# RS Aggarwal Class 9 Solutions Chapter-3.pdf

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# RS Aggarwal Class 9 Solutions Chapter-3.pdf

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### RS Aggarwal Class 9 Solutions Chapter-3.pdf

2. 2. CLASS IX WWW.Vedantu.com RS Aggarwal solutions Answer12. (ππ π β ππ π + πππ β ππ) (π3 π β π2 π + 5ππ β 5π) = (π3 π + 5ππ β π2 π β 5π) = [ππ(π2 + 5) β π(π2 + 5)] = (ππ β π)(π2 + 5) = π(π β 1)(π2 + 5) Answer13. π β ππ β πππ + ππ 8 β 4π β 2π3 + π4 = 8 β 2π3 β 4π + π4 = 2(4 β π3 ) β π(4 + π3 ) = (2 β π)(4 β π3 ) Answer14. ππ β πππ π + ππππ β πππ .π₯3 β 2π₯2 π¦ + 3π₯π¦2 β 6π¦3 = π₯2(π₯ β 2π¦) + 3π¦2 (π₯ β 2π¦) = (π₯ β 2π¦)(π₯2 + 3π¦2 ) Answer15. ππ β ππ + ππ β ππ pπ₯ β 5π + ππ β 5π₯ = ππ₯ β 5π₯ + ππ β 5π = π₯(π β 5) + π(π β 5) = (π₯ + π)(π β 5) Answer16. ππ + π β ππ β π .π₯2 + π¦ β π₯π¦ β π₯ = π₯2 β π₯π¦ + π¦ β π₯ = π₯(π₯ β π¦) + (β1)(π₯ β π¦) = (π₯ β 1)(π₯ β π¦) Answer17. (ππ β π)π β ππ + π (3π β 1)2 β 6π + 2 = (3π)2 + (1)2 β 2.3π. 1 β 6π + 2 = 9π2 + 1 β 6π β 6π + 2 = 9π2 β 12π + 3 = 9π2 β 9π β 3π + 3 = 9π(π β 1) β 3(π β 1) = (π β 1)(9π β 3) = 3(π β 1)(3π β 1) Answer18. (ππ β π)π β ππ + ππ (2π₯ β 3)2 β 8π₯ + 12 = (2π₯ β 3)2 β 4(2π₯ β 3) = (2π₯ β 3)[(2π₯ β 3) β 4] = (2π₯ β 3)(2π₯ β 7) Answer19. ππ + π β πππ β π .π3 + π β 3π2 β 3 = π(π2 + 1) β 3(π2 + 1) = (π2 + 1)(π β 3) Answer20. πππ β πππ β πππ + πππ 3ππ₯ β 6ππ¦ β 8ππ¦ + 4ππ₯ = 3π(π₯ β 2π¦) + 4π(β2π¦ + π₯) = (π₯ β 2π¦)(3π + 4π) Answer21. ππππ + ππ π + ππ π + ππ aππ₯2 + π2 π₯ + π2 π₯ + ππ = ππ₯(ππ₯ + π) + π(ππ₯ + π) = (ππ₯ + π)(ππ₯ + π)
3. 3. CLASS IX WWW.Vedantu.com RS Aggarwal solutions Answer22. ππ β ππ + ππ + π β π β π .π₯3 β π₯2 + ππ₯ + π₯ β π β 1 = π₯3 β π₯2 + π₯(π + 1) β 1(π + 1) = π₯3 β π₯2 + [(π₯ β 1)(π + 1)] = π₯2(π₯ β 1) + [(π₯ β 1)(π + 1)] = (π₯ β 1)[π₯2 + (π + 1)] = (π₯ β 1)(π₯2 + π + 1) Answer23. ππ + ππ β πππ β π 2π₯ + 4π¦ β 8π₯π¦ β 1 = 2π₯ β 1 + 4π¦ β 8π₯π¦ = 1(2π₯ β 1) β 4π¦[(β1) + 2π₯)] = 1(2π₯ β 1) β 4π¦(2π₯ β 1) = (2π₯ β 1)(1 β 4π¦) Answer24. ππ(ππ + ππ) β ππ(ππ + ππ ) aπ(π₯2 + π¦2) β π₯π¦(π2 + π2) = πππ₯2 + πππ¦2 β π2 π₯π¦ β π2 π₯π¦ = πππ₯2 β π2 π₯π¦ + πππ¦2 β π2 π₯π¦ = ππ₯(ππ₯ β ππ¦) β ππ¦[(βππ¦) + ππ₯] = ππ₯(ππ₯ β ππ¦) β ππ¦(ππ₯ β ππ¦) = (ππ₯ β ππ¦)(ππ₯ β ππ¦) Answer25. ππ + ππ(π + π) + ππ .π2 + ππ(π + 1) + π3 = π2 + ππ2 + ππ + π3 = π(π + π2) + π(π + π2 ) = (π + π)(π + π2 ) Answer26. ππ + ππ(π β ππ) β πππ .π3 + ππ(1 β 2π) β 2π2 = π3 + ππ β 2π2 π β 2π2 = π(π2 + π) β 2π(π2 + π) = (π β 2π)(π2 + π) Answer27. πππ + ππ β πππ β ππ 2π2 + ππ β 2ππ β ππ = 2π2 β 2ππ + ππ β ππ = 2π(π β π) β π(π β π) = (π β π)(2π β π) Answer28. (ππ + ππ)π + (ππ β ππ)π (ππ₯ + ππ¦)2 + (ππ₯ β ππ¦)2 = (ππ₯)2 + (ππ¦)2 + 2. ππ₯. ππ¦ + (ππ₯)2 + (ππ¦)2 β 2. ππ₯. ππ¦ = π2 π₯2 + π2 π¦2 + π2 π₯2 + π2 π¦2 + (2πππ₯π¦ β 2πππ₯π¦)- = π2 π₯2 + π2 π¦2 + π2 π₯2 + π2 π¦2 = a2(π₯2 + π¦2) + π2 (π₯2 + π¦2 ) = (π₯2 + π¦2 )(π2 + π2 ) Answer29. π(π + π β π) β ππ a(π + π β π) β ππ = π2 + ππ β ππ β ππ = π(π + π) β π(π + π) = (π + π)(π β π) Answer30. π(π β ππ β π) + πππ a(π β 2π β π) + 2ππ = π2 β 2ππ β ππ + 2ππ = π2 β ππ β 2ππ + 2ππ = π(π β π) β 2π(π β π) = (π β π)(π β 2π)
4. 4. CLASS IX WWW.Vedantu.com RS Aggarwal solutions Answer31. ππ ππ + (πππ + π)π + π .π2 π₯2 + (ππ₯2 + 1)π₯ + π = π2 π₯2 + ππ₯3 + π₯ + π = ππ₯2(π + π₯) + 1(π + π₯) = (π + π₯)(ππ₯2 + 1) Answer32. ππ(ππ + π) + π(ππ + ππ) aπ(π₯2 + 1) + π₯(π2 + π2) = πππ₯2 + ππ + π₯π2 + π₯π2 = πππ₯2 + π₯π2 + π₯π2 + ππ = ππ₯(ππ₯ + π) + π(ππ₯ + π) = (ππ₯ + π)(ππ₯ + π) Answer33. ππ β (π + π)π + ππ .π₯2 β (π + π)π₯ + ππ = π₯2 β ππ₯ β ππ₯ + ππ = π₯(π₯ β π) β π(π₯ β π) = (π₯ β π)(π₯ β π) Answer34. ππ + π ππ β π β ππ + π π .π₯2 + 1 π₯2 β 2 β 3π₯ + 3 π₯ = π₯2 + 1 π₯2 β 2. π₯. 1 π₯ β 3 (π₯ β 1 π₯ ) = (π₯ β 1 π₯ ) 2 β 3 (π₯ β 1 π₯ ) = (π₯ β 1 π₯ ) (π₯ β 1 π₯ β 3)
7. 7. CLASS IX WWW.Vedantu.com RS Aggarwal solutions = (π₯ β π¦)2 β π§2 = (π₯ β π¦ β π§)(π₯ β π¦ + π§) Answer.23. ππ + πππ + ππ β ππ + πππ β ππ .π₯2 + 2π₯π¦ + π¦2 β π2 + 2ππ β π2 = (π₯2 + 2π₯π¦ + π¦2) β (π2 β 2ππ + π2) = (π₯ + π¦)2 β (π β π)2 = {(π₯ + π¦) β (π β π)}{(π₯ + π¦) + (π β π)} = π₯ + π¦ β π + π)(π₯ + π¦ + π β π) Answer.24. ππππ β πππ + π β ππππ 25π₯2 β 10π₯ + 1 β 36π¦2 = (25π₯2 β 10π₯ + 1) β (6π¦)2 = (5π₯ β 1)2 β (6π¦)2 = (5π₯ β 1 β 6π¦)(5π₯ β 1 + 6π¦) Answer.25. π β π β ππ + ππ a β π β π2 + π2 = π β π β 1(π2 β π2) = (π β π) β 1(π β π)(π + π) = (π β π)(1 β π β π) Answer.26. ππ β ππ β πππ + πππ .π2 β π2 β 4ππ + 4π2 = (π2 β 4ππ + 4π2) β π2 = (π β 2π)2 β π2 = (π β 2π β π)(π β 2π + π) Answer.27. π β ππ + πππ β ππ 9β π2 + 2ππ β π2 = (3)2 β (π2 β 2ππ + π2) = (3)2 β (π β π)2 = {3 β (π β π)}{3 + (π β π)} = (3 β π + π)(3 + π β π) Answer.28. ππ β πππ β π + π .π₯3 β 5π₯2 β π₯ + 5 = π₯2(π₯ β 5) β 1(π₯ β 5) = (π₯ β 5)(π₯2 β 1) = (π₯ β 5)(π₯ β 1)(π₯ + 1) Answer.29. π + πππ β (ππ + ππ) 1+2ππ β (π2 + π2) = 1 β {(β2ππ} + (π2 + π2)} = 1 β (π β π)2 = {1 β (π β π)}{1 + (π β π)} = (1 β π + π)(1 + π β π) Answer.30. πππ + ππ + π β ππππ 9π2 + 6π + 1 β 36π2 = (9π2 + 6π + 1) β (6π)2 = (3π + 1)2 β (6π)2 = (3π + 1 β 6π)(3π + 1 + 6π) Answer.31. ππ βππ + ππ β π .π₯2 βπ¦2 + 6π¦ β 9 = π₯2 β (π¦2 β 6π¦ + 9) = π₯2 β(π¦ β 3)2 = {π₯ β (π¦ β 3)}{π₯ + (π¦ β 3)} = (π₯ β π¦ + 3)(π₯ + π¦ β 3)
