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ANSWERS/HINTS 355 3. 61m; 5th line at a distance of 22.5 m 4. 2 : 7 5. 3 1:1; , ✁ ✂ ✄ ☎ 0 2 6. x = 6, y = 3 7. (3, – 10) ✆ ✝ 8. ✂ 2 , ✁ 20 ✂ ☎ 7 7 ✄ ✆ ✝ 9. 7 , 13✁ ✁ 1, (0,5), ✄ ☎ ✄ ☎ 1, 2 2 ✂ ✆ ✝ ✆ ✝ 10. 24 sq. units EXERCISE 7.3 1. (i) 21 sq. 2 units (ii) 32 sq. units 2. (i) k = 4 (ii) k = 3 3. 1 sq. unit; 1 : 4 4. 28 sq. units EXERCISE 7.4 (Optional)* 1. 2 : 9 2. x + 3y – 7 = 0 3. (3, – 2) 4. (1, 0), (1, 4 ) 5. (i) (4, 6), (3, 2), (6, 5); taking AD and AB as coordinate axes (ii) (12, 2), (13, 6), (10, 3); taking CB and CD as coordinate axes. 9 sq. 2 9 sq. units; areas are the same in both the cases. 2 units, 6. 15 sq. 32 7. (i) units; 1 : 16 7 , ✁ 9 D✄ ☎ 2 2 (ii) 11 , ✁ 11 P✄ ☎ 3 3 (iii) (v) 11 , 11 , ✁ 11 ✁ , 11 Q✄ R ☎ ✄ ☎ ✆ ✝ ✆ ✝ 3 3 3 3 x1 x2 x3 , y1 y ✞ ✞ ✞ ✞ ✟ ✠ 2 y3 ☛ ✡ 3 3 (iv) P, Q, R are the same point. 8. Rhombus ✆ ✝ ✝ ✆ EXERCISE 8.1 ☞ ✌ 1. (i) 7 , 24 sin A = cosA = 25 25 (ii) 24 , 7 sin C = cosC = 25 25 2. 0 3. 7 , 3 cos A = tan A = 4 7 4. 15 , 17 sin A = secA = 17 8
356 MATHEMATICS 5. 5 , 12 , 5 , 12 , 13 sin cos = tan cot cosec = 13 13 12 5 5 ✁ ✁ ✁ 7. (i) 49 64 (ii) 49 64 8. Yes 9. (i) 1 (ii) 0 10. 12 , 5 , 12 sinP = cosP = tanP = 13 13 5 11. (i) False (ii) True (iii) False (iv) False (v) False 1. (i) 1 (ii) 2 (iii) EXERCISE 8.2 3 2 6 ✂ 8 (iv) 43 24 3 ✄ 11 2. (i) A (ii) D (iii) A (iv) C 3. ☎ A = 45°, ☎ B = 15° 4. (i) False (ii) True (iii) False (iv) False (v) True EXERCISE 8.3 1. (i) 1 (ii) 1 (iii) 0 (iv) 0 3. ☎ A = 36° 5. ☎ A = 22° 7. cos 23° + sin 15° EXERCISE 8.4 (v) 67 12 1. 1 1 sin A = , tan A = , secA = 2 1+cot A cot A 2 1+cot A cot A 2. 2 sec A – 1 , 1 , 2 sinA = cosA = tanA = sec A –1 sec A sec A 1 , secA cot A = cosec A = 2 2 sec A – 1 sec A – 1 3. (i) 1 (ii) 1 4. (i) B (ii) C (iii) D (iv) D EXERCISE 9.1 1. 10 m 2. 8 3m 3. 3m, 2 3m 4. 10 3 m
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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
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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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Page 280 and 281: STATISTICS 275 of the students. In
Page 282 and 283: STATISTICS 277 14.4 Median of Group
Page 284 and 285: STATISTICS 279 From the table above
Page 286 and 287: STATISTICS 281 Table 14.14 Marks ob
Page 288 and 289: STATISTICS 283 Example 7 : A survey
Page 290 and 291: STATISTICS 285 Solution : Class int
Page 292 and 293: STATISTICS 287 EXERCISE 14.3 1. The
Page 294 and 295: STATISTICS 289 5. The following tab
Page 296 and 297: STATISTICS 291 Remark : Note that b
Page 298 and 299: STATISTICS 293 EXERCISE 14.4 1. The
Page 300 and 301: PROBABILITY 15 The theory of probab
Page 302 and 303: PROBABILITY 297 For another example
Page 304 and 305: PROBABILITY 299 Solution : Kritika
Page 306 and 307: PROBABILITY 301 That is, the probab
Page 308 and 309: PROBABILITY 303 Example 7 : There a
Page 310 and 311: PROBABILITY 305 Let E be the event
Page 312 and 313: PROBABILITY 307 4 6 5 4 6 5 1 2 3 4
Page 314 and 315: PROBABILITY 309 10. A piggy bank co
Page 316 and 317: PROBABILITY 311 25. Which of the fo
Page 318 and 319: ANSWERS/HINTS EXERCISE 1.1 1. (i) 4
Page 320 and 321: ANSWERS/HINTS 347 his daughter. To
Page 322 and 323: ANSWERS/HINTS 349 EXERCISE 3.5 1. (
Page 324 and 325: ANSWERS/HINTS 351 5. Marks in mathe
Page 326 and 327: ANSWERS/HINTS 353 n 4. 12. By putti
Page 330 and 331: ANSWERS/HINTS 357 5. 40 3 m 6. 19 3
Page 332 and 333: ANSWERS/HINTS 359 EXERCISE 13.3 1.
Page 334 and 335: ANSWERS/HINTS 361 EXERCISE 15.1 1.
Page 336 and 337: ANSWERS/HINTS 363 ✆ ✂ ✂ ✂ 2
Page 338 and 339: PROOFS IN MATHEMATICS A1 A1.1 Intro
Page 340 and 341: PROOFS IN MATHEMATICS 315 (v) Some
Page 342 and 343: PROOFS IN MATHEMATICS 317 used to d
Page 344 and 345: PROOFS IN MATHEMATICS 319 Number of
Page 346 and 347: PROOFS IN MATHEMATICS 321 4. Using
Page 348 and 349: PROOFS IN MATHEMATICS 323 Remark :
Page 350 and 351: PROOFS IN MATHEMATICS 325 For examp
Page 352 and 353: PROOFS IN MATHEMATICS 327 In genera
Page 354 and 355: PROOFS IN MATHEMATICS 329 2. Write
Page 356 and 357: PROOFS IN MATHEMATICS 331 Since np
Page 358 and 359: PROOFS IN MATHEMATICS 333 EXERCISE
Page 360 and 361: MATHEMATICAL MODELLING 335 (v) Pred
Page 362 and 363: MATHEMATICAL MODELLING 337 For inst
Page 364 and 365: MATHEMATICAL MODELLING 339 A2.3 Som
Page 366 and 367: MATHEMATICAL MODELLING 341 Example
Page 368 and 369: MATHEMATICAL MODELLING 343 3. A T.V
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