11 x1 t15 06 roots & coefficients (2012)

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<p> 1. Roots and Coefficients 2. Roots and Coefficients Quadratics ax 2 bx c 0 3. Roots and Coefficients Quadratics ax 2 bx c 0b a 4. Roots and Coefficients Quadratics ax 2 bx c 0bc aa 5. Roots and Coefficients Quadraticsax 2 bx c 0 bc aaCubicsax 3 bx 2 cx d 0 6. Roots and Coefficients Quadraticsax 2 bx c 0 bc aaCubicsax 3 bx 2 cx d 0 b a 7. Roots and Coefficients Quadraticsax 2 bx c 0 bc aaCubicsax 3 bx 2 cx d 0 bc aa 8. Roots and Coefficients Quadraticsax 2 bx c 0 bc aaCubicsax 3 bx 2 cx d 0 bc aa d a 9. Roots and CoefficientsQuadraticsax 2 bx c 0 bc aa Cubics ax 3 bx 2 cx d 0bc aa d aQuartics ax 4 bx 3 cx 2 dx e 0 10. Roots and CoefficientsQuadraticsax 2 bx c 0 bc aa Cubics ax 3 bx 2 cx d 0bc aa d aQuartics ax 4 bx 3 cx 2 dx e 0 b a 11. Roots and CoefficientsQuadraticsax 2 bx c 0 bc aa Cubics ax 3 bx 2 cx d 0bc aa d aQuartics ax 4 bx 3 cx 2 dx e 0 b c a a 12. Roots and Coefficients Quadratics ax 2 bx c 0 bc aa Cubics ax 3 bx 2 cx d 0 b c a a d aQuartics ax 4 bx 3 cx 2 dx e 0 bc aad a 13. Roots and Coefficients Quadratics ax 2 bx c 0 bc aa Cubics ax 3 bx 2 cx d 0 b c a a d aQuartics ax 4 bx 3 cx 2 dx e 0 bc aade a a 14. For the polynomial equation;ax n bx n1 cx n2 dx n3 0 15. For the polynomial equation;ax n bx n1 cx n2 dx n3 0b a (sum of roots, one at a time) 16. For the polynomial equation;ax n bx n1 cx n2 dx n3 0b a(sum of roots, one at a time)c a (sum of roots, two at a time) 17. For the polynomial equation;ax n bx n1 cx n2 dx n3 0 b a (sum of roots, one at a time) c a (sum of roots, two at a time) d a (sum of roots, three at a time) 18. For the polynomial equation;ax n bx n1 cx n2 dx n3 0b a(sum of roots, one at a time)c a(sum of roots, two at a time)da (sum of roots, three at a time)e a (sum of roots, four at a time) 19. For the polynomial equation;ax n bx n1 cx n2 dx n3 0b a (sum of roots, one at a time)c a (sum of roots, two at a time)da (sum of roots, three at a time)e a (sum of roots, four at a time)Note: 2 22 20. e.g. (i ) If , and are the roots of 2 x 3 5 x 2 3 x 1 0, find the values of; a) 4 4 4 7 21. e.g. (i ) If , and are the roots of 2 x 3 5 x 2 3 x 1 0, find the values of; a) 4 4 4 75 2 22. e.g. (i ) If , and are the roots of 2 x 3 5 x 2 3 x 1 0, find the values of; a) 4 4 4 753 22 23. e.g. (i ) If , and are the roots of 2 x 3 5 x 2 3 x 1 0, find the values of; a) 4 4 4 753 1 22 2 24. e.g. (i ) If , and are the roots of 2 x 3 5 x 2 3 x 1 0, find the values of; a) 4 4 4 753 1 22 25 1 4 4 4 7 4 7 2 2 25. e.g. (i ) If , and are the roots of 2 x 3 5 x 2 3 x 1 0, find the values of; a) 4 4 4 75 31 2 225 1 4 4 4 7 4 7 2 227 2 26. e.g. (i ) If , and are the roots of 2 x 3 5 x 2 3 x 1 0, find the values of; a) 4 4 4 75 31 2 22 5 14 4 4 7 4 7 2 2 27 2 1 1 1b) 27. e.g. (i ) If , and are the roots of 2 x 3 5 x 2 3 x 1 0, find the values of; a) 4 4 4 75 31 2 22 5 14 4 4 7 4 7 2 2 27 2 1 1 1 b) 28. e.g. (i ) If , and are the roots of 2 x 3 5 x 2 3 x 1 0, find the values of; a) 4 4 4 75 31 2 22 5 14 4 4 7 4 7 2 2 27 2 1 1 1 b) 3 212 29. e.g. (i ) If , and are the roots of 2 x 3 5 x 2 3 x 1 0, find the values of; a) 4 4 4 75 31 2 22 5 14 4 4 7 4 7 2 2 27 2 1 1 1 b) 3 2 1 2 3 30. e.g. (i ) If , and are the roots of 2 x 3 5 x 2 3 x 1 0, find the values of; a) 4 4 4 753 1 22 2 5 14 4 4 7 4 7 2 2 27 2 1 1 1 b) c) 2 2 2 3 2 1 2 3 31. e.g. (i ) If , and are the roots of 2 x 3 5 x 2 3 x 1 0, find the values of; a) 4 4 4 753 1 22 2 5 14 4 4 7 4 7 2 227 2 1 1 1 b) c) 2 2 2 2 23 212 3 32. e.g. (i ) If , and are the roots of 2 x 3 5 x 2 3 x 1 0, find the values of; a) 4 4 4 753 1 22 2 5 14 4 4 7 4 7 2 227 2 1 1 1 b) c) 2 2 2 2 23 2 5 3 2 2 1 2 22 3 33. e.g. (i ) If , and are the roots of 2 x 3 5 x 2 3 x 1 0, find the values of; a) 4 4 4 753 1 22 2 5 14 4 4 7 4 7 2 227 2 1 1 1 b) c) 2 2 2 2 23 2 5 3 2 2 1 2 2372 3 4 34. 1988 Extension 1 HSC Q2c)If , and are the roots of x 3 x 1 0 find: 3(i) 35. 1988 Extension 1 HSC Q2c)If , and are the roots of x 3 x 1 0 find: 3(i) 0 36. 1988 Extension 1 HSC Q2c)If , and are the roots of x 3 x 1 0 find: 3(i) 0(ii) 37. 1988 Extension 1 HSC Q2c)If , and are the roots of x 3 x 1 0 find: 3(i) 0(ii) 1 38. 1988 Extension 1 HSC Q2c)If , and are the roots of x 3 x 1 0 find: 3(i) 0(ii) 1 1 1 1(iii) 39. 1988 Extension 1 HSC Q2c)If , and are the roots of x 3 x 1 0 find: 3(i) 0(ii) 1 1 1 1(iii) 1 1 1 40. 1988 Extension 1 HSC Q2c)If , and are the roots of x 3 x 1 0 find: 3(i) 0(ii) 1 1 1 1(iii) 1 1 1 3 1 41. 1988 Extension 1 HSC Q2c)If , and are the roots of x 3 x 1 0 find: 3(i) 0(ii) 1 1 1 1(iii) 1 1 1 3 1 3 42. 2003 Extension 1 HSC Q4c)It is known that two of the roots of the equation 2 x 3 x 2 kx 6 0 arereciprocals of each other.Find the value of k. 43. 2003 Extension 1 HSC Q4c)It is known that two of the roots of the equation 2 x 3 x 2 kx 6 0 arereciprocals of each other.Find the value of k.1Let the roots be , and 44. 2003 Extension 1 HSC Q4c)It is known that two of the roots of the equation 2 x 3 x 2 kx 6 0 arereciprocals of each other.Find the value of k.1Let the roots be , and 1 6 2 45. 2003 Extension 1 HSC Q4c)It is known that two of the roots of the equation 2 x 3 x 2 kx 6 0 arereciprocals of each other.Find the value of k.1Let the roots be , and 1 6 2 3 46. 2003 Extension 1 HSC Q4c)It is known that two of the roots of the equation 2 x 3 x 2 kx 6 0 arereciprocals of each other.Find the value of k.1Let the roots be , and 1 6 P 3 0 2 3 47. 