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TABLE OF CONTENTS 1 REAL NUMBERS 1 1.1 INTRODUCTION 1 1.2 IRRATIONAL NUMBERS 4 1.3 REPRESENTING IRRATIONAL NUMBERS ON NUMBER LINE 6 1.4 OPERATIONS ON REAL NUMBERS 7 2 POLYNOMIALS AND FACTORISATION 10 2.1 POLYNOMIALS IN ONE VARIABLE 10 2.2 ZEROES OF A POLYNOMIAL 14 2.3 DIVIDING POLYNOMIALS 16 2.4 FACTORIZING A POLYNOMIAL 19 2.5 ALGEBRAIC IDENTITIES 21 3 THE ELEMENTS OF GEOMETRY 23 3.1 EUCLID’S ELEMENTS 23 4 LINES AND ANGLES 26 4.1 BASIC TERMS IN GEOMETRY 26 4.2 PAIRS OF ANGLES 30 4.3 LINES AND TRANSVERSALS 33 4.4 ANGLE SUM PROPERTY OF A TRIANGLE 38 5 CO –ORDINATE GEOMETRY 41 5.1 INTRODUCTION 41 5.2 CARTESIAN SYSTEM 44 5.3 PLOTTING A POINT ON THE CARTESIAN PLANE WHEN ITS CO–ORDINATES ARE GIVEN 49

7 TRIANGLES 52 7.1 CRITERIA FOR CONGRUENCE OF TRIANGLES 52 7.2 SOME PROPERTIES OF A TRIANGLE 56 7.3 SOME MORE CRITERIA FOR CONGRUENCY OF TRIANGLES 58 7.4 INEQUALITIES IN A TRIANGLE 62 8 QUADRILATERALS 64 8.1 PROPERTIES OF A QUADRILATERAL 64 8.2 PARALLELOGRAM AND ITS PROPERTIES 67 8.3 DIAGONALS OF A PARALLELOGRAM 70 8.4 THE MIDPOINT THEOREM OF A TRIANGLE 73 9 STATISTICS 75 9.1 COLLECTION OF DATA 75 9.2 MEASURES OF CENTRAL TENDENCY 79 PROJECT BASED QUESTIONS 82 ADDITIONAL AS BASED PRACTICE QUESTIONS 84

CHAPTER 1 REAL NUMBERS EXERCISE 1.1 INTRODUCTION 1.1.1 Key Concepts i. In the number system, the different types of numbers are natural numbers (N), whole numbers (W), integers (Z), and rational numbers (Q). ii. Collection of negative numbers and whole numbers is called integers, Z = {. . . –3, –2, –1, 0, 1, 2, 3, . . . }. iii. Rational numbers (Q) contain all numbers that are in the form p , where 'p' and 'q' are integers q and q 0. iv. Natural numbers, whole numbers, integers, and rational numbers can be written in the form p , q where 'p' and 'q' are integers and q 0. v. Rational numbers do not have a unique representation in the form p , where 'p' and 'q' are q integers and q 0, as every rational number can have many equivalent rational numbers. For example: 3 , 15 , 27 are all equivalent fractions. 5 25 45 p vi. When we represent a rational number q on a number line, we assume that q 0 and that 'p' and 'q' have no common factors other than the universal factor ‘1’ (i.e., 'p' and 'q' are co–primes). vii. Decimal representation of fractions can be terminating, non–terminating, non–recurring or non–terminating recurring. viii. If the denominator of a fraction can be expressed as a product of prime factors 2, 2 or 5, or 2 and 5 only, then it is a terminating decimal. 1.1.2 Additional Questions Objective Questions 1. [AS1] A rational number which is neither a natural number nor a whole number nor an integer is . (A) 0 (B) –2 (C) 7 (D) 3 4 EXERCISE 1.1. INTRODUCTION 1

2. [AS1] A rational number between the two rational numbers 7 and 2 is . 9 3 (A) 1 (B) 13 2 18 (C) 13 (D) 13 9 3 3. [AS1] The decimal value of 7 is . 8 (A) 0.875 (B) 0.0875 (D) 87.5 (C) 8.75 4. [AS1] The fraction 6 can be expressed as a decimal. 18 (B) Non–terminating repeating (A) Terminating (D)None of these (C)Non–terminating non –repeating 5. [AS1] The decimal form of the fraction 172 is . 625 (B) 2.752 (D) 0.02752 (A) 27.52 (C) 0.2752 Very Short Answer Type Questions . 6 [AS3] Fill in the blanks. (i) The set of natural numbers including zero is called the set of (ii) A set of rational numbers can be written in the form p , where 'p', 'q' ∈ Z, q 0 and 'p', 'q' q are . [AS3] Choose the correct answer. (iii) The set of integers substitute for . (A) N (B) W (C) Q (D)None of these [AS3] Answer the following questions in one sentence. (iv) Which one among 7, 9, 10, 0, 11, 13 and 27, is not a natural number? EXERCISE 1.1. INTRODUCTION 2

(v) Which one among –2, –7, –9, –11 and 8, is not a negative integer? Short Answer Type Questions 7(i) [AS1] Find two rational numbers between 4 and 5. (ii) [AS1] Find two rational numbers between 4 and 3 . 3 2 8(i) [AS1] Express 3.16666 . . . in the form of p , where 'p', 'q' are integers and q 0. q (ii) [AS1] Express 5.795 in the form of p , where 'p', 'q' are integers and q 0. q 9(i) [AS2] Without actually dividing, ﬁnd whether 7 is a terminating decimal or not. 16 Verify it by actual division. (ii) [AS2] Without actually dividing, ﬁnd whether 120 is a terminating decimal or not. 432 Verify it by actual division. 10(i) [AS3] Express 13 in decimal form. 8 (ii) [AS3] Express 21 in the decimal form. 36 11(i) [AS5] Represent the following rational numbers on a number line. a) 3 4 b) 17 4 (ii) [AS5] Represent the following rational numbers on a number line. −13 , −5 , 0, 1 , 9 , 23 8 8 4 8 8 EXERCISE 1.1. INTRODUCTION 3

