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TRIANGLES 137 Now, BD = 1.2 m × 4 = 4.8 m. Note that in ABE and CDE, ✁ B = ✁ D (Each is of 90° because lamp-post as well as the girl are standing vertical to the ground) and ✁ E = ✁ E (Same angle) So, ABE ~ CDE (AA similarity criterion) Therefore, BE DE = AB CD i.e., 4.8 + x x = 3.6 0.9 i.e., 4.8 + x =4x i.e., 3x = 4.8 i.e., x = 1.6 (90 cm = 90 100 So, the shadow of the girl after walking for 4 seconds is 1.6 m long. Example 8 : In Fig. 6.33, CM and RN are respectively the medians of ABC and PQR. If ABC ~ PQR, prove that : (i) AMC ~ PNR (ii) CM RN ✂ AB PQ m = 0.9 m) (iii) CMB ~ RNQ Fig. 6.33 Solution : (i) ABC ~ PQR (Given) So, AB PQ = BC CA ✂ (1) QR RP and ✁ A = ✁ P, ✁ B = ✁ Q and ✁ C = ✁ R (2) But So, from (1), AB = 2 AM and PQ = 2 PN (As CM and RN are medians) 2AM 2PN = CA RP
138 MATHEMATICS i.e., AM PN = CA RP Also, MAC = NPR [From (2)] (4) So, from (3) and (4), (3) ✁ AMC ~ ✁ PNR (SAS similarity) (5) (ii) From (5), But Therefore, (iii) Again, Therefore, CM RN = CA RP CA RP = AB PQ CM RN = AB PQ AB PQ = BC QR CM RN = BC QR (6) [From (1)] (7) [From (6) and (7)] (8) [From (1)] [From (8)] (9) Also, CM RN = AB PQ ✂ 2 BM 2 QN i.e., CM RN = BM QN (10) i.e., CM RN = BC BM ✄ [From (9) and (10)] QR QN Therefore, ✁ CMB ~ ✁ RNQ (SSS similarity) [Note : You can also prove part (iii) by following the same method as used for proving part (i).] EXERCISE 6.3 1. State which pairs of triangles in Fig. 6.34 are similar. Write the similarity criterion used by you for answering the question and also write the pairs of similar triangles in the symbolic form :
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Foreword Preface Contents 1. Real N
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xi 9. Some Applications of Trigonom
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REAL NUMBERS 1 1.1 Introduction In
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REAL NUMBERS 3 moment we write down
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REAL NUMBERS 5 This algorithm works
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REAL NUMBERS 7 EXERCISE 1.1 1. Use
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REAL NUMBERS 9 An equivalent versio
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REAL NUMBERS 11 So, HCF (6, 72, 120
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3 = a b ✂ REAL NUMBERS 13 Substit
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REAL NUMBERS 15 1.5 Revisiting Rati
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REAL NUMBERS 17 We are now ready to
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REAL NUMBERS 19 You have seen that
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POLYNOMIALS 21 a cubic polynomial a
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POLYNOMIALS 23 Table 2.1 x - 2 -1 0
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POLYNOMIALS 25 Case (iii) : Here, t
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POLYNOMIALS 27 Note that 0 is the o
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a = k, b =- k(✞ + ✟) and c = k
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POLYNOMIALS 31 Example 4 : Find a q
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POLYNOMIALS 33 EXERCISE 2.2 1. Find
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POLYNOMIALS 35 Solution : Note that
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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
Page 92 and 93: QUADRATIC EQUATIONS 87 Example 15 :
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Page 96 and 97: QUADRATIC EQUATIONS 91 EXERCISE 4.4
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Page 106 and 107: ARITHMETIC PROGRESSIONS 101 Now, lo
Page 108 and 109: ARITHMETIC PROGRESSIONS 103 Example
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Page 112 and 113: ARITHMETIC PROGRESSIONS 107 17. Fin
Page 114 and 115: ARITHMETIC PROGRESSIONS 109 Here, a
Page 116 and 117: ARITHMETIC PROGRESSIONS 111 Therefo
Page 118 and 119: ARITHMETIC PROGRESSIONS 113 4. How
Page 120 and 121: ARITHMETIC PROGRESSIONS 115 EXERCIS
Page 122 and 123: TRIANGLES 6 6.1 Introduction You ar
Page 124 and 125: TRIANGLES 119 What can you say abou
Page 126 and 127: TRIANGLES 121 From the above, you c
Page 128 and 129: TRIANGLES 123 6.3 Similarity of Tri
Page 130 and 131: TRIANGLES 125 Therefore, from (1),
Page 132 and 133: TRIANGLES 127 Example 2 : ABCD is a
Page 134 and 135: TRIANGLES 129 5. In Fig. 6.20, DE |
Page 136 and 137: TRIANGLES 131 Let rays BP and CQ in
Page 138 and 139: TRIANGLES 133 Cut DP = AB and DQ =
Page 140 and 141: TRIANGLES 135 Now, PQ || EF and ABC
Page 144 and 145: TRIANGLES 139 Fig. 6.34 2. In Fig.
Page 146 and 147: TRIANGLES 141 11. In Fig. 6.40, E i
Page 148 and 149: TRIANGLES 143 Example 9 : In Fig. 6
Page 150 and 151: TRIANGLES 145 So, from (1) and (2),
Page 152 and 153: TRIANGLES 147 So, ✁ ✂ ✂ ✂ B
Page 154 and 155: TRIANGLES 149 or, BL 2 = ✄ AC 2
Page 156 and 157: TRIANGLES 151 8. In Fig. 6.54, O is
Page 158 and 159: TRIANGLES 153 (ii) AB 2 = AD 2 - BC
Page 160 and 161: COORDINATE GEOMETRY 7 7.1 Introduct
Page 162 and 163: COORDINATE GEOMETRY 157 Therefore,
Page 164 and 165: COORDINATE GEOMETRY 159 Also, PQ 2
Page 166 and 167: COORDINATE GEOMETRY 161 So, the req
Page 168 and 169: COORDINATE GEOMETRY 163 Draw AR, PS
Page 170 and 171: COORDINATE GEOMETRY 165 Therefore,
Page 172 and 173: COORDINATE GEOMETRY 167 EXERCISE 7.
Page 174 and 175: COORDINATE GEOMETRY 169 Example 11
Page 176 and 177: COORDINATE GEOMETRY 171 EXERCISE 7.
Page 178 and 179: INTRODUCTION TO TRIGONOMETRY 8 8.1
Page 180 and 181: INTRODUCTION TO TRIGONOMETRY 175 No
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Page 184 and 185: INTRODUCTION TO TRIGONOMETRY 179 Th
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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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