2003 Extension 1 HSC Q4c)It is known that two of the roots of the equation 2 x 3 x 2 kx 6 0 arereciprocals of each other.Find the value of k.1Let the roots be , and 1 6 P 3 0 2 32 3 3 k 3 6 032 48. 2003 Extension 1 HSC Q4c)It is known that two of the roots of the equation 2 x 3 x 2 kx 6 0 arereciprocals of each other.Find the value of k.1Let the roots be , and 1 6 P 3 0 2 32 3 3 k 3 6 032 54 9 3k 6 0 49. 2003 Extension 1 HSC Q4c)It is known that two of the roots of the equation 2 x 3 x 2 kx 6 0 arereciprocals of each other.Find the value of k.1Let the roots be , and 1 6 P 3 0 2 32 3 3 k 3 6 032 54 9 3k 6 0 3k 39 50. 2003 Extension 1 HSC Q4c)It is known that two of the roots of the equation 2 x 3 x 2 kx 6 0 arereciprocals of each other.Find the value of k.1Let the roots be , and 1 6 P 3 0 2 32 3 3 k 3 6 032 54 9 3k 6 0 3k 39 k 13 51. 2006 Extension 1 HSC Q4a)The cubic polynomial P x x 3 rx 2 sx t , where r, s and t are realnumbers, has three real zeros, 1, and (i) Find the value of r 52. 2006 Extension 1 HSC Q4a)The cubic polynomial P x x 3 rx 2 sx t , where r, s and t are realnumbers, has three real zeros, 1, and (i) Find the value of r 1 r 53. 2006 Extension 1 HSC Q4a)The cubic polynomial P x x 3 rx 2 sx t , where r, s and t are realnumbers, has three real zeros, 1, and (i) Find the value of r 1 r r 1 54. 2006 Extension 1 HSC Q4a)The cubic polynomial P x x 3 rx 2 sx t , where r, s and t are realnumbers, has three real zeros, 1, and (i) Find the value of r 1 r r 1(ii) Find the value of s + t 55. 2006 Extension 1 HSC Q4a)The cubic polynomial P x x 3 rx 2 sx t , where r, s and t are realnumbers, has three real zeros, 1, and (i) Find the value of r 1 r r 1(ii) Find the value of s + t 1 1 s 56. 2006 Extension 1 HSC Q4a)The cubic polynomial P x x 3 rx 2 sx t , where r, s and t are realnumbers, has three real zeros, 1, and (i) Find the value of r 1 r r 1(ii) Find the value of s + t 1 1 s s 2 57. 2006 Extension 1 HSC Q4a)The cubic polynomial P x x 3 rx 2 sx t , where r, s and t are realnumbers, has three real zeros, 1, and (i) Find the value of r 1 r r 1(ii) Find the value of s + t 1 1 s1 t s 2 58. 2006 Extension 1 HSC Q4a)The cubic polynomial P x x 3 rx 2 sx t , where r, s and t are realnumbers, has three real zeros, 1, and (i) Find the value of r 1 r r 1(ii) Find the value of s + t 1 1 s1 t s 2t 2 59. 2006 Extension 1 HSC Q4a)The cubic polynomial P x x 3 rx 2 sx t , where r, s and t are realnumbers, has three real zeros, 1, and (i) Find the value of r 1 r r 1(ii) Find the value of s + t 1 1 s 1 t s 2 t 2 s t 0 60. Exercise 4F; 2, 4, 5ac, 6ac, 8, 10a, 13, 15, 16ad, 17, 18, 19</p>