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REAL NUMBERS 19 You have seen that : A NOTE TO THE READER HCF ( p, q, r) × LCM (p, q, r) p × q × r, where p, q, r are positive integers (see Example 8). However, the following results hold good for three numbers p, q and r : LCM (p, q, r) = pqrHCF( p, q, ✁ ✁ r ✁ ) HCF( p, q) HCF( q, r) HCF( p, r) ✁ ✁ HCF (p, q, r) = pqrLCM( p, q, ✂ ✂ r ✂ ) LCM( p, q) LCM( q, r) LCM( p, r) ✂ ✂
POLYNOMIALS 2 2.1 Introduction In Class IX, you have studied polynomials in one variable and their degrees. Recall that if p(x) is a polynomial in x, the highest power of x in p(x) is called the degree of the polynomial p(x). For example, 4x + 2 is a polynomial in the variable x of degree 1, 2y 2 – 3y + 4 is a polynomial in the variable y of degree 2, 5x 3 – 4x 2 + x – 2 3 is a polynomial in the variable x of degree 3 and 7u 6 4 2 – u 4u u ✁ 8 is a polynomial 2 in the variable u of degree 6. Expressions like not polynomials. 1 , 2 x 1 x , x 2 ✂ ✄ ☎ 1 etc., are 2x 3 A polynomial of degree 1 is called a linear polynomial. For example, 2x – 3, 2 3 ✄ x 5, ✄ y 2 , x ✁ , 3z + 4, 2 u 1, etc., are all linear polynomials. Polynomials 11 3 such as 2x + 5 – x 2 , x 3 + 1, etc., are not linear polynomials. A polynomial of degree 2 is called a quadratic polynomial. The name ‘quadratic’ 2 2 has been derived from the word ‘quadrate’, which means ‘square’. 2x 3 ✁ x , 5 y 2 2 u – 2, 2 x 3 x, 2 2 2 2 1 2u 5, 5 v v, 4z ✁ ✁ are some examples of 3 3 7 quadratic polynomials (whose coefficients are real numbers). ✆ ✄ More generally, any quadratic polynomial in x is of the form ax 2 + bx + c, where a, b, c are real numbers and a 0. A polynomial of degree 3 is called a cubic polynomial. Some ✝ examples of ☎
Page 2 and 3: Foreword Preface Contents 1. Real N
Page 4 and 5: xi 9. Some Applications of Trigonom
Page 6 and 7: REAL NUMBERS 1 1.1 Introduction In
Page 8 and 9: REAL NUMBERS 3 moment we write down
Page 10 and 11: REAL NUMBERS 5 This algorithm works
Page 12 and 13: REAL NUMBERS 7 EXERCISE 1.1 1. Use
Page 14 and 15: REAL NUMBERS 9 An equivalent versio
Page 16 and 17: REAL NUMBERS 11 So, HCF (6, 72, 120
Page 18 and 19: 3 = a b ✂ REAL NUMBERS 13 Substit
Page 20 and 21: REAL NUMBERS 15 1.5 Revisiting Rati
Page 22 and 23: REAL NUMBERS 17 We are now ready to
Page 26 and 27: POLYNOMIALS 21 a cubic polynomial a
Page 28 and 29: POLYNOMIALS 23 Table 2.1 x - 2 -1 0
Page 30 and 31: POLYNOMIALS 25 Case (iii) : Here, t
Page 32 and 33: POLYNOMIALS 27 Note that 0 is the o
Page 34 and 35: a = k, b =- k(✞ + ✟) and c = k
Page 36 and 37: POLYNOMIALS 31 Example 4 : Find a q
Page 38 and 39: POLYNOMIALS 33 EXERCISE 2.2 1. Find
Page 40 and 41: POLYNOMIALS 35 Solution : Note that
Page 42 and 43: POLYNOMIALS 37 3. If the zeroes of
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PAIR OF LINEAR EQUATIONS IN TWO VAR
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QUADRATIC EQUATIONS 71 Sridharachar
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QUADRATIC EQUATIONS 73 (ii) Since x
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QUADRATIC EQUATIONS 75 Note that we
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QUADRATIC EQUATIONS 77 Therefore, (
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QUADRATIC EQUATIONS 79 So, 9x 2 - 1
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QUADRATIC EQUATIONS 81 Therefore, t
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QUADRATIC EQUATIONS 83 If b 2 - 4ac
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QUADRATIC EQUATIONS 85 x = 3 25 2 B
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QUADRATIC EQUATIONS 87 Example 15 :
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QUADRATIC EQUATIONS 89 b b b If b 2
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QUADRATIC EQUATIONS 91 EXERCISE 4.4
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ARITHMETIC PROGRESSIONS 5 5.1 Intro
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ARITHMETIC PROGRESSIONS 95 In the e
