FOUNDATION MATHEMATICS

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2 FOUNDATION MATHEMATICS CLASS - IX Module - Sr. No. Chapters Page No.. Number System 60. Polynomials 6. Co-ordinate Geometry 6 4. Linear Equations in Two 7 7 Variables ETOOS EDUCATION PVT. LTD. Corporate Office : Plot No. 8, Near Union Bank of India, Rajeev Gandhi Nagar, Kota 400, Rajasthan, India

3 NUMBER SYSTEM Chapter Contents Chapter Introduction Rational number in Decimal Representation Representation of rational numbers on a number line Representation of number on the number line by means of magnifying glass Conversion of terminating and nonterminating decimal numbers into the form of p/q Irrational numbers Real numbers Complex number Representation of Irrational numbers on a number line Geometrical representation of Square root of a real numbers on a number line Exponents of real numbers Laws of Integral Exponents Laws of Exponents Radicals & surds Conjugate surd square roots of binomial quadratic surds EXERCISE - EXERCISE - EXERCISE - CHAPTER TEST ANSWER KEY INTRODUCTION "God gave us the natural number, all else is the work of man". It was exclaimed by L. Kronecker (8-89), the reputed German Mathematician. This statement reveals in a nut shell the significant role of the universe of numbers played in the evolution of human thought. N : The set of natural numbers, 6, N W,,,4,... 99,00,..., 0, W : The set of whole numbers, Z : The set of Integers, Q : The set of rationals, R : The set of Real Numbers. e Z + +, In our day to life we deal with different types of numbers which can be broady classified as follows. Q R

4 MATHEMATICS MATHEMATICS NUMBER SYSTEM Ex. Express each of the following mixed recurring decimals in the form p q : (i) 4. (ii).7 Sol. (i) Let x = 4. 0 x = 4. [Multiplying both sides of x by 0] 0 x = x = x = x = x = 89 9 x = (ii) Let x =.7 ; 0 x = 7. 0 x = x = x = x = x = x = 8 x = 8 0 = p form = q (Complete numbers) (number formed by Non - repeating digit) No.of 9 as no.of repeating digits after that write no.of 0 as no.of non repeating digits. IMPORTANT KEY POINTS ADDITION & SUBTRACTION OF RECURRING DECIMALS Addition and subtraction of recurring decimals can be done in two ways either by converting them into vulgar fractions and then operating them or alternatively as follows :- Add Step- : Express the number without bar as Step- : Write the number as one above other i.e Step- : Divide the number into two parts. In the first part i.e., left side write as many digits as there will be integral value with non recurring decimal. In the right side write as many digits as the LCM of the number of recurring digits in the given decimal number e.g.,.7/ (Since.7 is the integral + non recurring part) 8.6/66 (LCM of and is 6) Step-4 : Now add or subtract as usual Step- : Put the bar over the digits which are on the right side in the resultant value Thus, Study with Best Etoos Faculties of Kota

5 MATHEMATICS NUMBER SYSTEM MATHEMATICS Sol. as x = x = a + b + a b a + b a b a + b + a b a + b + a b a + b + a b ( ) ( ) a + b a b a + b + a b + (a + b)(a b) = 4b = a + a 4b a + a 4b 4b = b bx = a + a 4b bx a = squaring both sides (bx a) = a 4b a 4b 4b x + a 4abx = a 4b 4b x 4abx + 4b = 0 b x abx + b = 0. Simplify the each of the following DPP - (i) (ii). Simplify the following expression (i) 4 7 (ii) (iii). Express each one of the following with rational denominator : (i) 6 4 (ii) (iii) (iv) b b b a 4. Simplify the each of the following (i) (ii) (ii) 40 India s No. Online Coaching Institute

