange was Rs. 10. Determine 3 possible number of apples and oranges in the bag. 9. Check whether (0,0),(0,1), (2,-1) and (0,2) are solutions of x + y = 2. 10. Anil and Sunil are brothers who carry some money in their pockets and the total money with them will always be Rs. 20. Find the money carried by Sunil if Anil has i) Rs. 2 ii) Rs. 5.50 iii) Rs. 17 iii) Rs. 11 11. Ravi eats 2 idlis and 4 vadas in a hotel. The waiter gave Ravi a bill of Rs. 100. What could be the cost of idlis and vadas. 12. During a festival season, a shoe company gives offers on two types of shoes. One pair of Type A shoes are sold at Rs. 100 and pair of Type B shoes are sold at Rs. 200. A customer walks in to buy Rs. 1000 worth of shoes. Give four combinations of number of shoes of Type A, Type B the customer can buy with Rs. 1000. 36