8. 8. CLASS IX WWW.Vedantu.com RS Aggarwal solutions Answer.32. πππ β πππ β ππ β ππ 4π₯2 β 9π¦2 β 2π₯ β 3π¦ = (2π₯)2 β (3π¦)2 β 1(2π₯ + 3π¦) = (2π₯ β 3π¦)(2π₯ + 3π¦) β 1(2π₯ + 3π¦) = (2π₯ + 3π¦)(2π₯ β 3π¦ β 1) Answer.33. πππ + ππ β ππ β ππππ 9π2 + 3π β 8π β 64π2 = (3π)2 β (8π)2 + 3π β 8π = (3π β 8π)(3π + 8π) + 1(3π β 8π) = (3π β 8π)(3π + 8π + 1) Answer.34. ππ + π ππ β π .π₯2 + 1 π₯2 β 3 = (π₯2 + 1 π₯2 β 2. π₯. 1 π₯ ) β 1 = (π₯ + 1 π₯ ) 2 β 12 = (π₯ + 1 π₯ β 1) (π₯ + 1 π₯ + 1) Answer.35. ππ β π + π ππ β ππ .π₯2 β 2 + 1 π₯2 β π¦2 = (π₯2 β 2 + 1 π₯2 ) β (π¦)2 = (π₯ β 1 π₯ ) 2 β (π¦)2 = (π₯ β 1 π₯ β π¦)(π₯ β 1 π₯ + π¦) Answer.36. ππ + π ππ .π₯4 + 4 π₯4 = π₯4 + 4 π₯4 + 2. π₯2 . 2 π₯2 β 2. π₯2 . 2 π₯2 = (π₯4 + 4 π₯4 + 2. π₯2 . 2 π₯2 ) β 4 =(π₯2 + 2 π₯2 ) β (2)2 = (π₯2 + 2 π₯2 β 2) (π₯2 + 2 π₯2 + 2) Answer.37. ππ β π .π₯8 β 1 = (π₯4)2 β 12 = (π₯4 β 1)(π₯4 + 1) = (π₯2 β 1)(π₯2 + 1)(π₯4 + 1 + 2. π₯2 . 1 β 2. π₯2 . 1) = (π₯ β 1)(π₯ + 1)(π₯2 + 1) {(π₯2 + 1)2 β (β2π₯) 2 } = (π₯ β 1)(π₯ + 1)(π₯2 + 1)(π₯2 + 1 β β2π₯)(π₯2 + 1 + β2π₯) Answer.38. ππππ β π 16π₯4 β 1 = (4π₯2 )2 β 12 = (4π₯2 β 1)(4π₯2 + 1) = (2π₯ β 1)(2π₯ + 1)(4π₯2 + 1) Answer.39. ππππ β ππ 81π₯4 β π¦4 = (9π₯2)2 β (π¦2)2 = (9π₯2 β π¦2 )(9π₯2 + π¦2 ) = (3π₯ β π¦)(3π₯ + π¦)(9π₯2 + π¦2 ) Answer.40. ππ β πππ .π₯4 β 625 = (π₯2)2 β (25)2 = (π₯2 β 25)(π₯2 + 25) = (π₯ β 5)(π₯ + 5)(π₯2 + 25)
9. 9. CLASS IX WWW.Vedantu.com RS Aggarwal solutions
11. 11. CLASS IX WWW.Vedantu.com RS Aggarwal solutions .π₯2 + 3β3π₯ + 6 = π₯2 + 2β3π₯ + β3π₯ + 6 = π₯(π₯ + 2β3) + β3(π₯ + 2β3) = (π₯ + 2β3)(π₯ + β3) Answer.12. ππ + πβππ + ππ .π₯2 + 6β6π₯ + 48 = π₯2 + 4β6π₯ + 2β6π₯ + 48 = π₯(π₯ + 4β6) + 2β6(π₯ + 4β6) = (π₯ + 4β6)(π₯ + 2β6) Answer.13. ππ + πβππ + ππ .π₯2 + 5β5π₯ + 30 = π₯2 + 3β5π₯ + 2β5π₯ + 30 = π₯(π₯ + 3β5) + 2β5(π₯ + 3β5) = (π₯ + 3β5)(π₯ + 2β5) Answer.14. ππ β πππ β πππ .π₯2 β 24π₯ β 180 = π₯2 β 30π₯ + 6π₯ β 180 = π₯(π₯ β 30) + 6(π₯ β 30) = (π₯ β 30)(π₯ + 6) Answer.15. ππ β πππ β πππ .π₯2 β 32π₯ β 105 = π₯2 β 35π₯ + 3π₯ β 105 = π₯(π₯ β 35) + 3(π₯ β 35) = (π₯ β 35)(π₯ + 3) Answer.16. ππ β πππ β ππ .π₯2 β 11π₯ β 80 = π₯2 β 16π₯ + 5π₯ β 80 = π₯(π₯ β 16) + 5(π₯ β 16) = (π₯ β 16)(π₯ + 5) Answer.17. π β π β ππ .6 β π₯ β π₯2 = βπ₯2 β 3π₯ + 2π₯ + 6 = βπ₯(π₯ + 3) + 2(π₯ + 3) = (βπ₯ + 2)(π₯ + 3) = (2 β π₯)(π₯ + 3) Answer.18. ππ β βππ β π .π₯2 β β3π₯ β 6 = π₯2 β 2β3π₯ + β3π₯ β 6 = π₯(π₯ β 2β3) + β3(π₯ β 2β3) = (π₯ β 2β3)(π₯ + β3) Answer.19. ππ + ππ β ππ .40 + 3π₯ β π₯2 = βπ₯2 + 8π₯ β 5π₯ + 40 =βπ₯(π₯ β 8) β 5(π₯ β 8) = (π₯ β 8)(βπ₯ β 5) = (8 β π₯)(π₯ + 5)
13. 13. CLASS IX WWW.Vedantu.com RS Aggarwal solutions 2π₯2 + 11π₯ β 21 = 2π₯2 + 14π₯ β 3π₯ β 21 = 2π₯(π₯ + 7) β 3(π₯ + 7) = (π₯ + 7)(2π₯ β 3) Answer.31. ππππ + ππ β π 15π₯2 + 2π₯ β 8 = 15π₯2 + 12π₯ β 10π₯ β 8 = 3π₯(5π₯ + 4) β 2(5π₯ + 4) = (5π₯ + 4)(3π₯ β 2) Answer.32. ππππ + ππ β π 21π₯2 + 5π₯ β 6 = 21π₯2 + 14π₯ β 9π₯ β 6 = 7π₯(3π₯ + 2) β 3(3π₯ + 2) = (3π₯ + 2)(7π₯ β 3) Answer.33. ππππ β πππ + ππ 24π₯2 β 41π₯ + 12 = 24π₯2 β 32π₯ β 9π₯ + 12 = 8π₯(3π₯ β 4) β 3(3π₯ β 4) = (3π₯ β 4)(8π₯ β 3) Answer.34. πππ β πππ + π 3π₯2 β 14π₯ + 8 = 3π₯2 β 12π₯ β 2π₯ + 8 = 3π₯(π₯ β 4) β 2(π₯ β 4) = (π₯ β 4)(3π₯ β 2) Answer.35. πππ + ππ β ππ 2π₯2 + 3π₯ β 90 = 2π₯2 + 15π₯ β 12π₯ β 90 = π₯(2π₯ + 15) β 6(2π₯ + 15) = (2π₯ + 15)(π₯ β 6) Answer.36. βπππ + ππ β πβπ β5π₯2 + 2π₯ β 3β5 = β5π₯2 + 5π₯ β 3π₯ β 3β5 = β5π₯(π₯ + β5) β 3(π₯ + β5) = (π₯ + β5)(β5π₯ β 3) Answer.37. πβπππ + π β πβπ 2β3π₯2 + π₯ β 5β3 = 2β3π₯2 + 6π₯ β 5π₯ β 5β3 = 2β3π₯(π₯ + β3) β 5(π₯ + β3) = (π₯ + β3)(2β3π₯ β 5) Answer.38. πππ + πβπππ + π 7π₯2 + 2β14π₯ + 2 = 7π₯2 + β14π₯ + β14π₯ + 2 = β7π₯(β7π₯ + β2) + β2(β7π₯ + β2) = (β7π₯ + β2)(β7π₯ + β2) Answer.39. πβπππ β πππ + πβπ 6β3π₯2 β 47π₯ + 5β3 = 6β3π₯2 β 45π₯ β 2π₯ + 5β3 = 3β3π₯(2π₯ β 5β3) β 1(2π₯ β 5β3) = (2π₯ β 5β3)(3β3π₯ β 1) Answer.40. πβπππ + πππ + πβπ