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ARITHMETIC PROGRESSIONS 97 Similarl
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ARITHMETIC PROGRESSIONS 99 (iv) a 2
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ARITHMETIC PROGRESSIONS 101 Now, lo
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ARITHMETIC PROGRESSIONS 103 Example
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ARITHMETIC PROGRESSIONS 105 It is a
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ARITHMETIC PROGRESSIONS 107 17. Fin
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ARITHMETIC PROGRESSIONS 109 Here, a
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ARITHMETIC PROGRESSIONS 111 Therefo
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ARITHMETIC PROGRESSIONS 113 4. How
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ARITHMETIC PROGRESSIONS 115 EXERCIS
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TRIANGLES 6 6.1 Introduction You ar
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TRIANGLES 119 What can you say abou
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TRIANGLES 121 From the above, you c
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TRIANGLES 123 6.3 Similarity of Tri
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TRIANGLES 125 Therefore, from (1),
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TRIANGLES 127 Example 2 : ABCD is a
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TRIANGLES 129 5. In Fig. 6.20, DE |
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TRIANGLES 131 Let rays BP and CQ in
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TRIANGLES 133 Cut DP = AB and DQ =
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TRIANGLES 135 Now, PQ || EF and ABC
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TRIANGLES 137 Now, BD = 1.2 m × 4
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TRIANGLES 139 Fig. 6.34 2. In Fig.
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TRIANGLES 141 11. In Fig. 6.40, E i
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TRIANGLES 143 Example 9 : In Fig. 6
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TRIANGLES 145 So, from (1) and (2),
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TRIANGLES 147 So, ✁ ✂ ✂ ✂ B
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TRIANGLES 149 or, BL 2 = ✄ AC 2
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TRIANGLES 151 8. In Fig. 6.54, O is
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TRIANGLES 153 (ii) AB 2 = AD 2 - BC
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COORDINATE GEOMETRY 7 7.1 Introduct
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COORDINATE GEOMETRY 157 Therefore,
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COORDINATE GEOMETRY 159 Also, PQ 2
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COORDINATE GEOMETRY 161 So, the req
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COORDINATE GEOMETRY 163 Draw AR, PS
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COORDINATE GEOMETRY 165 Therefore,
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COORDINATE GEOMETRY 167 EXERCISE 7.
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COORDINATE GEOMETRY 169 Example 11
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COORDINATE GEOMETRY 171 EXERCISE 7.
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INTRODUCTION TO TRIGONOMETRY 8 8.1
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INTRODUCTION TO TRIGONOMETRY 175 No
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INTRODUCTION TO TRIGONOMETRY 177 Fr
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INTRODUCTION TO TRIGONOMETRY 179 Th
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INTRODUCTION TO TRIGONOMETRY 181 EX
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INTRODUCTION TO TRIGONOMETRY 183 As
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INTRODUCTION TO TRIGONOMETRY 185 Ta
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INTRODUCTION TO TRIGONOMETRY 187 1.
