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AREAS RELATED TO CIRCLES 231 10. An umbrella has 8 ribs which are equally spaced (see Fig. 12.13). Assuming umbrella to be a flat circle of radius 45 cm, find the area between the two consecutive ribs of the umbrella. 11. A car has two wipers which do not overlap. Each wiper has a blade of length 25 cm sweeping through an angle of 115°. Find the total area cleaned at each sweep of the blades. 12. To warn ships for underwater rocks, a lighthouse spreads a red coloured light over a sector of angle 80° to a distance of 16.5 km. Find the area of the sea over which the ships are warned. (Use = 3.14) 13. A round table cover has six equal designs as shown in Fig. 12.14. If the radius of the cover is 28 cm, find the cost of making the designs at the rate of Rs 0.35 per cm 2 . (Use 3 = 1.7) Fig. 12.13 14. Tick the correct answer in the following : Fig. 12.14 Area of a sector of angle p (in degrees) of a circle with radius R is p (A) 2 ✁ ✂ R (B) 180 p 180 2 p ✂ R (C) 2 R ✁ ✂ (D) ✁ 360 p 720 ✁ 2 R ✂ 2 12.4 Areas of Combinations of Plane Figures So far, we have calculated the areas of different figures separately. Let us now try to calculate the areas of some combinations of plane figures. We come across these types of figures in our daily life and also in the form of various interesting designs. Flower beds, drain covers, window designs, designs on table covers, are some of such examples. We illustrate the process of calculating areas of these figures through some examples. Example 4 : In Fig. 12.15, two circular flower beds have been shown on two sides of a square lawn ABCD of side 56 m. If the centre of each circular flower bed is the point of intersection O of the diagonals of the square lawn, find the sum of the areas of the lawn and the flower beds. Fig. 12.15
232 MATHEMATICS Solution : Area of the square lawn ABCD = 56 × 56 m 2 (1) Let OA = OB = x metres So, x 2 + x 2 =56 2 or, 2x 2 = 56 × 56 or, x 2 = 28 × 56 (2) Now, area of sector OAB = 90 360 2 ✁ x = 1 4 2 x ✁ = 1 22 4 7 28 56 m 2 [From (2)] (3) Also, area of ✂ OAB = 1 2 56 56 m 4 ✄ (☎ AOB = 90°) (4) ✄ So, area of flower bed AB = Similarly, area of the other flower bed Therefore, total area = = = = ✆ ✠ ☛ 1 22 1 ✝ 28 56 56 56 ✞ m ✞ ✞ ✟ ✞ ✞ 4 7 4 1 ✌ ✍ 22 28 56 2 ✎ ✎ ✏ m 4 7 ✑ ✓ 1 8 28 56 m 4 7 1 8 28 56 m 4 7 ✆ ✠ ☛ 2 2 ✒ ✔ 2 ✡ 2 ☞ [From (3) and (4)] (5) ✄ ✄ ✄ (6) 1 8 56 56 28 ✞ 56 ✕ ✞ ✞ ✞ 4 7 = 1 8✝ 2 28 56 m ✕ ✞ ✞ ✞ ✡☞ [From (1), (5) and (6)] 4 7 2 ✌ ✍ 2 28 56 2 ✎ ✖ ✖ m 7 7 ✑ ✒ 2 ✓ ✔ = 18 28 56 m 4032m ✗ 7 2 2
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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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Page 212 and 213: CIRCLES 207 You might have seen a p
Page 214 and 215: CIRCLES 209 Take a point Q on XY ot
Page 216 and 217: CIRCLES 211 Theorem 10.2 : The leng
Page 218 and 219: CIRCLES 213 Now, TPR + RPO = 90° =
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Page 240 and 241: AREAS RELATED TO CIRCLES 235 2. Fin
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Page 246 and 247: SURFACE AREAS AND VOLUMES 241 First
Page 248 and 249: SURFACE AREAS AND VOLUMES 243 Examp
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Page 260 and 261: SURFACE AREAS AND VOLUMES 255 So, t
Page 262 and 263: SURFACE AREAS AND VOLUMES 257 Now,
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Page 266 and 267: STATISTICS 261 x = n ✁ fx i 1 n i
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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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