14. 14. CLASS IX WWW.Vedantu.com RS Aggarwal solutions 5β5π₯2 + 20π₯ + 3β5 = 5β5π₯2 + 15π₯ + 5π₯ + 3β5 = 5π₯(β5π₯ + 3) + β5(β5π₯ + 3) = (β5π₯ + 3)(5π₯ + β5) Answer.41. βπππ + πππ + πβπ .β3π₯2 + 10π₯ + 8β3 = β3π₯2 + 6π₯ + 4π₯ + 8β3 = β3π₯(π₯ + 2β3) + 4(π₯ + 2β3) = (π₯ + 2β3)(β3π₯ + 4) Answer.42. βπππ + ππ + βπ .β2π₯2 + 3π₯ + β2 = β2π₯2 + 2π₯ + π₯ + β2 = β2π₯(π₯ + β2) + 1(π₯ + β2) = (π₯ + β2)(β2π₯ + 1) Answer.43. πππ + πβππ + π 2π₯2 + 3β3π₯ + 3 = 2π₯2 + 2β3π₯ + β3π₯ + 3 = 2π₯(π₯ + β3) + β3(π₯ + β3) = (π₯ + β3)(2π₯ + β3) Answer.44. ππππ β π β ππ 15π₯2 β π₯ β 28 = 15π₯2 β 21π₯ + 20π₯ β 28 = 3π₯(5π₯ β 7) + 4(5π₯ β 7) = (5π₯ β 7)(3π₯ + 4) Answer.45. πππ β ππ β ππ 6π₯2 β 5π₯ β 21 = 6π₯2 β 14π₯ + 9π₯ β 21 = 2π₯(3π₯ β 7) + 3(3π₯ β 7) = (3π₯ β 7)(2π₯ + 3) Answer.46. πππ β ππ β ππ 2π₯2 β 7π₯ β 15 = 2π₯2 β 10π₯ + 3π₯ β 15 = 2π₯(π₯ β 5) + 3(π₯ β 5) = (π₯ β 5)(2π₯ + 3) Answer.47. πππ β πππ β ππ 5π₯2 β 16π₯ β 21 = 5π₯2 β 21π₯ + 5π₯ β 21 = π₯(5π₯ β 21) + 1(5π₯ β 21) = (5π₯ β 21)(π₯ + 1) Answer.48. πππ β πππ β ππ 6π₯2 β 11π₯ β 35 = 6π₯2 β 21π₯ + 10π₯ β 35 = 3π₯(2π₯ β 7) + 5(2π₯ β 7) = (2π₯ β 7)(3π₯ + 5) Answer.49. πππ β ππ β ππ 9π₯2 β 3π₯ β 20 = 9π₯2 β 15π₯ + 12π₯ β 20 = 3π₯(3π₯ β 5) + 4(3π₯ β 5) = (3π₯ β 5)(3π₯ + 4) Answer.50. ππππ β ππ β π 10π₯2 β 9π₯ β 7 = 10π₯2 β 14π₯ + 5π₯ β 7 = 2π₯(5π₯ β 7) + 1(5π₯ β 7)
15. 15. CLASS IX WWW.Vedantu.com RS Aggarwal solutions = (5π₯ β 7)(2π₯ + 1) Answer.51. ππ β ππ + π ππ .π₯2 β 2π₯ + 7 16 = 16π₯2 β 32π₯ + 7 16 = 16π₯2 β 28π₯ β 4π₯ + 7 16 = 4π₯ ( 4π₯ β 7 )β 1 ( 4π₯ β 7 ) 16 = ( 4π₯ β 7 )( 4π₯ β 1 ) 16 = ( 4π₯ β 7 16 ) (4π₯ β 1) = ( π₯ 4 β 7 16 ) (4π₯ β 1) Answer.52. π π ππ β ππ β π . 1 3 π₯2 β 2π₯ β 9 = π₯2 β 6π₯ β 27 3 = π₯2 β 9π₯ + 3π₯ β 27 3 = π₯ ( π₯ β 9 ) + 3 ( π₯ β 9 ) 3 = ( π₯ β 9 ) ( π₯ + 3 ) 3 = ( π₯ + 3 3 ) (π₯ β 9) = (1 + π₯ 3 ) (π₯ β 9) Answer.53. ππ + ππ ππ π + π ππ .π₯2 + 12 35 π₯ + 1 35 = 35π₯2 + 12π₯ + 1 35 = 35π₯2 + 7π₯ + 5π₯ + 1 35 = 7π₯ ( 5π₯+ 1 ) + 1 ( 5π₯+ 1 ) 35 = ( 5π₯+ 1 ) ( 7π₯ + 1 ) 35 = (5π₯ + 1) ( 7π₯ + 1 35 ) = (5π₯ + 1) ( π₯ 5 + 1 35 ) Answer.54. ππππ β ππ + π ππ 21π₯2 β 2π₯ + 1 21 = 21π₯2 β π₯ β π₯ + 1 21 = 21π₯ (π₯ β 1 21 ) β 1(π₯ β 1 21 ) = (π₯ β 1 21 ) (21π₯ β 1) Answer.55. π π ππ + πππ + ππ . 3 2 π₯2 + 16π₯ + 10 = 3 2 π₯2 + π₯ + 15π₯ + 10 = π₯ 2 (3π₯ + 2) + 5(3π₯ + 2) = (3π₯ + 2) ( π₯ 2 + 5)
16. 16. CLASS IX WWW.Vedantu.com RS Aggarwal solutions Answer.56. π π ππ β ππ π π β ππ . 2 3 π₯2 β 17 3 π₯ β 28 = 2π₯2 β 17π₯β 84 3 = 2π₯2 β 24π₯ +7π₯ β 84 3 = 2π₯ ( π₯ β 12 ) + 7 ( π₯ β 12 ) 3 = ( π₯ β 12 ) ( 2π₯ +7 ) 3 = ( π₯ β 12 3 )(2π₯ + 7) = ( π₯ 3 β 1 4 ) (2π₯ + 7) Answer.57. π π ππ β ππ π π + π . 3 5 π₯2 β 19 5 π₯ + 4 = 3π₯2 β 19π₯ + 20 5 = 3π₯2 β 15π₯ β 4π₯ + 20 5 = 3π₯ ( π₯ β 5 ) β 4 ( π₯ β 5 ) 5 = ( π₯ β 5 ) ( 3π₯ β 4 ) 5 = ( π₯ β 5 5 )(3π₯ β 4) = ( π₯ 5 β 1) (3π₯ β 4) Answer.58. πππ β π + π π 2π₯2 β π₯ + 1 8 = 16π₯2 β 8π₯ + 1 8 = 16π₯2 β 4π₯ β 4π₯ + 1 8 = 4π₯ ( 4π₯ β 1 ) β 1( 4π₯ β 1) 8 = ( 4π₯ β 1 8 )(4π₯ β 1) = ( π₯ 2 β 1 8 ) (4π₯ β 1) Answer.59. π(π + π)π β π(π + π) β π Lππ‘ (π₯ + π¦) = π 2(π₯ + π¦)2 β 9(π₯ + π¦) β 5 = 2π2 β 9π β 5 = 2π2 β 10π + π β 5 = 2π(π β 5) + 1(π β 5) = (π β 5)(2π + 1) βΈ« 2(π₯ + π¦)2 β 9(π₯ + π¦) β 5 = (π₯ + π¦ β 5){2(π₯ + π¦) + 1} [β΅ π = (π₯ + π¦) ] = (π₯ + π¦ β 5)(2π₯ + 2π¦ + 1) Answer.60. π(ππ β π)π β π(ππ β π) β ππ Lππ‘ (2π β π) = π 9(2π β π)2 β 4(2π β π) β 13 = 9π2 β 4π β 13 = 9π2 β 13π + 9π β 13 = π(9π β 13) + 1(9π β 13) = (9π β 13)(π + 1) βΈ« 9(2π β π)2 β 4(2π β π) β 13 = {9(2π β π) β 13}{(2π β π) + 1} [β΅ π = (2π β π) ] = (18π β 9π β 13)(2π β π + 1) Answer.61. π(π β ππ)π β ππ(π β ππ) + ππ
17. 17. CLASS IX WWW.Vedantu.com RS Aggarwal solutions Lππ‘ (π₯ β 2π¦) = π 7(π₯ β 2π¦)2 β 25(π₯ β 2π¦) + 12 = 7π2 β 25π + 12 = 7π2 β 21π β 4π + 12 = 7π(π β 3) β 4(π β 3) = (π β 3)(7π β 4) βΈ« 7(π₯ β 2π¦)2 β 25(π₯ β 2π¦) + 12 = (π₯ β 2π¦ β 3){7(π₯ β 2π¦) β 4} [β΅ π = (π₯ β 2π¦) ] = (π₯ β 2π¦ β 3)(7π₯ β 14π¦ β 4) Answer.62. ππ (ππ + π π ) π β (ππ + π π ) β π Lππ‘ (3π₯ + 1 π₯ ) = π 10 (3π₯ + 1 π₯ ) 2 β (3π₯ + 1 π₯ ) β 3 = 10π2 β π β 3 = 10π2 β 6π + 5π β 3 = 2π(5π β 3) + 1(5π β 3) = (5π β 3)(2π + 1) βΈ« 10 (3π₯ + 1 π₯ ) 2 β (3π₯ + 1 π₯ ) β 3 = {5 (3π₯ + 1 π₯ ) β 3}{2 (3π₯ + 1 π₯ ) + 1}[β΅ π = (3π₯ + 1 π₯ )] = (15π₯ + 5 π₯ β 3) (6π₯ + 2 π₯ + 1) Answer.63. π (ππ β π π ) π + π(ππ β π π ) β ππ Lππ‘ (2π₯ β 3 π₯ ) = π 6(2π₯ β 3 π₯ ) 2 + 7 (2π₯ β 3 π₯ ) β 20 = 6π2 + 7π β 20 = 6π2 + 15π β 8π β 20 = 3π(2π + 5) β 4(2π + 5) = (2π + 5)(3π β 4) βΈ« 6 (2π₯ β 3 π₯ ) 2 + 7 (2π₯ β 3 π₯ ) β 20 = {2 (2π₯ β 3 π₯ ) + 5}{3 (2π₯ β 3 π₯ ) β 4}[β΅ π = (2π₯ β 3 π₯ )] = (4π₯ β 6 π₯ + 5) (6π₯ β 9 π₯ β 4) Answer.64. (π + ππ)π + πππ(π + ππ) + πππ Lππ‘ (π + 2π) = π .(π + 2π)2 + 101(π + 2π) + 100 = π2 + 101π + 100 = π2 + 100π + π + 100 = π(π + 100) + 1(π + 100) = (π + 100)(π + 1) βΈ« (π + 2π)2 + 101(π + 2π) + 100 = {(π + 2π) + 100}{(π + 2π) + 1} [β΅ π = (π + 2π) ] = (π + 2π + 100)(π + 2π + 1) Answer.65. πππ + πππ β π Lππ‘ π₯2 = π¦ 4π₯4 + 7π₯2 β 2 = 4π¦2 + 7π¦ β 2 = 4π¦2 + 8π¦ β π¦ β 2 = 4π¦(π¦ + 2) β 1(π¦ + 2) = (π¦ + 2)(4π¦ β 1) βΈ« 4π₯4 + 7π₯2 β 2 = (π₯2 + 2)(4π₯2 β 1) = (π₯2 + 2){(2π₯)2 β 12} = (π₯2 + 2)(2π₯ β 1)(2π₯ + 1)