6 MATHEMATICS MATHEMATICS NUMBER SYSTEM Things To Remember In this chapter, you have studied the following points:. A number r is called a rational number, if it can be written in the form p, where p and q are integers and q 0. q. A number s is called a irrational number, if it cannot be written in the form p, where p and q are integers and q 0. q. The decimal expansion of a rational number is either terminating or non-terminating recurring. Moreover, a number whose decimal expansion is terminating or non-terminating recurring is rational. 4. The decimal expansion of an irrational number is non-terminating non-recurring. Moreover, a number whose decimal expansion is non-terminating non-recurring is irrational.. All the rational and irrational numbers make up the collection of real numbers. 6. There is a unique real number corresponding to every point on the number line. Also, corresponding to each real number, there is a unique point on the number line. 7. If r is rational and s is irrational, then r + s and r s are irrational numbers, and rs and r s are irrational numbers, r For positive real numbers a and b, the following identities hold: (i) ab a b (ii) a b (iii) a b a b a b (iv) (v) a b a ab b a b a b a b a b 9. To rationalise the denominator of a b we multiply this by a b where a and b are integers. a b 0. Let a > 0 be a real number and p and q be rational numbers. Then (i) a p. a q = a p + q (ii) (a p ) q = a p q a (iii) a p pq p p a q (iv) a b ab p Study with Best Etoos Faculties of Kota 4

7 MATHEMATICS NUMBER SYSTEM MATHEMATICS NCERT BASED SOME IMPORTANT QUESTIONS. Is zero a rational number? Can you write it in the form p q, where p and q are integers and q 0? Ans. Yes, zero is a rational number. We can write zero in the form p q whose p and q are integers and q 0. so, 0 can be written as 0 = 0 = 0 etc.. Find five rational numbers between and 4. Ans. Let a = 4, b =, n = d = b a n + = 4 = + 0 so, Rational number are a + d, a + d, a + d.... Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number. Ans. No, 4 = is a rational number. 4. Show how can be represented on the number line. Ans. Using Pythagoras Theorem: = + Taking positive square root we get B 0 A C 4. Mark a point 'A' representing units on number line.. Now construct AB of unit length perpendicular to OA. Join OB. Now taking O as centre and OB as radius draw an arc, intersecting number line at point C. 4. Point C represents on number line. [length (OB) = length (OC)]. Write the following in decimal form and say what kind of decimal expansion each has : Ans. (ii) (ii) (iii) (iii) 4 (vi) = (Non terminating Repeating) 4 = 8 8 = 4. (terminating decimal) (vi) Terminating decimal 400 India s No. Online Coaching Institute

8 MATHEMATICS MATHEMATICS NUMBER SYSTEM OBJECTIVE TYPE. How many prime number are there between 0 and 0 :- (A) 9 (B) 0 (C) 8 (D). Two irrational numbers between and. are :- (A) and (B) and (C) and 7 (D) None of these. The exponential form of is :- (A) 6 / (B) 6 / (C) 6 /4 (D) 6 4. The rational form of.687 is :- (A) 4 6 (B) EXERCISE 4 6 (C) 4 6 (D) The rational form of.74 is :- (A) (B) 7 99 (C) (D) The value of 0.4 is :- (A) (B) (C) (D) Which of the following is not a rational number :- (A) (B) 4 (C) 9 (D) is equal to :- + + / (A)/ (B) /7 (C) (D) 9. The number is :- + (A) Rational (B) Irrational (C) Both (D) Can't say 0. If x = x then x x equals :- (A) 6 (B) (C) (D). The value of. :- (A) 4 9 (B) 46 9 (C) 47 9 (D) None Study with Best Etoos Faculties of Kota 47

9 MATHEMATICS NUMBER SYSTEM MATHEMATICS TRUE / FALSE TYPE QUESTION. The sum of two rational numbers in rational. The sum of two irrational numbers is irrational.. The product of two rational numbers is rational. 4. The product of two irrational numbers is irrational. EXERCISE. The sum of a rational number and an irrational number is irrational. 6. The product of a non zero rational number and an irrational number is a rational number. 7. Every real number is rational. 8. is irrational and 7 is rational. 9. Every rational number must be a whole number. 0. The number zero is both positive and negative.. The sum of two prime numbers is always even.. The product of two odd numbers is always odd. FILL IN THE BLANKS. Every point on the number line corresponds to a... number which may be either... or The decimal form of an irrational number is neither... nor.... The decimal representation of the rational number 6 is is a / an... number 7. The decimal equivalent to 7 8. The decimal equivalent to is The common fraction equivalent to is The common fraction equivalent to 0.4 is.... Every real number is either... number or... number. VERY SHORT ANSWER TYPE QUESTION. Express the following in the form of p/q. is... (i). (ii).4 (iii).0 (iv). (v) 0.6. Examine, whether the following numbers are rational or an irrational : (i) 7 (ii) 4 (iii) + (iv) + (v) + (vi) ( - ) (vii) ( - )( + ) (viii) ( + ) (ix) (x) 0 8 (xi) 4 India s No. Online Coaching Institute