EXERCISE 1.2 IRRATIONAL NUMBERS 1.2.1 Key Concepts i. There are inﬁnitely many rational numbers between two given numbers. ii. If a decimal is terminating or non–terminating and recurring, then it can be written in the form of p , which is a rational number. q p iii. If a decimal is non–terminating and non–recurring, it cannot be written in the form of q and √ these types of numbers are called irrational numbers. For example, 2 = 1.4142135. . . ; √ 3 = 1.7320508075689. . . √ iv. If ‘n’ is a natural number other than a perfect square then n is an irrational number. √ (For example, 3 is not a perfect square and hence 3 is an irrational number.) v. We can also represent irrational numbers on the number line. vi. All irrational and rational numbers put together are called real numbers, denoted by ‘R’. vii. If a and b are two positive numbers such that a × b is not a perfect square of a rational √ number then ab is an irrational number between a and b. 1.2.2 Additional Questions Objective Questions 1. [AS1√] A rational number among the following is (B) √3 343 . (A) 21 (C) √4 164 √ (D) 17 2. [AS1] An irrational number among the following is . (A) √3 125 (B) √4 16 (C) √5 1024 √ (D) 32 3. [AS1] An irrational number between 7and 8 is . √ (B) 7 (A) 56 8 (C) 8 (D)None of these 7 EXERCISE 1.2. IRRATIONAL NUMBERS 4

4. [AS1] An irrational number between 1 and 2 is . (A) 21 (B) 31 (C) 31/2 (D) 51/2 5. [AS1] A rational number among the following is . √ (B) √3 216 (A) 75 (D) √5 300 (C) √4 32 Short Answer Type Questions √ 6(i) [AS1] Find the value of 7 by division method up to 4 decimal points. √ (ii) [AS1] Find the value of 11 up to 4 decimal places. 7(i) [AS1] Find an irrational number between the two numbers 8 and 12. (ii) [AS1] Find an irrational number between the pairs of numbers given. a) 11, 13 b) 20, 32 8 [AS1] Identify the irrational numbers among the following: 9 [AS1] Identify the rational numbers among the following: √√√√ √ √ √ √ √ √ √ 1, 3, 4, 14, 15, 16, 18, 20, 30, 36, 48 Long Answer Type Questions √ 10 [AS1] Find the value of 6 correct to two places of decimal. √ 11 [AS1] Find the value of 5 correct to 5 places of decimals. EXERCISE 1.2. IRRATIONAL NUMBERS 5

EXERCISE 1.3 REPRESENTING IRRATIONAL NUMBERS ON NUMBER LINE 1.3.1 Key Concepts i. We can represent real numbers on the number line through the method of successive magniﬁcation. Long Answer Type Questions √ 1 [AS5] Construct the square root spiral up to 7. 2 [AS5] Visualize 3.26 on a number line up to 4 decimals. EXERCISE 1.3. REPRESENTING IRRATIONAL NUMBERS ON NUMBER LINE 6

EXERCISE 1.4 OPERATIONS ON REAL NUMBERS 1.4.1 Key Concepts i. Rational numbers satisfy the commutative, associative and distributive laws of addition and multiplication. ii. Rational numbers are closed with respect to addition, subtraction and multiplication. iii. Irrational numbers are not closed with respect to addition, subtraction, multiplication and division. iv. If 'q' is rational and 's' is irrational, then q + s, q – s, qs and q are irrational numbers. s v. If the product of two irrational numbers is a rational number then each of the two are the rationalizing factors (R.F.) of the other. Also notice that the R.F. of a given irrational number is not unique. It is convenient to use the simplest of all R.F.s of a given irrational number. vi. Let a > 0 be a real number and ‘n’ be a positive integer, if bn = a, for some positive real number b, then b is called nth root of ‘a’ and we write √n a = b. vii. If ‘n’ is a positive integer greater than 1 and ‘a’ is a positive rational number but not nth power of any rational number then √n a or a 1 is called a surd of nth order. In general, we say the n positive nth root of ‘a’ is called a surd or a radical. Properties of irrational numbers: √ √√ (a) ab = √a. b √a (b) a = ; if b 0 √b √ √ (c) √b (d) ( a + b)( a − b) = a − b (e) √ √ (a + b)(a − b) = a2 − b √ √ √√ √ √ √ bc √ ( a + b)( c + d) = ac + ad + + bd √ √b)2 √ (f) (a + = a + 2 ab + b 1.4.2 Additional Questions Objective Questions 1. [AS1] √√ √ . 7 2 + 8 + 11 2 = √ √ (A) 20 2 (B) 18 2 (D) 72 √ (C)18 10 EXERCISE 1.4. OPERATIONS ON REAL NUMBERS 7

√√ . 2. [AS1] 4 3 × 2 2 = √ (B) 8 23 √ (A) 8 32 √ (D)8 6 √ (C)6 6 √ . 3. [AS1] The rationalising factor of 6 − 4 3 is √ (A) √6 (B) 6 + 4 3 √ 43 (D)6.4 3 √ (C)4 3 − 6 . 4. [AS1] √7 43 in exponential form is 7 (A) 4(3)(7) (B) 4 3 (C) 4 3 (D) 410 7 . 4 (B) 4 213 5. [AS1] The radical form of (21)3 is √ (D)4 21 × 3 (A) 3 214 (C) 4 √ 3 21 Very Short Answer Type Questions 6 [AS3] Fill in the blanks. (i) The surd 7 √3 3 in exponential form is . (ii) The exponential form of 2√1 21 = . 1 (iii) The radical form of (250) 3 is . (iv) The radical form of 1 is . (347) 7 (v) The exponential form of 1√2 3 is . Short Answer Type Questions 7(i) [AS1] Simplify: √ √ √ √ 35 +7 5 +6 5 + 125 EXERCISE 1.4. OPERATIONS ON REAL NUMBERS 8