18. 18. CLASS IX WWW.Vedantu.com RS Aggarwal solutions Answer.66. {(πππ)π β π} .{(999)2 β 1} = {(999)2 β (1)2 } = (999 β 1)(999 + 1) = 998 Γ 1000 = 998000
19. 19. CLASS IX WWW.Vedantu.com RS Aggarwal solutions EXERCISE β 3D Formula Used -: (π + π + π)π = ππ + ππ + ππ + πππ + πππ + πππ Answer.1. (i) (π + ππ + ππ)π .(π + 2π + 5π)2 = (π)2 + (2π)2 + (5π)2 + 2(π)(2π) + 2(2π)(5π) + 2(5π)(π) = π2 + 4π2 + 25π2 + 4ππ + 20ππ + 10ππ (ii)(ππ β π + π)π .(2π β π + π)2 = (2π)2 + (βπ)2 + (π)2 + 2(2π)(βπ) + 2(βπ)(π) + 2(π)(2π) = 4π2 + π2 + π2 β 4ππ β 2ππ + 4ππ (iii) (π β ππ β ππ)π .(π β 2π β 3π)2 = (π)2 + (β2π)2 + (β3π)2 + 2(π)(β2π) + 2(β2π)(β3π) + 2(β3π)(π) = π2 + 4π2 + 9π2 β 4ππ + 12ππ β 6ππ Answer.2. (i) (ππ β ππ β ππ)π .(2π β 5π β 7π)2 = (2π)2 + (β5π)2 + (β7π)2 + 2(2π)(β5π) + 2(β5π)(β7π) + 2(β7π)(2π) = 4π2 + 25π2 + 49π2 β 20ππ + 70ππ β 28ππ (ii) (βππ + ππ β ππ)π .(β3π + 4π β 5π)2 = (β3π)2 + (4π)2 + (β5π)2 + 2(β3π)(4π) + 2(4π)(β5π) + 2(β5π)(β3π) = 9π2 + 16π2 + 25π2 β 24ππ β 40ππ + 30ππ (iii)( π π π β π π π + π) π .( 1 2 π β 1 4 π + 2) 2 = ( 1 2 π) 2 + (β 1 4 π) 2 + (2)2 + 2 ( 1 2 π) (β 1 4 π) + 2 (β 1 4 π) (2) + 2(2) ( 1 2 π) = π2 4 + π2 16 + 4 β ππ 4 β π + 2π Answer.3. πππ + πππ + ππππ + ππππ β ππππ β ππππ 4π₯2 + 9π¦2 + 16π§2 + 12π₯π¦ β 24π¦π§ β 16π§π₯ = (2π₯)2 + (3π¦)2 + (β4π§)2 + {2(2π₯)(3π¦)} + {2(3π¦)(β4π§)} + {2(β4π§)(2π₯)} = {(2π₯) + (3π¦) + (β4π§)}2 = (2π₯ + 3π¦ β 4π§)2 = (2π₯ + 3π¦ β 4π§)(2π₯ + 3π¦ β 4π§) Answer.4. πππ + ππππ + πππ β ππππ + ππππ β ππππ 9π₯2 + 16π¦2 + 4π§2 β 24π₯π¦ + 16π¦π§ β 12π§π₯ = (β3π₯)2 + (4π¦)2 + (2π§)2 + {2(β3π₯)(4π¦)} + {2(4π¦)(2π§)} + {2(2π§)(β3π₯)} = {(β3π₯) + (4π¦) + (2π§)}2 = (β3π₯ + 4π¦ + 2π§)2 = (β3π₯ + 4π¦ + 2π§)(β3π₯ + 4π¦ + 2π§) Answer.5. ππππ + πππ + πππ β ππππ β ππππ + ππππ 25π₯2 + 4π¦2 + 9π§2 β 20π₯π¦ β 12π¦π§ + 30π§π₯ = (5π₯)2 + (β2π¦)2 + (3π§)2 + {2(5π₯)(β2π¦)} + {2(β2π¦)(3π§)} + {2(3π§)(5π₯)} = {(5π₯) + (β2π¦) + (3π§)}2 = (5π₯ β 2π¦ + 3π§)2 = (5π₯ β 2π¦ + 3π§)(5π₯ β 2π¦ + 3π§) Answer.6. ππππ + πππ + πππ β ππππ β ππππ + ππππ 16π₯2 + 4π¦2 + 9π§2 β 16π₯π¦ β 12π¦π§ + 24π§π₯ = (4π₯)2 + (β2π¦)2 + (3π§)2 + {2(4π₯)(β2π¦)} + {2(β2π¦)(3π§)} + {2(3π§)(4π₯)} = {(4π₯) + (β2π¦) + (3π§)}2 = (4π₯ β 2π¦ + 3π§)2
20. 20. CLASS IX WWW.Vedantu.com RS Aggarwal solutions = (4π₯ β 2π¦ + 3π§)(4π₯ β 2π¦ + 3π§) Answer.7. (i) (ππ)π (99)2 = (100 β 1)2 = (100)2 + (1)2 β 2 Γ 100 Γ 1 = 10000 + 1 β 200 = 9801 (ii) (πππ)π (995)2 = (1000 β 5)2 = (1000)2 + (5)2 β 2 Γ 1000 Γ 5 = 1000000 + 25 β 10000 = 990025 (iii) (πππ)π (107)2 = (100 + 7)2 = (100)2 + (7)2 + 2 Γ 100 Γ 7 = 10000 + 49 + 1400 = 11449
21. 21. CLASS IX WWW.Vedantu.com RS Aggarwal solutions EXERCISE -3E Formulae Used: (i) (π + π)π = ππ + ππ + πππ(π + π) (ii)(π β π)π = ππ β ππ β πππ(π β π) Answer.1. (i) (ππ + π)π .(3π₯ + 2)3 = (3π₯)3 + (2)3 + 3(3π₯)(2)(3π₯ + 2) = 27π₯3 + 8 + 18π₯(3π₯) + 18π₯(2) = 27π₯3 + 8 + 54π₯2 + 36π₯ (ii) (ππ + π ππ ) π .(3π + 1 4π ) 3 = (3π)3 + ( 1 4π ) 3 + 3(3π) ( 1 4π ) (3π + 1 4π ) = 27π3 + 1 64π3 + 9π 4π (3π) + 9π 4π ( 1 4π ) = 27π3 + 1 64π3 + 27π2 4π + 9π 16π2 (iii) (π + π π π) π .(1 + 2 3 π) 3 = (1)3 + ( 2 3 π) 3 + 3(1) ( 2 3 π) (1 + 2 3 π) = 13 + 8 27 π3 + 2π(1) + 2π ( 2 3 π) = 1 + 8 27 π3 + 2π + 4 3 π2 = 1 + 8 27 π3 + 4 3 π2 + 2π Answer.2. (i)(ππ β ππ)π (5π β 3π)3 = (5π)3 β (3π)3 β 3(5π)(3π)(5π β 3π) = 125π3 β 27π3 β 45ππ(5π) + 45ππ(3π) = 125π3 β 27π3 β 225π2 π + 135ππ2 (ii)(ππ β π π ) π .(3π₯ β 5 π₯ ) 3 = (3π₯)3 β ( 5 π₯ ) 3 β 3(3π₯) ( 5 π₯ ) (3π₯ β 5 π₯ ) = 27π₯3 β 125 π₯3 β 45(3π₯) + 45 ( 5 π₯ ) = 27π₯3 β 125 π₯3 β 135π₯ + 225 π₯ (iii)( π π π β π) π .( 4 5 π β 2) 3 = ( 4 5 π) 3 β (2)3 β 3 ( 4 5 π) (2) ( 4 5 π β 2) = 64 125 π3 β 8 β 24 5 π ( 4 5 π) + 24 5 π(2) = 64 125 π3 β 8 β 96 25 π2 + 48 5 π Answer.3. πππ + ππππ + ππππ π + πππππ 8π3 + 27π3 + 36π2 π + 54ππ2 = (2π)3 + (3π)3 + 3(2π)(3π)(2π) + 3(2π)(3π)(3π)
22. 22. CLASS IX WWW.Vedantu.com RS Aggarwal solutions = (2π)3 + (3π)3 + 3(2π)(3π)(2π + 3π) = (2π + 3π)3 = (2π + 3π)(2π + 3π)(2π + 3π) Answer.4. ππππ β ππππ β πππππ π + ππππππ 64π3 β 27π3 β 144π2 π + 108ππ2 = (4π)3 β (3π)3 β 3(4π)(3π)(4π) + 3(4π)(3π)(3π) = (4π)3 β (3π)3 β 3(4π)(3π)(4π β 3π) = (4π β 3π)3 = (4π β 3π)(4π β 3π)(4π β 3π) Answer.5. π + ππ πππ ππ + ππ π + ππππ ππ 1 + 27 125 π3 + 9π 5 + 27π2 25 = (1)3 + ( 3 5 π) 3 + 3(1) ( 3 5 π) (1) + 3(1) ( 3 5 π) ( 3 5 π) = (1)3 + ( 3 5 π) 3 + 3(1) ( 3 5 π) (1 + 3 5 π) = (1 + 3 5 π) 3 = (1 + 3 5 π) (1 + 3 5 π) (1 + 3 5 π) Answer.6. πππππ β ππππ β πππππ π + ππππππ 125π₯3 β 27π¦3 β 225π₯2 π¦ + 135π₯π¦2 = (5π₯)3 β (3π¦)3 β 3(5π₯)(3π¦)(5π₯) + 3(5π₯)(3π¦)(3π¦) = (5π₯)3 β (3π¦)3 β 3(5π₯)(3π¦)(5π₯ β 3π¦) = (5π₯ β 3π¦)3 = (5π₯ β 3π¦)(5π₯ β 3π¦)(5π₯ β 3π¦) Answer.7. ππ ππ β πππ πππ + ππππ π β ππ .π3 π₯3 β 3π2 ππ₯2 + 3ππ2 π₯ β π3 = π3 π₯3 β π3 β 3π2 ππ₯2 + 3ππ2 π₯ = (ππ₯)3 β (π)3 β 3(ππ₯)(π)(ππ₯) + 3(ππ₯)(π)(π) = (ππ₯)3 β (π)3 β 3(ππ₯)(π)(ππ₯ β π) = (ππ₯ β π)3 = (ππ₯ β π)(ππ₯ β π)(ππ₯ β π) Answer.8. ππ πππ ππ β ππ ππ ππ + ππ π π β π . 64 125 π3 β 96 25 π2 + 48 5 π β 8 = 64 125 π3 β 8 β 96 25 π2 + 48 5 π = ( 4 5 π) 3 β (2)3 β 3 ( 4 5 π) (2) ( 4 5 π) + 3 ( 4 5 π) (2)(2) = ( 4 5 π) 3 β (2)3 β 3 ( 4 5 π) (2) ( 4 5 π β 2) = ( 4 5 π β 2) 3 = ( 4 5 π β 2) ( 4 5 π β 2) ( 4 5 π β 2) Answer.9. ππ β πππ(π β π) β ππ
