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QUADRATIC EQUATIONS 91 EXERCISE 4.4 1. Find the nature of the roots of the following quadratic equations. If the real roots exist, find them: (i) 2x 2 – 3x + 5 = 0 (ii) 3x 2 – 4 3 x + 4 = 0 (iii) 2x 2 – 6x + 3 = 0 2. Find the values of k for each of the following quadratic equations, so that they have two equal roots. (i) 2x 2 + kx + 3 = 0 (ii) kx (x – 2) + 6 = 0 3. Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800 m 2 ? If so, find its length and breadth. 4. Is the following situation possible? If so, determine their present ages. The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48. 5. Is it possible to design a rectangular park of perimeter 80 m and area 400 m 2 ? If so, find its length and breadth. 4.6 Summary In this chapter, you have studied the following points: 1. A quadratic equation in the variable x is of the form ax 2 + bx + c = 0, where a, b, c are real numbers and a 0. 2. A real number ✁ is said to be a root of the quadratic equation ax 2 + bx + c = 0, if a✁ 2 + b✁ + c = 0. The zeroes of the quadratic polynomial ax 2 + bx + c and the roots of the quadratic equation ax 2 + bx + c = 0 are the same. 3. If we can factorise ax 2 + bx + c, a 0, into a product of two linear factors, then the roots of the quadratic equation ax 2 + bx + c = 0 can be found by equating each factor to zero. 4. A quadratic equation can also be solved by the method of completing the square. 5. Quadratic formula: The roots of a quadratic equation ax 2 + bx + c = 0 are given by 2 b b 4ac , provided b 2 – ☎ 4ac 0. 2a ✂ ✄ ✂ 6. A quadratic equation ax 2 + bx + c = 0 has (i) two distinct real roots, if b 2 – 4ac > 0, (ii) two equal roots (i.e., coincident roots), if b 2 – 4ac = 0, and (iii) no real roots, if b 2 – 4ac < 0.
92 MATHEMATICS A NOTE TO THE READER In case of word problems, the obtained solutions should always be verified with the conditions of the original problem and not in the equations formed (see Examples 11, 13, 19 of Chapter 3 and Examples 10, 11, 12 of Chapter 4).
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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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Page 76 and 77: QUADRATIC EQUATIONS 71 Sridharachar
Page 78 and 79: QUADRATIC EQUATIONS 73 (ii) Since x
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Page 84 and 85: QUADRATIC EQUATIONS 79 So, 9x 2 - 1
Page 86 and 87: QUADRATIC EQUATIONS 81 Therefore, t
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Page 90 and 91: QUADRATIC EQUATIONS 85 x = 3 25 2 B
Page 92 and 93: QUADRATIC EQUATIONS 87 Example 15 :
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Page 98 and 99: ARITHMETIC PROGRESSIONS 5 5.1 Intro
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Page 114 and 115: ARITHMETIC PROGRESSIONS 109 Here, a
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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
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Page 136 and 137: TRIANGLES 131 Let rays BP and CQ in
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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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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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