√√√√√√ (ii) [AS1] Simplify: 7 3 + 2 7 + 3 3 + 8 7 + 2 3 + 9 3 √√ 8(i) [AS1] Simplify: 13 × 11 √ √ √√ √ √ (ii) [AS1] Simplify: 5 + 7 − 3 5 + 7 + 3 √ √√ √ 9(i) [AS1] Simplify: ( 2 + 3 7)( 2 − 3 7) (ii) [AS1] Simplify: √ − √5)2 (7 10(i) [AS1] Rationalize the denominator of √2 . 33 (ii) [AS1] Rationalize the denominator of 1√ . 2+3 5 12 11(i) [AS1] Simplify: 33 × 3 5 (ii) [AS1] Simplify: 2 ÷ 3 53 57 2 5 12(i) [AS1] Simplify: 43 7 (ii) [AS1] Simplify: 2 × 1 33 43 √ 13(i) [AS2] Check whether 16 − 2 is rational or not. √ (ii) [AS2] Check whether 8 3 is irrational or not. Long Answer Type Questions √ 14 [AS1] Rationalize the denominator of 5+2 √3 . 7+4 3 15 [AS1] Simplify: √ 4 √ + √ 3 √ 3 3 −2 2 3 3 +2 2 16 [AS1] If √ +5 then ﬁnd the value of x + 1 . x=2 6 x 17 [AS1] Find the values of ‘a’ and ’ b’ where √ = a + √ . 3+2 √3 b3 3−2 3 18 [AS1] Find the value of √√ √ =1.414. correct to two places of decimals, given 2 3+ √2 + 3− √2 3− 2 3+ 2 EXERCISE 1.4. OPERATIONS ON REAL NUMBERS 9

CHAPTER 2 POLYNOMIALS AND FACTORISATION EXERCISE 2.1 POLYNOMIALS IN ONE VARIABLE 2.1.1 Key Concepts i. A polynomial p(x) in one variable 'x' of degree 'n' is an expression of the form p(x) = anxn + an−1xn−1 + ....... + a2x2 + a1x + a0 where, a0, a1, a2, a3, .......an are constants, n is called the degree of the polynomial if an 0 and anxn, an−1xn−1, .......a2x2 and a1x are called the terms of the polynomial p(x). ii. Polynomials are classiﬁed as monomials, binomials, trinomials etc. according to the number of terms in them. iii. Polynomials are also named as linear polynomials, quadratic polynomials, cubic polynomials and so on according to their degree. 2.1.2 Additional Questions Objective Questions 1. [AS1] One of the following which is not a polynomial is . (A) 3x 2 + 7x − 5 (B) 3x − 2 + 7 x (C)7x3 − 4x2 + 8x − 5 (D)8x5 − 7 2. [AS1] The degree of the polynomial 6x 5 − 4x4 + 3x7 + 2x − 8 is . (A) 5 (B) 4 (C) 7 (D) 1 EXERCISE 2.1. POLYNOMIALS IN ONE VARIABLE 10

3. [AS1] The coefﬁcient of x4 in 7x 5 − 3x3 + 6x4− 5x + 21 is . (A) –5 (B) 7 (C) –3 (D) 6 4. [AS1] The degree of a cubic polynomial is . (A) 0 (B) 1 (C) 2 (D) 3 5. [AS1] A polynomial containing 3 terms is called a . (A) Trinomial (B) Binomial (C) Monomial (D)None of these Very Short Answer Type Questions 6 [AS1] State true or false. (i) y2 – 81 is a polynomial of degree 2 in the variable y. (ii) 8x3 − 2 + 7x − 9 is a polynomial. [] [] 4x3 [AS1] Choose the correct answer. (iii) Which of the following is a polynomial? (A) 7x3 − 9 + 4 x2 (B) 7x3 − 9x2 + 7 + 3 x (C) x3 − 9x2 + 7x + 4 7 (D)None of these EXERCISE 2.1. POLYNOMIALS IN ONE VARIABLE 11

(iv) Which of the following is not a polynomial? (A) x3 + 4 (B) 7x2 + 9 3 2 (C) a x2 + bx + √ (D) 9 √ +7 c x 7 [AS1] Fill in the blanks. . (i) The degree of the polynomial 5x4 + 7x3 y2 + 9x2 y3 is √ √ (ii) 7 is the coefﬁcient of in the polynomial 9x4 + 7 x3 + 9x2 + 7x + 5 . 2 2 (iii) The degree of the term with coefﬁcient 7 in the polynomial 13y3 + 9y2 + 5y + 7 is . [AS1] Choose the correct answer. (iv) Which of the following is a second degree polynomial in one variable? (A) 9x2 − 19xy + 17y2 (B) 9x2 − 19x + 17 (C)9x2 − 3x + 4 + 7y + 5y2 (D)None of these [AS1] Answer the following question in one sentence. (v) What is the degree of the polynomial 3x2 + 7xy + 4y2 + 5yz + 9z2 + 6xyz ? 8 (i) [AS3] State true or false. z 2 + z + 7 is a polynomial in one variable, z. z [ ] (ii) [AS3] Answer the following questions in one sentence. Give an example of the standard form of a polynomial of degree 2 in two variables. 9 [AS3] Choose the correct answer. (i) Which of the following is a polynomial in one variable z? (A) 3x2 + 4xy + 9y2 + 10z (B) 2z2 + 9z + 11 (C)7z2 + yz + 11 (D)9y2 + 7yz + 11z2 (ii) Which of the following is not a polynomial in one variable? (A) 7x2 + 4x + 9a (B) 3a2 + 6a + 7 (C)19p4 + 21p3 + 7p2 (D)q2 + 11q + 28 EXERCISE 2.1. POLYNOMIALS IN ONE VARIABLE 12

