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COORDINATE GEOMETRY 157 Therefore, QT = 2 units and PT = RS = 2 units. Now, using the Pythagoras theorem, we have So, PQ 2 =PT 2 + QT 2 =2 2 + 2 2 = 8 PQ = 2 2 units How will we find the distance between two points in two different quadrants? Consider the points P(6, 4) and Q(–5, –3) (see Fig. 7.4). Draw QS perpendicular to the x-axis. Also draw a perpendicular PT from the point P on QS (extended) to meet y-axis at the point R. Fig. 7.3 Fig. 7.4 Then PT = 11 units and QT = 7 units. (Why?) Using the Pythagoras Theorem to the right triangle PTQ, we get PQ = 2 2 11 7 = 170 units.
158 MATHEMATICS Let us now find the distance between any two points P(x 1 , y 1 ) and Q(x 2 , y 2 ). Draw PR and QS perpendicular to the x-axis. A perpendicular from the point P on QS is drawn to meet it at the point T (see Fig. 7.5). Then, OR = x 1 , OS = x 2 . So, RS = x 2 – x 1 = PT. Also, SQ = y 2 , ST = PR = y 1 . So, QT = y 2 – y 1 . Now, applying the Pythagoras theorem in PQ 2 =PT 2 + QT 2 PTQ, we get =(x 2 – x 1 ) 2 + (y 2 – y 1 ) 2 Fig. 7.5 2 2 Therefore, PQ = ✂ ✁ x2 x1 ✂ ✁ y2 y1 ✄ ☎ ✄ Note that since distance is always non-negative, we take only the positive square root. So, the distance between the points P(x 1 , y 1 ) and Q(x 2 , y 2 ) is 2 2 PQ = ✝ ✆ x2– x1 ✝ + ✆ y2– y , 1 which is called the distance formula. Remarks : 1. In particular, the distance of a point P(x, y) from the origin O(0, 0) is given by 2 2 OP = ✞ x y . 2 2 2. We can also ✟ ✠ ✟ ✠ write, PQ = x1 x2 y ✄ ☎ ✄ 1 y . (Why?) 2 Example 1 : Do the points (3, 2), (–2, –3) and (2, 3) form a triangle? If so, name the type of triangle formed. Solution : Let us apply the distance formula to find the distances PQ, QR and PR, where P(3, 2), Q(–2, –3) and R(2, 3) are the given points. We have ✞ ✞ ✞ ✡ ✞ ✡ = 7.07 (approx.) PQ = (3 2 2) (2 2 3) 2 5 2 5 50 ✞ ✡ ✞ ✡ = 7.21 (approx.) QR = 2 (–2 – 2) 2 (–3 – 3) 2 (– 4) 2 (– 6) 52 ✞ ✡ ✞ ☛ ✡ = 1.41 (approx.) PR = 2 (3–2) 2 (2–3) 2 1 2 ( 1) 2 Since the sum of any two of these distances is greater than the third distance, therefore, the points P, Q and R form a triangle.
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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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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),
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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 142 and 143: TRIANGLES 137 Now, BD = 1.2 m × 4
Page 144 and 145: TRIANGLES 139 Fig. 6.34 2. In Fig.
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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
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Page 158 and 159: TRIANGLES 153 (ii) AB 2 = AD 2 - BC
Page 160 and 161: COORDINATE GEOMETRY 7 7.1 Introduct
Page 164 and 165: COORDINATE GEOMETRY 159 Also, PQ 2
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Page 178 and 179: INTRODUCTION TO TRIGONOMETRY 8 8.1
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Page 192 and 193: INTRODUCTION TO TRIGONOMETRY 187 1.
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Page 200 and 201: 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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