23. 23. CLASS IX WWW.Vedantu.com RS Aggarwal solutions .π3 β 12π(π β 4) β 64 = (π)3 β (4)3 β 3(π)(4)(π β 4) = (π β 4)3 = (π β 4)(π β 4)(π β 4) Answer.10. (i) (πππ)π (103)3 = (100 + 3)3 = (100)3 + (3)3 + 3 Γ 100 Γ 3 Γ (100 + 3) = 1000000 + 27 + (900 Γ 103) = 1000027 + 92700 = 1092727 (ii) (ππ)π (99)3 = (100 β 1)3 = (100)3 β (1)3 β 3 Γ 100 Γ 1 Γ (100 β 1) = 1000000 β 1 β (300 Γ 99) = 999999 β 29700 = 970299
24. 24. CLASS IX WWW.Vedantu.com RS Aggarwal solutions EXERCISE β 3F Formulae Used: (i) (ππ + ππ) = (π + π)(ππ β ππ + ππ) (ii) (ππ β ππ) = (π β π)(ππ + ππ + ππ ) Answer.1. ππ + ππ .π₯3 + 27 = (π₯)3 + (3)3 = (π₯ + 3)(π₯2 β 3π₯ + 9) Answer.2. ππππ + ππππ .27π3 + 64π3 = (3π)3 + (4π)3 =(3π + 4π)(9π2 β 12ππ + 16π2 ) Answer.3. πππππ + π π .125π3 + 1 8 = (5π)3 + ( 1 8 ) 3 = (5π + 1 8 ) (25π2 β 5π 2 + 1 4 ) Answer.4. πππππ + π πππ .216π₯3 + 1 125 = (6π₯)3 + ( 1 5 ) 3 = (6π₯ + 1 5 ) (36π₯2 β 6π₯ 5 + 1 25 ) Answer.5. ππππ + πππ .16π₯4 + 54π₯ = 2π₯(8π₯3 + 27) = 2π₯{(2π₯)3 + (3)3 } = 2π₯(2π₯ + 3)(4π₯2 β 6π₯ + 6) Answer.6. πππ + ππππ .7π3 + 56π3 = 7(π3 + 8π3) = 7{(π)3 + (2π)3 } = 7(π + 2π)(π2 β 2ππ + 4π2 ) Answer.7. ππ + ππ .π₯5 + π₯2 = π₯2 (π₯3 + 1) = π₯2 {(π₯)3 + (1)3 } = π₯2 (π₯ + 1)(π₯2 β π₯ + 1) Answer.8. ππ + π. πππ .π3 + 0.008 = (π)3 + (0.2)3 = (π + 0.2)(π2 β 0.2π + 0.04) Answer.9. π β ππππ .1 β 27π3 = (1)3 β (3π)3 = (1 β 3π)(1 + 3π + 9π2 ) Answer.10. ππππ β πππ .64π3 β 343 = (4π)3 β (7)3 = (4π β 7)(16π2 + 28π + 49)
25. 25. CLASS IX WWW.Vedantu.com RS Aggarwal solutions Answer.11. ππ β πππ .π₯3 β 512 = (π₯)3 β (8)3 = (π₯ β 8)(π₯2 + 8π₯ + 64) Answer.12. ππ β π. πππ .π3 β 0.064 = (π)3 β (0.4)3 = (π β 0.4)(π2 + 0.4π + 0.16) Answer.13. πππ β π ππππ .8π₯3 β 1 27π¦3 = (2π₯)3 β ( 1 3π¦ ) 3 = (2π₯ β 1 3π¦ ) (4π₯2 + 2π₯ 3π¦ + 1 9π¦2 ) Answer.14. ππ πππ β πππ . π₯3 216 β 8π¦3 = ( π₯ 6 ) 3 β (2π¦)3 = ( π₯ 6 + 2π¦) ( π₯2 36 + 2π₯π¦ 6 + 4π¦2) = ( π₯ 6 + 2π¦) ( π₯2 36 + π₯π¦ 3 + 4π¦2) Answer.15. π β ππππ .π₯ β 8π₯π¦3 = π₯(1 β 8π¦3 ) = π₯{(1)3 β (2π¦)3 } = π₯(1 β 2π¦)(1 + 2π¦ + 4π¦2 ) Answer.16. ππππ β ππππ .32π₯4 β 500π₯ = 4π₯(8π₯3 β 125) = 4π₯{(2π₯)3 β (5)3 } = 4π₯(2π₯ β 5)(4π₯2 + 10π₯ + 25) Answer.17. πππ π β ππππ ππ 3π7 π β 81π4 π4 = 3π4 π(π3 β 27π3 ) = 3π4 π{(π)3 β (3π)3 } = 3π4 π(π β 3π)(π2 + 3ππ + 9π2 ) Answer.18. ππ ππ β ππ .π₯4 π¦4 β π₯π¦ = π₯π¦(π₯3 π¦3 β 1) = π₯π¦{(π₯π¦)3 β (1)3 } = π₯π¦(π₯π¦ β 1)(π₯2 π¦2 + π₯π¦ + 1) Answer.19. πππ ππ β ππ 8π₯2 π¦3 β π₯5 = π₯2(8π¦3 β π₯3) = π₯2 {(2π¦)3 β (π₯)3 } = π₯2 (2π¦ β π₯)(4π¦2 + 2π₯π¦ + π₯2 ) Answer.20. ππππ β πππ .1029 β 3π₯3 = 3(343 β π₯3 ) = 3{(7)3 β (π₯)3}
26. 26. CLASS IX WWW.Vedantu.com RS Aggarwal solutions = 3(7 β π₯)(49 + 7π₯ + π₯2 ) Answer.21. ππ β πππ .π₯6 β 729 = (π₯2 )3 β (9)3 = (π₯2 β 9){(π₯2)2 + 9π₯2 + 81)} = (π₯ β 3)(π₯ + 3)(π₯4 + 9π₯2 + 81) = (π₯ β 3)(π₯ + 3){(π₯4 + 9π₯2 + 9π₯2 + 81) β (3π₯)2} = (π₯ β 3)(π₯ + 3){(π₯2 + 9)2 β (3π₯)2} = (π₯ β 3)(π₯ + 3)(π₯2 + 3π₯ + 9)(π₯2 β 3π₯ + 9) Answer.22. ππ β ππ .π₯9 β π¦9 = (π₯3)3 β (π¦3)3 = (π₯3 β π¦3){(π₯3)2 + π₯3 π¦3 + (π¦3)2} = {(π₯)3 β (π¦)3 }(π₯6 + π₯3 π¦3 + π¦6 ) = (π₯ β π¦)(π₯2 + π₯π¦ + π¦2 )(π₯6 + π₯3 π¦3 + π¦6 ) Answer.23. (π + π)π β (π β π)π Lππ‘ (π + π) = π₯ πππ (π β π) = π¦ .(π₯3 β π¦3) = (π₯ β π¦)(π₯2 + π₯π¦ + π¦2 ) Nππ€, ππ’π‘π‘πππ π£πππ’π ππ π₯ πππ π¦ . ππ»π = (π₯ β π¦)(π₯2 + π₯π¦ + π¦2 ) = {(π + π) β (π β π)}{(π + π)2 + (π + π)(π β π) + (π β π)2 } = (π + π β π + π){π2 + π2 + 2ππ + π2 β ππ + ππ β π2 + π2 + π2 β 2ππ} = (2π)(3π2 + π2 ) Sπ, .(π + π)3 β (π β π)3 = 2π(3π2 + π2 ) Answer.24. πππ β ππ β πππ + πππ 8π3 β π3 β 4ππ₯ + 2ππ₯ = (2π)3 β (π)3 β 2π₯(2π β π) = (2π β π)(4π2 + 2ππ + π2) β 2π₯(2π β π) = (2π β π)(4π2 + 2ππ + π2 β 2π₯) Answer.25. ππ + πππ π + ππππ + ππ β π .π3 + 3π2 π + 3ππ2 + π3 β 8 = {π3 + π3 + 3ππ(π + π)} β (2)3 = (π + π)3 β (2)3 = (π + π β 2)[(π + π)2 + 2(π + π) + 4] Answer.26. ππ β π ππ β ππ + π π .π3 β 1 π3 β 2π + 2 π = (π3 β 1 π3 ) β (2π + 2 π ) = (π β 1 π ) (π2 + 1 + 1 π2 ) β 2 (π β 1 π ) = (π β 1 π ) (π2 + 1 + 1 π2 β 2) = (π β 1 π ) (π2 β 1 + 1 π2 ) Answer.27. πππ + ππππ β ππ β πππ 2π3 + 16π3 β 5π β 10π = 2(π3 + 8π3 ) β 5(π + 2π) = 2(π + 2π)(π2 β 2ππ + 4π2) β 5(π + 2π) = (π + 2π){2(π2 β 2ππ + 4π2) β 5} = (π + 2π)(2π2 β 4ππ + 8π2 β 5) Answer.28. ππ + ππ .π6 + π6 = (π2)3 + (π2)3