(iii) Which of the following is a polynomial in one variable? (A) 9a2 − 7ab + 9b2 (B) 7p3 + 9p2q − 11pq2 + 13q3 (C)11y4 − 11y3a + 9y2 + 7y + 5 (D)None of these [AS3] Answer the following questions in one sentence. (iv) Is the polynomial 4x 2 + 9xz + 5z2 a polynomial in one variable? Short Answer Type Questions 10(i) [AS1] Identify and write the quadratic polynomials from the following: 8x3 − 7x2 + 4x + 3 ; 7y2 + 6xy + 4y2 + 3 ; a3 + 3a2b + 3ab2 + b3 ; 8p2 + 6pq + 7q2 ; 9a3 + 8a2b + 5ab2 + 7b3 (ii) [AS1] Write three cubic polynomials in 2 and 3 variables each. EXERCISE 2.1. POLYNOMIALS IN ONE VARIABLE 13

EXERCISE 2.2 ZEROES OF A POLYNOMIAL 2.2.1 Key Concepts i. A real number ‘a’ is a zero of a polynomial p(x) if p (a) = 0. In this case, ‘a’ is also called a root of the polynomial equation p(x) = 0. ii. Every linear polynomial in one variable has a unique zero. iii. A non–zero constant polynomial has no zero. 2.2.2 Additional Questions Objective Questions 1. [AS1] The zero of the polynomial 3x – 2 is . (A) 2 (B) 2 3 (C) 3 (D) 3 2 2. The zero of ax + 3 is 1, then a = . (A) –1 (B) 3 (C) 1 (D) –3 3. [AS1] The zeroes of the polynomial x2 – 5x + 6 are . (A) 2, 3 (B) 5, 6 (C) –2, 3 (D) 5, –6 4. [AS1] The number of zeroes of a cubic polynomial is . (A) 1 (B) 3 (C) 2 (D) 0 EXERCISE 2.2. ZEROES OF A POLYNOMIAL 14

5. [AS1] If p(x) = x2+ 4x + 3 then p(3) = . (A) 24 (B) –24 (C) 15 (D) –15 Short Answer Type Questions 6(i) [AS1] Find the value of the polynomial 7x3 – 2x2 + 8, when x = 3 . 2 (ii) [AS1] Find p(–2) and p(3) for the polynomial, p(y) = 6y3 − 5y2 + 8y + 12 . 7(i) [AS2] Verify whether the value of x given is zero of the polynomial or not: p(x) = 5x2 − 7x + 2 ; x = 1. (ii) [AS2] Verify whether the values of 'x' given are the zeroes of the given polynomial or not: p (x) = 2x4 − 9x3 + 14x2 − 9x + 2 ; x = 1, 2. Long Answer Type Questions 8 [AS1] Find the zeroes of the polynomial x3 + 4x2 + x − 6 . 9 [AS1] Find the value of ‘l’ and ‘m’ of the polynomial x4− x3 + lx 2 + mx + 4 so that its two zeroes are the same as the zeros of x 2 − x − 2 . EXERCISE 2.2. ZEROES OF A POLYNOMIAL 15

EXERCISE 2.3 DIVIDING POLYNOMIALS 2.3.1 Key Concepts i. Remainder Theorem: If p(x) is any polynomial of degree greater than or equal to 1 and p(x) is divided by the linear polynomial, x – a = 0, then the remainder is p(a). 2.3.2 Additional Questions Objective Questions 1. [AS1] The quotient of (6x + 8) ÷ (3x + 4) is . (A) 2 (B) 9x + 12 (C)3x +12 (D)3x + 8 2. [AS1] The remainder when p(x) is divided by (x –a) is . (A) p(x –a) (B) x(a) (C) p(a) (D) None of these 3. [AS1] The remainder when 3x + 4 is divided by x – 2 is . (A) 6 (B) 10 (C) 8 (D) 14 4. [AS1] A factor of x2 – 5x + 6 is . (A) x + 3 (B) x + 2 (C)x + 6 (D)x – 2 EXERCISE 2.3. DIVIDING POLYNOMIALS 16

5. [AS1] A multiple of the expression (x – 4) is . (A) x2 − 6x + 8 (B) x2 − 2x − 4 (C) x2 − 5x + 7 (D) x2 − 3x + 6 Very Short Answer Type Questions 6 [AS1] Fill in the blanks. . (i) 3x2 + 4x ÷ x = [AS1] Answer the following questions in one sentence. (ii) What is the remainder when x3 − 2x2 + 4x + 7 is divided by x – 2? (iii) 2x2 + 5x ÷ (2x + 5) = (iv) x2 + 5x + 6 ÷ =x+3 (v) ÷ (x − 2) = (x − 1) Short Answer Type Questions 7(i) [AS1] Find the remainder when x3 − 3x2 + 4x − 5 is divided by: a) x + 2 b) x – 3 (ii) [AS1] Find the remainder when 3x3 − 2x2 + x + 2 is divided by: a) x – 1 b) 2x – 1 8(i) [AS2] Verify whether or not x – 2 and x + 3 are the factors of the polynomial: x 4 + 5x3 + 5x2 − 5x − 6. (ii) [AS2] Verify if x3 − 3x2 − 10x + 24 is a multiple of (x + 2) and (x + 3). Long Answer Type Questions 9 [AS1] Find the remainder when x4 + 4x3 + 5x2 − 6x + 7 is divided by: a) x – 3 b) x – 1 c) x + 1 d) x – 2 e) x + 2 EXERCISE 2.3. DIVIDING POLYNOMIALS 17

10 [AS1] Find the values of b and c, if the division of x2 + bx + c by: (i) x – 1 leaves a remainder 0. (ii) x + 2 leaves a remainder 12. 11 [AS2] Find the remainder when 4x3 − 3x + 9 is divided by 2x – 3 and verify the result by actual division. EXERCISE 2.3. DIVIDING POLYNOMIALS 18