10 MATHEMATICS NUMBER SYSTEM MATHEMATICS EXERCISE. Arrange in ascending order 6 4 7,, 48 (A) 4 6, 48, 7 (B) ,, 7 (C) 6 7, 48, 4 (D) None of these. Arrange in descending order 8, 8, 0 (A) 8, 8, 0 (B) 0, 8, 8 (C) 8, 0, 8 (D) None of these. The value of x in 4 x + = is : (A) 6 (B) (C) 4 (D) None of these 4. The value of ( ) is :- (A) (B) (C) ± (D) None of these. Which one of the following is a rational number? FOR COMPETITIVE EXAMS (A) (B) (C) + (D) 6. The value of x in 4x 7 = 0 is :- (A) (B) 44 (C) (D) None of these 7. Which of the following is not an improper fraction :- (A) 4 (B) (C) (D) 7 8. If x, y, z are positive real numbers, the value of x.y y.z z.x is :- (A) xyz (B) xyz a /ab b /bc c /ca 9. The value of x x x.. b c a x x x is :- (C) (D) xyz (A) (B) 0 (C) x /abc (D) x /(ab+bc+ca) b (b+c-a) c (c+a-b) a (a+b-c) 0. The value of : x x x.. c a b x x x.. = (A) x abc (B) x a+b+c (C) x ab+bc+ca (D) (A) 4 (B). If pqr =, then p + q + q + r + r + p (C) is equal to :- (D) None (A) 0 (B) pq (C) pq (D). The simplified form of 4 x 7/. y / x. y is : India s No. Online Coaching Institute

11 MATHEMATICS NUMBER SYSTEM MATHEMATICS OBJECTIVE TYPE CHAPTER TEST Time : 60 Min Marks : 0. If x= + 8 and y = 8 then = x y (A) 4 (B) 4 (C) 8 (D) 8. If 7 = a + b 7 then (a,b) = 7 (A) (8, ) (B) ( 8, -) (C) ( 8, ) (D) (8, ). (A) 8 (B) 8 (C) 6 (D) 6 4. If x = and y =, the value of x y is : x y (A) 4 (B) 4 4 (C) 6 4 (D) 6 4. Which one is greatest in the following : (A) (B) (C) 4 (D) 6. The value of is : (A) /8 (B) /6 (C) / (D) None 7. If x and x the value of x + xy + y is : (A) 99 (B) 00 (C) (D) 0 8. Simplify : (A) (B) 0 (C) 0 (D) Which of the following is smallest? (A) 4 (B) 4 (C) 4 (D) 0 The product of and is : (A) 6 7 (B) 6 67 (C) 6 7 (D) India s No. Online Coaching Institute

12 MATHEMATICS NUMBER SYSTEM MATHEMATICS ANSWER KEY DPP- (i) (ii) 4 90 (iii) (iv) (v) 7 9 (vi) DPP- 4 (i) 4 (ii) 9 (iii). (i) 0 (ii) DPP- 4 (iv) (v) 9. (i) (ii) 6 (iii). (i) 4 (ii) 7 4. Simplify : (i) 8 (ii) 0 (iii) 0 (iii) 7 (iv) a b a EXERCISE OBJECTIVE ANSWER KEY EXERCISE - Que Ans. B A C A C D A B B A C C B A A Que Ans. A D B D A D D B C A D A B C A EXERCISE TRUE & FALSE TYPE QUESTIONS. True. False. True 4. False. True 6. False 7. False 8. True 9. False 0. False. False. True FILL IN THE BALANKS. Real, Rational number, irrational number 4. Terminating, recurring Rational Rational, irrational 8 India s No. Online Coaching Institute

CLASS-IX MATHEMATICS. For. Pre-Foundation Course CAREER POINT

CLASS-IX MATHEMATICS. For. Pre-Foundation Course CAREER POINT CLASS-IX MATHEMATICS For Pre-Foundation Course CAREER POINT CONTENTS S. No. CHAPTERS PAGE NO. 0. Number System... 0 3 0. Polynomials... 39 53 03. Co-ordinate Geometry... 54 04. Introduction to Euclid's