27. 27. CLASS IX WWW.Vedantu.com RS Aggarwal solutions = (π2 + π2 ){(π2)2 β π2 π2 + (π2)2 } = (π2 + π2 )(π4 β π2 π2 + π4 ) Answer.29. πππ β πππ .π12 β π12 = (π6)2 β (π6)2 = (π6 β π6)(π6 + π6) = [(π3)2 β(π3)2][(π2)3 + (π2)3] = (π3 β π3)(π3 + π3)[(π2 + π2 )(π4 β π2 π2 + π4 )] = (π β π)(π2 + ππ + π2 )(π + π)(π2 β ππ + π2 )(π2 + π2 )(π4 β π2 π2 + π4 ) = (π β π)(π + π)(π2 + π2)(π2 + ππ + π2)(π2 β ππ + π2)(π4 β π2 π2 + π4) Answer.30. ππ β πππ β π Lππ‘ π₯3 = π .π₯6 β 7π₯3 β 8 = π2 β 7π β 8 = π2 β 8π + π β 8 = π(π β 8) + 1(π β 8) = (π + 1)(π β 8) Sπ, π₯6 β 7π₯3 β 8 = (π + 1)(π β 8) = (π₯3 + 1)(π₯3 β 8) [β΅ π = π₯3 ] = [(π₯)3 + (1)3][(π₯)3 β (2)3 ] = (π₯ + 1)(π₯2 β π₯ + 1)(π₯ β 2)(π₯2 + π₯ + 4) = (π₯ β 2)(π₯ + 1)(π₯2 β π₯ + 1)(π₯2 + π₯ + 4) Answer.31. ππ β πππ + ππ + π .π₯3 β 3π₯2 + 3π₯ + 7 = (π₯3 β 3π₯2 + 3π₯ β 1) + 8 = [π₯3 β 13 β 3π₯(π₯ β 1)] + (2)3 = (π₯ β 1)3 + (2)3 = [(π₯ β 1) + 2][(π₯ β 1)2 β (π₯ β 1) Γ (2) + (2)2 ] = (π₯ + 1)[(π₯2 β 2π₯ + 1) β 2π₯ + 2 + 4] = (π₯ + 1)(π₯2 β 4π₯ + 7) Answer.32. (π + π)π + (π β π)π .(π₯ + 1)3 + (π₯ β 1)3 = [(π₯ + 1) + (π₯ β 1)][(π₯ + 1)2 β (π₯ + 1)(π₯ β 1) + (π₯ β 1)2] = (2π₯)[(π₯2 + 2π₯ + 1) β π₯2 + 1 + (π₯2 β 2π₯ + 1)] = 2π₯[π₯2 + 2π₯ + 1 β π₯2 + 1 + π₯2 + 1 β 2π₯] = 2π₯(π₯2 + 3) Answer.33. (ππ + π)π + (π β π)π .(2π + 1)3 + (π β 1)3 = [(2π + 1) + (π β 1)][(2π + 1)2 β (2π + 1)(π β 1) + (π β 1)2] = (3π)[(4π2 + 4π + 1) β 2π2 + 2π β π + 1 + (π2 + 1 β 2π)] = 3π[4π2 + 4π + 1 β 2π2 + π + 1 + π2 + 1 β 2π] = 3π(3π2 + 3π + 3) = 9π(π2 + π + 1) Answer.34. π(π + π)π β ππ(π β π)π 8(π₯ + π¦)3 β 27(π₯ β π¦)3 = {2(π₯ + π¦)}3 β {3(π₯ β π¦)}3 = {2(π₯ + π¦) β 3(π₯ β π¦)}[{2(π₯ + π¦)}2 + {2(π₯ + π¦)}{3(π₯ β π¦)} + {3(π₯ β π¦)}2 ] = [2π₯ + 2π¦ β 3π₯ + 3π¦][{4(π₯2 + 2π₯π¦ + π¦2)} + 6(π₯2 β π¦2) + {9(π₯2 β 2π₯π¦ + π¦2 )}] = (βπ₯ + 5π¦)[4π₯2 + 8π₯π¦ + 4π¦2 + 6π₯2 β 6π¦2 + 9π₯2 β 18π₯π¦ + 9π¦2 ] = (βπ₯ + 5π¦)(19π₯2 β 10π₯π¦ + 7π¦2)
28. 28. CLASS IX WWW.Vedantu.com RS Aggarwal solutions Answer.35. (π + π)π + (π β π)π .(π₯ + 2)3 + (π₯ β 2)3 = [(π₯ + 2) + (π₯ β 2)][(π₯ + 2)2 β (π₯ + 2)(π₯ β 2) + (π₯ β 2)2] = (2π₯)[(π₯2 + 4π₯ + 4) β π₯2 + 4 + (π₯2 β 4π₯ + 4)] = 2π₯[π₯2 + 4π₯ + 4 β π₯2 + 4 + π₯2 β 4π₯ + 4] = 2π₯(π₯2 + 12) Answer.36. (π + π)π β (π β π)π .(π₯ + 2)3 β (π₯ β 2)3 = [(π₯ + 2) β (π₯ β 2)][(π₯ + 2)2 + (π₯ + 2)(π₯ β 2) + (π₯ β 2)2] = (4)[(π₯2 + 4π₯ + 4) + π₯2 β 4 + (π₯2 β 4π₯ + 4)] = 4[π₯2 + 4π₯ + 4 + π₯2 β 4 + π₯2 β 4π₯ + 4] = 4(3π₯2 + 4) Answer.37. Prove that : π.ππΓπ.ππΓπ.ππ+π.ππΓπ.ππΓπ.ππ π.ππΓπ.ππβπ.ππΓπ.ππ+π.ππΓπ.ππ = π L.H.S.β 0.85Γ0.85Γ0.85+0.15Γ0.15Γ0.15 0.85Γ0.85β0.85Γ0.15+0.15Γ0.15 = (0.85)3+(0.15)3 0.85Γ0.85β0.85Γ0.15+0.15Γ0.15 = (0.85+0.15){(0.85)2β0.85Γ0.15+(0.15)2 (0.85)2β0.85Γ0.15+(0.15)2 = (0.85 + 0.15) = 1 = R.H.S Hππππ, πΏ. π». π = π. π». π Answer.38. Prove that : ππΓππΓππβπΓπΓπ ππΓππ+ππΓπ+πΓπ = ππ L.H.S.β 59Γ59Γ59β9Γ9Γ9 59Γ59+59Γ9+9Γ9 = (59)3 β(9)3 59Γ59+59Γ9+9Γ9 = (59β9){(59)2+59Γ9+(9)2} 59Γ59+59Γ9+9Γ9 = (59 β 9) = 50 = R.H.S Hππππ, πΏ. π». π = π. π». π
29. 29. CLASS IX WWW.Vedantu.com RS Aggarwal solutions EXERCISE β 3G Formula Used : (ππ + ππ + ππ β ππππ) = (π + π + π)(ππ + ππ + ππ β ππ β ππ β ππ) Answer.1. (π + π β π)(ππ + ππ + ππ β ππ + ππ + ππ) .(π₯ + π¦ β π§)(π₯2 + π¦2 + π§2 β π₯π¦ + π¦π§ + π§π₯) = {π₯ + π¦ + (βπ§)}{π₯2 + π¦2 + (βπ§)2 β π₯π¦ β π¦(βπ§) β (βπ§)π₯} = {π₯3 + π¦3 + (βπ§)3 β 3π₯π¦(βπ§)} = (π₯3 + π¦3 β π§3 + 3π₯π¦π§) Answer.2. (π β π β π)(ππ + ππ + ππ + ππ β ππ + ππ) .(π₯ β π¦ β π§)(π₯2 + π¦2 + π§2 + π₯π¦ β π¦π§ + π§π₯) = {π₯ + (βπ¦) + (βπ§)}{π₯2 + (βπ¦)2 + (βπ§)2 β π₯(βπ¦) β (βπ¦)(βπ§) β (βπ§)π₯} = {π₯3 + (βπ¦)3 + (βπ§)3 β 3π₯(βπ¦)(βπ§)} = (π₯3 β π¦3 β π§3 β 3π₯π¦π§) Answer.3. (π β ππ + π)(ππ + πππ + πππ + ππ β ππ + π) .(π₯ β 2π¦ + 3)(π₯2 + 4π¦2 + 9 + 2π₯π¦ + 6π¦ β 3π₯) = {π₯ + (β2π¦) + 3}{π₯2 + (β2π¦)2 + 32 β π₯(β2π¦) β (β2π¦)(3) β 3(π₯)} = {π₯3 + (β2π¦)3 + (3)3 β 3π₯(β2π¦)(3)} = (π₯3 β 8π¦3 + 27 + 18π₯π¦) Answer.4. (ππ β ππ + π)(πππ + ππππ + ππππ + πππ β πππ + ππ) .(3π₯ β 5π¦ + 4)(9π₯2 + 25π¦2 + 15π₯π¦ + 20π¦ β 12π₯ + 16) = {3π₯ + (β5π¦) + 4}{(3π₯)2 + (β5π¦)2 + 42 β 3π₯(β5π¦) β (β5π¦)(4) β 4(3π₯)} = {(3π₯)3 + (β5π¦)3 + (4)3 β 3(3π₯)(β5π¦)(4)} = (27π₯3 β 125π¦3 + 64 + 180π₯π¦) Answer.5. πππππ + ππ + ππππ β πππππ 125π3 + π3 + 64π3 β 60πππ = (5π)3 + (π)3 + (4π)3 β 3 Γ (5π) Γ (π) Γ (4π) = (5π + π + 4π)[(5π)2 + (π)2 + (4π)2 β (5π)(π) β (π)(4π) β (4π)(5π)] = (5π + π + 4π)(25π2 + π2 + 16π2 β 5ππ β 4ππ β 20ππ) Answer.6. ππ + πππ + ππππ β πππππ π3 + 8π3 + 64π3 β 24πππ = (π)3 + (2π)3 + (4π)3 β 3 Γ (π) Γ (2π) Γ (4π) = (π + 2π + 4π)[(π)2 + (2π)2 + (4π)2 β (π)(2π) β (2π)(4π) β (4π)(π)] = (π + 2π + 4π)(π2 + 4π2 + 16π2 β 2ππ β 8ππ β 4ππ) Answer.7. π + ππ + πππ β ππππ 1 + π3 + 8π3 β 6πππ = (1)3 + (π)3 + (2π)3 β 3 Γ (1) Γ (π) Γ (2π) = (1 + π + 2π)[(1)2 + (π)2 + (2π)2 β (1)(π) β (π)(2π) β (2π)(1)] = (1 + π + 2π)(1 + π2 + 4π2 β π β 2π β 2π) Answer.8. πππ + ππππ + πππ β ππππππ 216 + 27π3 + 8π3 β 108πππ = (6)3 + (3π)3 + (2π)3 β 3 Γ (6) Γ (3π) Γ (2π) = (6 + 3π + 2π)[(6)2 + (3π)2 + (2π)2 β (6)(3π) β (3π)(2π) β (2π)(6)] = (6 + 3π + 2π)(36 + 9π2 + 4π2 β 18π β 6ππ β 12π) Answer.9. ππππ β ππ + πππ + πππππ 27π3 β π3 + 8π3 + 18πππ = (3π)3 + (βπ)3 + (2π)3 β 3 Γ (3π) Γ (βπ) Γ (2π) = (3π β π + 2π)[(3π)2 + (βπ)2 + (2π)2 β (3π)(βπ) β (βπ)(2π) β (2π)(3π)] = (3π β π + 2π)(9π2 + π2 + 4π2 + 3ππ + 2ππ β 6ππ)