EXERCISE 2.4 FACTORIZING A POLYNOMIAL 2.4.1 Key Concepts i. Factor Theorem: If (x – a) is a factor of the polynomial p(x), then p(a) = 0. Also, if p(a) = 0 then (x – a) is a factor of p(x). 2.4.2 Additional Questions Objective Questions 1. [AS1] If a polynomial p(x) is such that p(a) = 0, then a factor of p(x) is . (A) x + a (B) x – a (C) ax (D) x a 2. [AS1] One of the factors of 2x 2+ 7x − 5 is . (A) 2x + 1 (B) 2x –1 (C)x – 4 (D)None of these 3. [AS1] The factors of a polynomial are (x + 5) and (x – 4), then the co–efﬁcient of x in the polynomial is . (A) 1 (B) 0 (C) 9 (D) –9 4. [AS1] One of the factors of 2x2 + 7x + 5 is . (A) 2x + 7 (B) 2x – 7 (C)x + 1 (D)x – 5 EXERCISE 2.4. FACTORIZING A POLYNOMIAL 19

5. [AS1] If ax + b is a factor of p(x), then . (A) p(b) = 0 (B) p(a) = 0 (C) p( b ) = 0 (D) p( −b ) = 0 a a Short Answer Type Questions 6(i) [AS1] Factorize x2 + 7x + 12. (ii) [AS1] Factorize 5x2 + 31x − 28. 7(i) [AS2] Verify whether or not x – 3 is a factor of 2x 3 + 3x2 + 4x − 6. (ii) [AS2] Verify whether or not 2x – 1 is a factor of 6x4 + 29x3 + 30x2 − 11x − 6. Long Answer Type Questions 8 [AS1] Find the values of l and m so that lx4 + mx3 + 2x2 + 4 has (x − 2) and (x + 1) as factors. 9 [AS1] Factorize: x4 + 4x3 + 3x2 − 4x − 4 10 [AS2] If x + a is a factor of ax2 + 2a2 x + b3, then prove that a = b or a2 + ab + b2 = 0 . 11 [AS2] If a and b are unequal and x2 + ax + b and x2 + bx + a have a common factor, then show that a + b + 1 = 0. EXERCISE 2.4. FACTORIZING A POLYNOMIAL 20

EXERCISE 2.5 ALGEBRAIC IDENTITIES 2.5.1 Key Concepts Some algebraic identities are: i. (x + y)2 = x2 + 2xy + y2 ii. (x − y)2 = x2 − 2xy + y2 iii. x2 − y2 = (x + y)(x − y) iv. (x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx v. (x + y)3 = x3 + y3 + 3xy(x + y) vi. (x − y)3 = x3 − y3 − 3xy(x − y) vii. x3 + y3 + z3 − 3xyz = (x + y + z)(x2 + y2 + z2 − xy − yz − zx) viii. x3 + y3 = (x + y)(x2 − xy + y2) ix. x3 − y3 = (x − y)(x2 + xy + y2) 2.5.2 Additional Questions Objective Questions 1. [AS1] 1022 − 982 . (A) 10201 (C)199 × 2 (B) 597 (D)None of these 2. [AS1] (3x + 4)(3x – 4) = . (A) 9x 2 – 16 (B) 6x – 16 (C)9x – 8 (D)9x2 – 8 3. [AS1] (30 + 8) 2 = (30)2 + + (8)2 (A) 240 (C) 480 (B) 120 (D) 960 EXERCISE 2.5. ALGEBRAIC IDENTITIES 21

4. [AS1] 100 2 − 2(100)(2) + (2)2 = . (A) (102)2 (B) (102) (104) (C) 96 2 (D)98 2 5. [AS1] 1532 = (100 + + 3)2 (B) 2(50)(3) (D) 2(100)(3) (A) 50 (C) 2(100)(50) Short Answer Type Questions 6(i) [AS1] Find the products of the following: a) (3x + 4y) 9x2 − 12xy + 16y2 b) (2x + 5) (2x − 5) (ii) [AS1] Expand the following using an appropriate formula: a) (2a + b)2 b) (a + b − c)2 7(i) [AS1] Factorize the following using a suitable identity: 9a2b2 − 6abc + c2 (ii) [AS1] Factorize: x6 − 2x3 + 1 Long Answer Type Questions 8 [AS1] Evaluate: (i) 100 2− 992 (ii) (98)3 9 [AS1] If x + 1 = 3 , then ﬁnd the value of x3 + 1 and x2 + 1 . x x3 x2 10 [AS2] If a + b + c = 0, then show that a3 + b3 + c3 = 3abc. 11 [AS2] Write the geometrical proof for the identity (x + y)2 = x2 + 2xy + y2. 12 [AS5] Draw the geometrical ﬁgure for the identity (x + a)(x + b) = x2 + (a + b)x + ab. EXERCISE 2.5. ALGEBRAIC IDENTITIES 22

CHAPTER 3 THE ELEMENTS OF GEOMETRY EXERCISE 3.1 EUCLID’S ELEMENTS 3.1.1 Key Concepts i. Geometry (meaning geo – “Earth”, – metron “measurement”) is a branch of mathematics concerned with questions of shape, size, position of ﬁgures and properties of space. ii. The three building blocks of geometry are points, lines and planes, which are undeﬁned terms. iii. Ancient mathematicians including Euclid tried to deﬁne these undeﬁned terms. iv. Euclid developed a system of thought in his book “The Elements”, that serves as the foundation for the development of all subsequent mathematics. v. Some of Euclid’s axioms: a. Things which are equal to the same things are equal to one another. b. If equals are added to equals, the wholes are equal. c. If equals are subtracted from equals, the remainders are equal. d. Things which coincide with one another are equal to one another. e. The whole is greater than the part. f. Things which are double of the same things are equal to one another. g. Things which are halves of the same things are equal to one another. vi. Euclid’s postulates: a. Postulate –1: A straight line segment can be drawn by joining any two points in the same plane. b. Postulate –2: A straight line segment can be extended indeﬁnitely to produce a straight line. c. Postulate –3: Given any straight line segment, a circle can be drawn having the segment as radius and one end point as the centre. d. Postulate –4: All right angles are equal to one another. e. Postulate –5: If a straight line falling on two straight lines makes the interior angles on the same side of it and taken together, their sum is less than two right angles, then the two straight lines, if produced inﬁnitely, meet on that side on which the sum of the angles is less than two right angles. EXERCISE 3.1. EUCLID’S ELEMENTS 23