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INTRODUCTION TO TRIGONOMETRY 189 Ex
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INTRODUCTION TO TRIGONOMETRY 191 Is
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INTRODUCTION TO TRIGONOMETRY 193 =
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SOME APPLICATIONS OF TRIGONOMETRY 9
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CIRCLES 207 You might have seen a p
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CIRCLES 209 Take a point Q on XY ot
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CIRCLES 211 Theorem 10.2 : The leng
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CIRCLES 213 Now, TPR + RPO = 90° =
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CIRCLES 215 10.4 Summary In this ch
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CONSTRUCTIONS 217 Let us see how th
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CONSTRUCTIONS 219 Solution : Given
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CONSTRUCTIONS 221 Steps of Construc
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AREAS RELATED TO CIRCLES 12 12.1 In
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AREAS RELATED TO CIRCLES 225 You ca
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AREAS RELATED TO CIRCLES 227 In a w
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AREAS RELATED TO CIRCLES 229 Soluti
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AREAS RELATED TO CIRCLES 231 10. An
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AREAS RELATED TO CIRCLES 233 Altern
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AREAS RELATED TO CIRCLES 235 2. Fin
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AREAS RELATED TO CIRCLES 237 11. On
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SURFACE AREAS AND VOLUMES 13 13.1 I
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SURFACE AREAS AND VOLUMES 241 First
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SURFACE AREAS AND VOLUMES 243 Examp
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SURFACE AREAS AND VOLUMES 245 7. A
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SURFACE AREAS AND VOLUMES 247 Examp
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SURFACE AREAS AND VOLUMES 249 How a
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✁ ✂ ✂ ☎ ✄ SURFACE AREAS A
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SURFACE AREAS AND VOLUMES 253 How c
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SURFACE AREAS AND VOLUMES 255 So, t
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SURFACE AREAS AND VOLUMES 257 Now,
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SURFACE AREAS AND VOLUMES 259 3. Gi
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STATISTICS 261 x = n ✁ fx i 1 n i
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STATISTICS 263 Table 14.3 Class int
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STATISTICS 265 Substituting the val
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STATISTICS 267 Example 2 : The tabl
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STATISTICS 269 Example 3 : The dist
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STATISTICS 271 4. Thirty women were
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STATISTICS 273 Clearly, 2 is the nu
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STATISTICS 275 of the students. In
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STATISTICS 277 14.4 Median of Group
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STATISTICS 279 From the table above
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STATISTICS 281 Table 14.14 Marks ob
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STATISTICS 283 Example 7 : A survey
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STATISTICS 285 Solution : Class int
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STATISTICS 287 EXERCISE 14.3 1. The
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STATISTICS 289 5. The following tab
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STATISTICS 291 Remark : Note that b
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STATISTICS 293 EXERCISE 14.4 1. The
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PROBABILITY 15 The theory of probab
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PROBABILITY 297 For another example
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PROBABILITY 299 Solution : Kritika
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PROBABILITY 301 That is, the probab
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PROBABILITY 303 Example 7 : There a
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PROBABILITY 305 Let E be the event
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PROBABILITY 307 4 6 5 4 6 5 1 2 3 4
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PROBABILITY 309 10. A piggy bank co
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PROBABILITY 311 25. Which of the fo
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ANSWERS/HINTS EXERCISE 1.1 1. (i) 4
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ANSWERS/HINTS 347 his daughter. To
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ANSWERS/HINTS 349 EXERCISE 3.5 1. (
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ANSWERS/HINTS 351 5. Marks in mathe
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ANSWERS/HINTS 353 n 4. 12. By putti
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ANSWERS/HINTS 355 3. 61m; 5th line
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ANSWERS/HINTS 357 5. 40 3 m 6. 19 3
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ANSWERS/HINTS 359 EXERCISE 13.3 1.
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ANSWERS/HINTS 361 EXERCISE 15.1 1.
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ANSWERS/HINTS 363 ✆ ✂ ✂ ✂ 2
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PROOFS IN MATHEMATICS A1 A1.1 Intro
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PROOFS IN MATHEMATICS 315 (v) Some
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PROOFS IN MATHEMATICS 317 used to d
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PROOFS IN MATHEMATICS 319 Number of
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PROOFS IN MATHEMATICS 321 4. Using
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PROOFS IN MATHEMATICS 323 Remark :
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PROOFS IN MATHEMATICS 325 For examp
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PROOFS IN MATHEMATICS 327 In genera
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PROOFS IN MATHEMATICS 329 2. Write
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PROOFS IN MATHEMATICS 331 Since np
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PROOFS IN MATHEMATICS 333 EXERCISE
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MATHEMATICAL MODELLING 335 (v) Pred
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MATHEMATICAL MODELLING 337 For inst
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MATHEMATICAL MODELLING 339 A2.3 Som
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MATHEMATICAL MODELLING 341 Example
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MATHEMATICAL MODELLING 343 3. A T.V
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