30. 30. CLASS IX WWW.Vedantu.com RS Aggarwal solutions Answer.10. πππ + πππππ β ππππ + ππππππ 8π3 + 125π3 β 64π3 + 120πππ = (2π)3 + (5π)3 + (β4π)3 β 3 Γ (2π) Γ (5π) Γ (β4π) = (2π + 5π β 4π)[(2π)2 + (5π)2 + (β4π)2 β (2π)(5π) β (5π)(β4π) β (β4π)(2π)] = (2π + 5π β 4π)(4π2 + 25π2 + 16π2 β 10ππ + 20ππ + 8ππ) Answer.11. π β ππππ β πππππ β ππππππ 8 β 27π3 β 343π3 β 126πππ = (2)3 + (β3π)3 + (β7π)3 β 3 Γ (2) Γ (β3π) Γ (β7π) = (2 β 3π β 7π)[(2)2 + (β3π)2 + (β7π)2 β (2)(β3π) β (β3π)(β7π) β (β7π)(2)] = (2 β 3π β 7π)(4 + 9π2 + 49π2 + 6π β 21ππ + 14π) Answer.12. πππ β πππ β ππππ β ππππ 125 β 8π₯3 β 27π¦3 β 90π₯π¦ = (5)3 + (β2π₯)3 + (β3π¦)3 β 3 Γ (5) Γ (β2π₯) Γ (β3π¦) = (5 β 2π₯ β 3π¦)[(5)2 + (β2π₯)2 + (β3π¦)2 β (5)(β2π₯) β (β2π₯)(β3π¦) β (β3π¦)(5)] = (5 β 2π₯ β 3π¦)(25 + 4π₯2 + 9π¦2 + 10π₯ β 6π₯π¦ + 15π¦) Answer.13. πβπππ + ππβπππ + ππ β πππππ 2β2π3 + 16β2π3 + π3 β 12πππ = (β2π) 3 + (2β2π) 3 + (π)3 β 3(β2π)(2β2π)(π) = (β2π + 2β2π + π) [(β2π) 2 + (2β2π) 2 + (π)3 β (β2π)(2β2π) β (2β2π)π β π(β2π)] = (β2π + 2β2π + π)(2π2 + 8π2 + π2 β 4ππ β 2β2ππ β β2ππ) Answer.14. ππππ β ππ β ππ β ππππ 27π₯3 β π¦3 β π§3 β 9π₯π¦π§ = (3π₯)3 + (βπ¦)3 + (βπ§)3 β 3 Γ (3π₯) Γ (βπ¦) Γ (βπ§) = (3π₯ β π¦ β π§)[(3π₯)2 + (βπ¦)2 + (βπ§)2 β (3π₯)(βπ¦) β (βπ¦)(βπ§) β (βπ§)(3π₯)] = (3π₯ β π¦ β π§)(9π₯2 + π¦2 + π§2 + 3π₯π¦ β π₯π¦ + 3π§π₯) Answer.15. πβπππ + πβπππ + ππ β πβππππ 2β2π3 + 3β3π3 + π3 β 3β6πππ = (β2π) 3 + (β3π) 3 + (π)3 β 3(β2π)(β3π)(π) = (β2π + β3π + π) [(β2π) 2 + (β3π) 2 + (π)3 β (β2π)(β3π) β (β3π)π β π(β2π)] = (β2π + β3π + π)(2π2 + 9π2 + π2 β β6ππ β β3ππ β β2ππ) Answer.16. πβπππ β ππ β πβπππ β πβπππππ 3β3π3 β π3 β 5β5π3 β 3β15πππ = (β3π) 3 + (βπ)3 + (ββ5π) 3 β 3(β3π)(βπ)(ββ5π) = (β3π β π β β5π) [(β3π) 2 + (βπ)2 + (ββ5π) 3 β (β3π)(βπ) β (βπ)(ββ5π) β (ββ5π)(β3π)] = (β3π β π β β5π)(3π2 + π2 + 5π2 + β3ππ β β5ππ + β15ππ) Answer.17. (π β π)π + (π β π)π + (π β π)π πΏππ‘(π β π) = π₯, (π β π) = π¦ πππ (π β π) = π§, (π β π)3 + (π β π)3 + (π β π)3 = π₯3 + π¦3 + π§3 , π€βπππ (π₯ + π¦ + π§) = (π β π) + (π β π) + (π β π) = 0 = 3π₯π¦π§ [β΅ (π₯ + π¦ + π§) = 0 β (π₯3 + π¦3 + π§3 ) = 3π₯π¦π§] = 3(π β π)(π β π)(π β π) Answer.18. (π β ππ)π + (ππ β π)π + (π β π)π πΏππ‘(π β 3π) = π₯, (3π β π) = π¦ πππ (π β π) = π§, (π β 3π)3 + (3π β π)3 + (π β π)3 = π₯3 + π¦3 + π§3 , π€βπππ (π₯ + π¦ + π§) = (π β 3π) + (3π β π) + (π β π) = 0 = 3π₯π¦π§ [β΅ (π₯ + π¦ + π§) = 0 β (π₯3 + π¦3 + π§3) = 3π₯π¦π§] = 3(π β 3π)(3π β π)(π β π)
31. 31. CLASS IX WWW.Vedantu.com RS Aggarwal solutions Answer.19. (ππ β ππ)π + (ππ β ππ)π + (ππ β ππ)π πΏππ‘(3π β 2π) = π₯, (2π β 5π) = π¦ πππ (5π β 3π) = π§, (3π β 2π)3 + (2π β 5π)3 + (5π β 3π)3 = π₯3 + π¦3 + π§3 , π€βπππ (π₯ + π¦ + π§) = (3π β 2π) + (2π β 5π) + (5π β 3π) = 0 = 3π₯π¦π§ [β΅ (π₯ + π¦ + π§) = 0 β (π₯3 + π¦3 + π§3) = 3π₯π¦π§] = 3(3π β 2π)(2π β 5π)(5π β 3π) Answer.20. (ππ β ππ)π + (ππ β ππ)π + (ππ β ππ)π πΏππ‘(5π β 7π) = π₯, (7π β 9π) = π¦ πππ (9π β 5π) = π§, (5π β 7π)3 + (7π β 9π)3 + (9π β 5π)3 = π₯3 + π¦3 + π§3 , π€βπππ (π₯ + π¦ + π§) = (5π β 7π) + (7π β 9π) + (9π β 5π) = 0 = 3π₯π¦π§ [β΅ (π₯ + π¦ + π§) = 0 β (π₯3 + π¦3 + π§3) = 3π₯π¦π§] = 3(5π β 7π)(7π β 9π)(9π β 5π) Answer.21. ππ(π β π)π + ππ(π β π)π + ππ(π β π)π πΏππ‘π(π β π) = π₯, π(π β π) = π¦ πππ π(π β π) = π§, [π(π β π)]3 + [π(π β π)]3 + [π(π β π)]3 = π₯3 + π¦3 + π§3 , π€βπππ (π₯ + π¦ + π§) = π(π β π) + π(π β π) + π(π β π) = 0 = 3π₯π¦π§ [β΅ (π₯ + π¦ + π§) = 0 β (π₯3 + π¦3 + π§3 ) = 3π₯π¦π§] = 3πππ(π β π)(π β π)(π β π) Answer.22. (i) (βππ)π + ππ + ππ πΏππ‘ π₯ = (β12), π¦ = 7 πππ π§ = 5 (π₯ + π¦ + π§) = 0 β (π₯3 + π¦3 + π§3 ) = 3π₯π¦π§ β (β12)3 + 73 + 53 = 3 Γ (β12) Γ (7) Γ (5) β (β12)3 + 73 + 53 = β108 (ii) (ππ)π + (βππ)π + (βππ)π πΏππ‘ π₯ = 28 , π¦ = (β15) πππ π§ = (β13) (π₯ + π¦ + π§) = 0 β (π₯3 + π¦3 + π§3 ) = 3π₯π¦π§ β (28)3 + (β15)3 + (β13)3 = 3 Γ (28) Γ (β15) Γ (β13) β (28)3 + (β15)3 + (β13)3 = 16380 Answer.23. (π + π + π)π β ππ β ππ β ππ = π(π + π)(π + π)(π + π) πΏ. π». π β (π + π + π)3 β π3 β π3 β π3 = [(π + π) + π]3 β π3 β π3 β π3 = (π + π)3 + π3 + 3(π + π)π Γ [(π + π) + π] β π3 β π3 β π3 = π3 + π3 + 3ππ(π + π) + π3 + 3(π + π)π Γ [(π + π) + π] β π3 β π3 β π3 = 3ππ(π + π) + 3(π + π)π Γ [(π + π) + π] = 3(π + π)[ππ + π(π + π) + π2] = 3(π + π)[ππ + ππ + ππ + π2] = 3(π + b)[π(π + π) + π(π + π)] = 3(π + π)(π + π)(π + π) = π. π». π Answer.24. πΊππ£ππ, π + π + π = 0 πππ π, π , π πππ πππ πππ β π§πππ π2 ππ + π2 ππ + π2 ππ = 3 πΏ. π». π β (π3 + π3 + π3) πππ