3.1.2 Additional Questions Objective Questions 1. [AS3] If a pair of alternate angles formed are equal to each other then the lines are . (A) Parallel (B) Perpendicular (C) Intersecting (D)None of these 2. [AS3] Two distinct lines can have point(s) in common. (A) Two (B) Only one (C) Three (D)None of these 3. [AS3] The book called “The Elements” was written by . (A) Boudhayana (B) Euler (C) Euclid (D)None of these 4. [AS3] The boundaries of solid shapes are . (A) Points (C) Curves (B) Lines (D) Surfaces 5. [AS3] Straight lines parallel to the same straight line are each other. (A) Parallel to (B) Perpendicular to (C) Intersecting (D)None of these Short Answer Type Questions 6(i) [AS2] In the given ﬁgure, AC = XD, C and D are the mid–points of AB and XY respectively. Show that AB = XY. EXERCISE 3.1. EUCLID’S ELEMENTS 24

(ii) [AS2] Prove that an equilateral triangle can be constructed on any given line segment. 7(i) [AS2] Prove using Euclid’s postulate: Two distinct lines cannot have more than one point in common. (ii) [AS2] Prove using Euclid’s postulate: “If the alternate angles are equal when two straight lines are cut by a transversal, then the two lines are parallel.” Long Answer Type Questions 8 [AS2] Three points lie on a line and B lies between A and C, as shown below, then prove that AC– AB = BC. 9 [AS5] Draw an equilateral triangle with side 4.5 cm. 10 [AS5] Draw an equilateral triangle whose side is 3 cm. 11 [AS5] In the given ﬁgure, QM = 1 PQ, QN = 1 QR and PQ = QR. Show that QM = QN. 2 2 EXERCISE 3.1. EUCLID’S ELEMENTS 25

CHAPTER 4 LINES AND ANGLES EXERCISE 4.1 BASIC TERMS IN GEOMETRY 4.1.1 Key Concepts i. Ray – A ray is a part of a line. It begins at a point and goes on endlessly in a speciﬁed direction. ii. Line segment – A part of a line, with two end–points is known as line segment. iii. If three or more points lie on the same line, they are called collinear points, otherwise they are called non–collinear points. iv. An angle is formed when two rays originate from the same point. v. When three or more lines meet at a point, they are called concurrent lines and the point at which they meet is called the point of concurrence. 4.1.2 Additional Questions Objective Questions 1. [AS1] A part of a line with one end–point is called . (A) A line segment (B) A ray (C)A line (D)None of these 2. [AS1] The angle between the two hands at 3 o' clock is . ◦ ◦ (A) 90 (B) 60 (C) 30◦ (D) 0◦ EXERCISE 4.1. BASIC TERMS IN GEOMETRY 26

3. [AS1] If the angle between the hands of a clock is 30◦, then the possible time in the clock is . (A)2 o’ clock (D)3 o' clock (C)1 o’ clock (D)None of these 4. [AS1] A part of a line with two end–points is called . (A) A line (B) A ray (C)A line segment (D)None of these 5. [AS1] The union of two rays with one common point is . (A) An angle (B) A line segment (C)A ray (D)None of these Very Short Answer Type Questions 6 [AS4] Fill in the blanks. (i) A ray has end–point. [AS4] Answer the following questions in one sentence. (ii) What is the angle at B on the line AC known as? (iii) The points A, B, C and D in the given ﬁgure are called . EXERCISE 4.1. BASIC TERMS IN GEOMETRY 27

(iv) The angle represented between the hands of the clock given is a/ an angle. (v) Draw the hands in the clock given such that the angle between them is 90°. Short Answer Type Questions 7(i) [AS1] What is the angle between the two hands of a clock when the time is 2 o’ clock? Also, draw the hands of the clock. (ii) [AS1] What are the angles between the two hands of a clock when the time is: a) 3 o’ clock b) 5 o’ clock c) 7 o’ clock EXERCISE 4.1. BASIC TERMS IN GEOMETRY 28

8 [AS2] In the given ﬁgure, name (i) Any four line segments (ii) Any four collinear points 9 [AS4] Draw the hands on each of the clocks representing the angles as given under each clock. EXERCISE 4.1. BASIC TERMS IN GEOMETRY 29

EXERCISE 4.2 PAIRS OF ANGLES 4.2.1 Key Concepts i. Two angles are complementary if the sum of their angles equals 90o . o ii. Two angles are supplementary if the sum of their angles equals 180 . iii. Pairs of angles which have a common vertex, a common arm and non common arms that lie on either side of the common arm are called adjacent angles. iv. If the sum of two adjacent angles is 180 ◦, they are said to be a linear pair. v. Linear pair axiom: If a ray stands on a straight line, then the sum of the two adjacent angles so formed is 180°. vi. Converse of linear pair axiom: If the sum of two adjacent angles is 180°, then the non–common arms of the angles form a line. vii. Theorem: If two lines intersect each other, then the vertically opposite angles are equal. 4.2.2 Additional Questions angles is equal. (B) Opposite Objective Questions (D)None of these 1. [AS3] In a parallelogram, a pair of (A) Adjacent (C) Exterior 2. [AS3] The sum of the angles in a linear pair is . (A) 0◦ (B) 90◦ (C) 180◦ (D) 360◦ 3. [AS1] If one of the angles of a pair of supplementary angles is 87◦, then the other angle is . (A) 93 ◦ ◦ (C) 273◦ (B) 103 (D)None of these EXERCISE 4.2. PAIRS OF ANGLES 30

4. [AS1] If one of the angles of a pair of complementary angles is 62◦, then the other angle is . (A) 0 ◦ (B) 20 ◦ (C) 30◦ (D) 28 ◦ 5. [AS1] In a linear pair, if one angle is100 ,◦ then the other angle is . (A) 80◦ (B) ◦ (C) 50◦ 90 (D) 100◦ Short Answer Type Questions 6(i) [AS1] Find the value of ‘x’ in the following ﬁgure: (ii) [AS1] In the adjacent ﬁgure, AB is a straight line. Find the value of x and also ﬁnd ∠AOC, ∠CODand ∠BOD . 7(i) [AS2] In the given parallelogram ABCD, name the pairs of angles which form a linear pair and the pairs of supplementary angles. EXERCISE 4.2. PAIRS OF ANGLES 31

(ii) [AS2] In the following figure, write the angles which form linear pairs and the angles which are supplementary. Long Answer Type Questions 8 [AS2] In the given ﬁgure, ray OS stands on a line PQ. Ray OR and ray OT are the bisectors of ∠POS and ∠S OQ respectively. Prove that ∠ROT is a right angle. 9 [AS5] Draw a quadrilateral ABCD such that AB DC, AD BC and ∠A = 50°. Find the other angles. 10 [AS5] Given that ∠XYZ = 64° and XY is produced to a point P. A ray YQ bisects ∠ZY P. Draw a ﬁgure from the given information. Find ∠XYQ and reﬂex ∠QYP . EXERCISE 4.2. PAIRS OF ANGLES 32

EXERCISE 4.3 LINES AND TRANSVERSALS 4.3.1 Key Concepts i. Axiom of corresponding angles: If a transversal intersects two parallel lines, then each pair of corresponding angles are equal. ii. Converse of axiom of corresponding angles: If a transversal intersects two lines such that a pair of corresponding angles are equal, then the two lines are parallel to each other. iii. Theorem: If a transversal intersects two parallel lines, then each pair of alternate interior angles are equal. iv. Converse: If a transversal intersects two lines such that a pair of alternate interior angles are equal, then the two lines are parallel. v. Theorem: If a transversal intersects two parallel lines, then each pair of interior angles on the same side of the transversal are supplementary. vi. Converse: If a transversal intersects two lines such that a pair of interior angles on the same side of the transversal are supplementary, then the two lines are parallel. 4.3.2 Additional Questions Objective Questions 1. [AS1] If one of the interior angles formed by a pair of parallel lines intersected by a transversal is 135◦, then the other angle on the same side is . (A) 135◦ (B) 225◦ (C) 45◦ (D)None of these 2. [AS3] If two lines are intersected by another line, then the third line is called . (A) A transversal (B) An intersecting line (C) A parallel line (D) None of these EXERCISE 4.3. LINES AND TRANSVERSALS 33

3. [AS3] Two lines parallel to a given line are to each other. (A) Perpendicular (B) Intersecting (C) Parallel (D)Parallel or perpendicular 4. [AS3] If a pair of alternate interior angles formed when a line intersects a pair of lines are equal then the lines are . (A) Parallel (B) Perpendicular (C) Intersecting (D)None of these 5. If a pair of parallel lines is intersected by a transversal then the pair of corresponding angles formed are . (A) Not equal (B) Equal (C) Complementary (D)None of these Short Answer Type Questions 6(i) [AS1] In the given ﬁgure ’l’ and ’m’ are intersected by a transversal ’n’. Is l m? EXERCISE 4.3. LINES AND TRANSVERSALS 34

(ii) [AS1] In the adjacent ﬁgure, the lines ’l’ and ’m’ are parallel and ’n’ is a transversal. Fill in the following blanks. . . a. If ∠1 = 80° then ∠2 = . b. If ∠3 = 45° then ∠7 = c. If ∠2 = 90° then ∠8 = . d. If ∠4 = 100° then ∠8 = . e. If ∠6 = 70° then ∠2 = . f. If ∠5 = 60° then ∠2 = 7(i) [AS1] Find the values of x and y given that l m and n p. (ii) [AS1] In the adjacent ﬁgure, PQ RS UV and a : b = 3 : 7. Find a. EXERCISE 4.3. LINES AND TRANSVERSALS 35

8(i) [AS4] Name any two pairs of angles which are congruent in the following ﬁgure. (ii) [AS4] From the following ﬁgure, name any three pairs of angles that are congruent to each other. Long Answer Type Questions 9 [AS1] In the given ﬁgure, ‘l’ is parallel to m and PQ is transversal intersecting l and m. If ∠1 : ∠4 = 2 : 3, find the measures of all the angles. EXERCISE 4.3. LINES AND TRANSVERSALS 36

10 [AS2] In the adjoining ﬁgure, −→ −−→ and −−→ −P−→D. Show that ∠x + ∠y = ◦ . OA PC OB 180 11 [AS5] Draw ﬁgures for the statement: “If the two arms of one angle are respectively perpendicular to the two arms of another angle then the two angles are either equal or supplementary.” 12 [AS5] AB CD and ‘l’ is a transversal intersecting AB and CD at E and F respectively. If the adjacent angles formed are in the ratio 5 : 4, ﬁnd the measures of all the angles. EXERCISE 4.3. LINES AND TRANSVERSALS 37

EXERCISE 4.4 ANGLE SUM PROPERTY OF A TRIANGLE 4.4.1 Key Concepts i. Theorem: The sum of the angles of a triangle is 180◦ . ii. Theorem: If a side of a triangle is produced, then the exterior angle so formed is equal to the sum of the two interior opposite angles. 4.4.2 Additional Questions Objective Questions 1. [AS1] If two angles of a triangle are 50◦and 70◦, then the third angle is . (A) 120◦ (B) 150◦ (C) 60◦ (D) 300◦ 2. [AS1] The exterior angle of a triangle whose opposite interior angles are 30◦ and 40◦ is . (A) 70◦ (B) 110◦ (C) 250◦ (D)None of these 3. [AS1] The exterior angle of a triangle is 110◦. If one of the opposite interior angles is 48◦, then the other angle is . (A) 158◦ (B) 110◦ (C) 48◦ (D) 62◦ 4. [AS3] The sum of the angles of a triangle = . (A) 120◦ (B) 60◦ (C)Right angle (D)2 right angles EXERCISE 4.4. ANGLE SUM PROPERTY OF A TRIANGLE 38