32. 32. CLASS IX WWW.Vedantu.com RS Aggarwal solutions β 3πππ πππ [β΅ (π + π + π) = 0 β (π3 + π3 + π3 ) = 3πππ] β 3 β π. π». π Answer.25. πΊππ£ππ π + π + π = 9 πππ π2 + π2 + π2 = 35 (π + π + π) = 9 β (π + π + π)2 = 81 β (π2 + π2 + π2) + 2(ππ + ππ + ππ) = 81 β 35 + 2(ππ + ππ + ππ) = 81 β (ππ + ππ + ππ) = 23 βΈ« (π3 + π3 + π3 β 3πππ) = (π + π + π)(π2 + π2 + π2 β ππ β ππ β ππ) = (π + π + π)[(π2 + π2 + π2 ) β (ππ + ππ + ππ)] = 9 Γ (35 β 23) = 9 Γ 12 = 108
33. 33. CLASS IX WWW.Vedantu.com RS Aggarwal solutions MULTIPLE CHOICE QUESTION (MCQ) Answer.1. πΏππ‘ π(π₯) = 2π₯2 + ππ₯ πππππ, (π₯ + 1) ππ  π ππππ‘ππ f(β1) = 0 β2(β1)2 + π(β1) = 0 β 2 β π = 0 β π = 2 πͺππππππ πΆπππππ βΆ (π) Answer.2. (249)2 β(248)2 = (249 β 248)(249 + 248) = (1)(497) = 497 πͺππππππ πΆπππππ βΆ (π) Answer.3. π₯ π¦ + π¦ π₯ = β1 β π₯2 + π¦2 = βπ₯π¦ β π₯2 + π¦2 + π₯π¦ = 0 (π₯3 β π¦3 ) = (π₯ β π¦)(π₯2 + π¦2 + π₯π¦) = (π₯ β π¦) Γ 0 = 0 πͺππππππ πΆπππππ βΆ (π) Answer.4. π3 + π3 + π3 β 3πππ = (π + π + π)(π2 + π2 + π2 β ππ β ππ β ππ) = 0 Γ (π2 + π2 + π2 β ππ β ππ β ππ) [β΅ π + π + π = 0] = 0 π3 + π3 + π3 β 3πππ = 0 βΈ« π3 + π3 + π3 = 3πππ πͺππππππ πΆπππππ βΆ (π) Answer.5. (3π₯ + 1 2 ) (3π₯ β 1 2 ) = 9π₯2 β π {(3π₯)2 β ( 1 2 ) 2 } = 9π₯2 β π (9π₯2 β 1 4 ) = 9π₯2 β π ππ, π = 1 4 πͺππππππ πΆπππππ βΆ (π) Answer.6. (π₯ + 3)3 = π₯3 + 33 + 3 Γ (π₯2) Γ 3 + 3 Γ π₯ Γ (32 ) = π₯3 + 27 + 9π₯2 + 27π₯ πΆππππππππππ‘ ππ π₯ = 2 πͺππππππ πΆπππππ βΆ (π) Answer.7. (π₯ + π¦)3 β (π₯3 + π¦3) = (π₯ + π¦)3 β {(π₯ + π¦)(π₯2 β π₯π¦ + π¦2 )} = (π₯ + π¦){(π₯ + π¦)2 β (π₯2 β π₯π¦ + π¦2 )} = (π₯ + π¦){π₯2 + π¦2 + 2π₯π¦ β π₯2 + π₯π¦ β π¦2} = (π₯ + π¦)(3π₯π¦) ππ, 3π₯π¦ ππ  π ππππ‘ππ. πͺππππππ πΆπππππ βΆ (π)
34. 34. CLASS IX WWW.Vedantu.com RS Aggarwal solutions Answer.8. (25π₯2 β 1) + (1 + 5π₯)2 = 25π₯2 β 1 + 1 + 25π₯2 + 10π₯ = 50π₯2 + 10π₯ = 10π₯(5π₯ + 1) ππ, 10π₯ ππ  π ππππ‘ππ. πͺππππππ πΆπππππ βΆ (π) Answer.9. π(π₯) = π₯3 β 20π₯ + 5π πππππ, (π₯ + 5) ππ  π ππππ‘ππ. π(β5) = 0 β(β5)3 β 20(β5) + 5π = 0 ββ125 + 100 + 5π = 0 β5π = 25 βπ = 5 πͺππππππ πΆπππππ βΆ (π) Answer.10. πΏππ‘ π(π₯) = π₯3 + 10π₯2 + ππ₯ + π (π₯ + 2) = 0 β π₯ = β2 (π₯ β 1) = 0 β π₯ = 1 πππ€, π(β2) = 0 πππ π(1) = 0 [β΅ (π₯ + 2) πππ (π₯ β 1) πππ ππππ‘πππ .] ππ, π(β2) = 0 β (β2)3 + 10 Γ (β2)2 + (β2)π + π = 0 β β8 + 40 β 2π + π = 0 β 2π β π = 32 β¦ (π) π΄ππ, π(1) = 0 β(1)3 + 10 Γ (1)2 + π + π = 0 β1 + 11 + π + π = 0 βπ + π = β11 β¦ (ππ) π΄πππππ (π) πππ (ππ), β2π β π + π + π = 32 β 11 β3π = 21 βπ = 7 β¦ (πππ) πΌππππ (ππ) πππ (πππ), β7 + π = β11 βπ = β18 β΄ π = 7 πππ π = β18 πͺππππππ πΆπππππ βΆ (π) Answer.11. (πππ Γ ππ) = ? (104 Γ 96) = (100 + 4) Γ (100 β 4) = (100)2 β (4)2 = 10000 β 16 = 9984 πͺππππππ πΆπππππ βΆ (π) Answer.12. (πππ Γ πππ) = ? (305 Γ 308) = 305 Γ (300 + 8) = (305 Γ 300) + (305 Γ 8) = (91500 + 2440) = 93940 πͺππππππ πΆπππππ βΆ (π) Answer.13. (πππ Γ πππ) = ? (207 Γ 193) = 207 Γ (200 β 7)
35. 35. CLASS IX WWW.Vedantu.com RS Aggarwal solutions = (207 Γ 200) β (207 Γ 7) = (41400 β 1449) = 39951 πͺππππππ πΆπππππ βΆ (π) Answer.14. πππ + ππ + πππ + ππ + ππ + π 4π2 + π2 + 4ππ + 8π + 4π + 4 = (2π)2 + (π)2 + (2)2 + 2(2π)(π) + 2(π)(2) + 2(2)(2π) = (2π + π + 2)2 πͺππππππ πΆπππππ βΆ (π) Answer.15. (ππ β ππ β ππ) = ? (π₯2 β 4π₯ β 21) = π₯2 β 7π₯ + 3π₯ β 21 = π₯(π₯ β 7) + 3(π₯ β 7) = (π₯ β 7)(π₯ + 3) πͺππππππ πΆπππππ βΆ (π) Answer.16. (πππ + ππ β π) = ? (4π₯2 + 4π₯ β 3) = 4π₯2 + 6π₯ β 2π₯ β 3 = 2π₯(2π₯ + 3) β 1(2π₯ + 3) = (2π₯ + 3)(2π₯ β 1) πͺππππππ πΆπππππ βΆ (π) Answer.17. πππ + πππ + π = ? 6π₯2 + 17π₯ + 5 = 6π₯2 + 15π₯ + 2π₯ + 5 = 3π₯(2π₯ + 5) + 1(2π₯ + 5) = (2π₯ + 5)(3π₯ + 1) πͺππππππ πΆπππππ βΆ (π) Answer.18. πΏππ‘ π₯3 + 2π₯2 β π₯ β 2 = π(π₯) π(β1) = {(β1)3 + 2(β1)2 β (β1) β 2} = (β1 + 2 + 1 β 2) = 0 πͺππππππ πΆπππππ βΆ (π) Answer.19. πππ + πππ + ππ + π = ? 3π₯3 + 2π₯2 + 3π₯ + 2 = π₯2(3π₯ + 2) + 1(3π₯ + 2) = (3π₯ + 2)(π₯2 + 1) πͺππππππ πΆπππππ βΆ (π) Answer.20. π + π + π = 0 βπ3 + π3 + π3 β 3πππ = 0 βπ3 + π3 + π3 = 3πππ ( π2 ππ + π2 ππ + π2 ππ ) = 1 πππ (π3 + π3 + π3) = 1 πππ (3πππ) = 3 πͺππππππ πΆπππππ βΆ (π) Answer.21. (π₯3 + π¦3 + π§3 β 3π₯π¦π§) = (π₯ + π¦ + π§)(π₯2 + π¦2 + π§2 β π₯π¦ β π¦π§ β π§π₯) = (π₯ + π¦ + π§)(π₯2 + π¦2 + π§2 + 2π₯π¦ + 2π¦π₯ + 2π§π₯ β 3π₯π¦ β 3π¦π§ β 3π§π₯)
36. 36. CLASS IX WWW.Vedantu.com RS Aggarwal solutions = (π₯ + π¦ + π§)[(π₯2 + π¦2 + π§2 + 2π₯π¦ + 2π¦π₯ + 2π§π₯) β 3(π₯π¦ + π¦π§ + π§π₯)] = (π₯ + π¦ + π§)[(π₯ + π¦ + π§)2 β 3(π₯π¦ + π¦π§ + π§π₯)] = 9 Γ [(9)2 β 3 Γ (23)] = 9 Γ (81 β 69) = 9 Γ 12 = 108 πͺππππππ πΆπππππ βΆ (π) Answer.22. ( π π + π π ) = β1 β π π + π π = β1 β π2+π2 ππ = β1 βπ2 + π2 + ππ = 0 (π3 β π3) = (π β π)(π2 + ππ + π2 ) = (π β π) Γ 0 = 0 πͺππππππ πΆπππππ βΆ (π)