5. [AS3] The sum of the exterior angles of a triangle is . (A) 360◦ (B) 180◦ (C) 90◦ (D)None of these Short Answer Type Questions 6(i) [AS1] In ∆ABC, ∠A = 30◦ and ∠B = 45 ◦, f ind ∠C. (ii) [AS1] In ∆PQR, if ∠P = 3∠Q and ∠R = 2∠Q , ﬁnd all the three angles of ∆PQR. 7(i) [AS1] In the ﬁgure, ﬁnd the values of x and y. (ii) [AS1] In the following ﬁgure, ﬁnd the values of x, y and z. EXERCISE 4.4. ANGLE SUM PROPERTY OF A TRIANGLE 39

Long Answer Type Questions 8 [AS1] In the ﬁgure, D and E are the points on the sides AB and AC of ABC such that DE BC. If ∠B = 30◦ and ∠A = 40◦, ﬁnd the values of x, y and z. 9 [AS1] One of the exterior angles of a triangle is 105° and the interior opposite angles are in the ratio 2 : 5. Find the angles of the triangle. 10 [AS2] In the adjoining ﬁgure, the bisectors of ∠B and ∠C o f ∆ABC intersect at D. Show that ∠D = 90◦ + z . 2 EXERCISE 4.4. ANGLE SUM PROPERTY OF A TRIANGLE 40

CHAPTER 5 CO–ORDINATE GEOMETRY EXERCISE 5.1 INTRODUCTION 5.1.1 Key Concepts i. To locate the exact position of a point in a plane we need two references. ii. A point or an object in a plane is located with the help of two perpendicular number lines. One of them is horizontal (X–axis) and the other is vertical (Y–axis). iii. We locate the exact position of a point in a plane using the ordered pair (x, y). 5.1.2 Additional Questions Objective Questions 1. [AS3] The co–ordinate system was ﬁrst introduced by the mathematician . (A) George Cantor (B) Rene Descartes (C)Both (A) and (B) (D)Neither (A) nor (B) 2. [AS2] In the alphabet chart, the 7 thletter from the last is . (A) V (B) U (C) T (D) S 3. [AS2] In the alphabet chart, the alphabets which are equidistant from both the ends are . (A) M, N (B) K, L (C)O, P (D)N, O EXERCISE 5.1. INTRODUCTION 41

4. [AS2] In the set of integers, the 7 th integer to the left of zero is . (A) 14 (B) –14 (C) 7 (D) –7 5. [AS2] In the set of whole numbers, the 20th whole number from zero is . (A) –20 (B) 20 (C) 200 (D)None of these Very Short Answer Type Questions 6 [AS1] Fill in the blanks. (i) The study of co–ordinate geometry was developed by the French mathematician . [AS5] Answer the following questions in one sentence. (ii) Observe the chart of alphabet given and ﬁll in the blanks according to the position of the alphabet. a. The alphabet in 2nd row 3rd column is . 42 b. The position of the alphabet M is . c. O is the alphabet in row and column. d. The alphabet in 1stcolumn third row is . EXERCISE 5.1. INTRODUCTION

Long Answer Type Questions 7 [AS5] Observe the alphabet chart and ﬁll in the blanks in the table given, showing the positions of the letters in the chart. Alphabet Chart X A PMC F J TNBLV D OR HWK S UQEG I Complete the table. Row Column Position 3 3 (3, 3) Alphabet R L A I EXERCISE 5.1. INTRODUCTION 43

EXERCISE 5.2 CARTESIAN SYSTEM 5.2.1 Key Concepts i. The system in which points in a plane are represented in the form of coordinates is called Cartesian coordinate system. ii. The point of intersection of the axes is called the origin. iii. The ordered pair (x, y) is different from the ordered pair (y, x). iv. X–axis is denoted by the equation y = 0. v. Y–axis is denoted by the equation x = 0. 5.2.2 Additional Questions . Objective Questions (B) –5 1. [AS3] The abscissa of (3, –5 ) is (A) 3 (C) –8 (D) –2 2. [AS3] The point (–6, 7) lies in the quadrant. (A) Q1 (B) Q2 (D) Q4 (C) Q3 3. [AS3] The point that lies on the negative X–axis is ______. (A) (0, 3) (B) (3, 0) (C)(0, –3) (D)(–3, 0) 4. [AS3] The ordinate of the point (7, –6) is . (A) 7 (B) –13 (C) –6 (D) 13 EXERCISE 5.2. CARTESIAN SYSTEM 44

5. [AS3] The point that lies on the X–axis is . (A) (1, 2) (B) (2, 0) (C)(3, 2) (D)(0, 4) Very Short Answer Type Questions 6 [AS3] Fill in the blanks. (i) A point ‘P’ is at a distance of 3 units and 6 units from Y–axis and X–axis respectively measured along positive axes. Then the co–ordinates of P are . (ii) The abscissa and ordinate of a point are –1 and 7. Then the co–ordinates of the point are . (iii) Study the graph given and answer the question. The coordinates of A are . EXERCISE 5.2. CARTESIAN SYSTEM 45

(iv) Study the graph given and answer the question. The ordinate of B is . (v) Study the graph given and answer the question. The abscissa of C is . 46 Short Answer Type Questions 7(i) [AS3] State the abscissa and ordinate of the following points: a) (6, 8) b) (–3, 4) EXERCISE 5.2. CARTESIAN SYSTEM

(ii) [AS3] State the abscissa and ordinate of the following points and describe the position of the points. a) (4, –3) b) (–4, –7) 8(i) [AS5] Identify the abscissa and ordinates of the points marked in the graph given and write the quadrants in which they lie. (ii) [AS5] Identify the coordinates of the points given in the graph and ﬁll in the table. EXERCISE 5.2. CARTESIAN SYSTEM 47

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