11 x1 t15 06 roots & coefficients (2013)

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<ul><li> 1. Roots and Coefficients</li></ul> <p> 2. Roots and Coefficients Quadraticsax 2 bx c 0 3. Roots and Coefficients Quadraticsax 2 bx c 0b a 4. Roots and Coefficients Quadraticsax 2 bx c 0b ac a 5. Roots and Coefficients Quadraticsax 2 bx c 0b a Cubicsax 3 bx 2 cx d 0c a 6. Roots and Coefficients Quadraticsax 2 bx c 0b a Cubicsax 3 bx 2 cx d 0 b ac a 7. Roots and Coefficients Quadraticsax 2 bx c 0 c ab a Cubicsax 3 bx 2 cx d 0b ac a 8. Roots and Coefficients Quadraticsax 2 bx c 0 c ab a Cubicsax 3 bx 2 cx d 0b ad ac a 9. Roots and Coefficients Quadraticsax 2 bx c 0 c ab a Cubicsax 3 bx 2 cx d 0b aQuarticsd ac aax 4 bx 3 cx 2 dx e 0 10. Roots and Coefficients Quadraticsax 2 bx c 0 c ab a Cubicsax 3 bx 2 cx d 0b aQuarticsd ac aax 4 bx 3 cx 2 dx e 0b a 11. Roots and Coefficients Quadraticsax 2 bx c 0 c ab a Cubicsax 3 bx 2 cx d 0b aQuarticsd ac aax 4 bx 3 cx 2 dx e 0b ac a 12. Roots and Coefficients Quadraticsax 2 bx c 0 c ab a Cubicsax 3 bx 2 cx d 0b aQuarticsd ac aax 4 bx 3 cx 2 dx e 0c b a a d a 13. Roots and Coefficients Quadraticsax 2 bx c 0 c ab a Cubicsax 3 bx 2 cx d 0b aQuarticsd ac aax 4 bx 3 cx 2 dx e 0c b a a d e 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 0 b a(sum of roots, one at a time) 16. For the polynomial equation;ax n bx n1 cx n2 dx n3 0 b a c a(sum of roots, one at a time) (sum of roots, two at a time) 17. For the polynomial equation;ax n bx n1 cx n2 dx n3 0 b a c a d a(sum of roots, one at a time) (sum of roots, two at a time) (sum of roots, three at a time) 18. For the polynomial equation;ax n bx n1 cx n2 dx n3 0 b a c a d a e a(sum of roots, one at a time) (sum of roots, two at a time) (sum of roots, three at a time) (sum of roots, four at a time) 19. For the polynomial equation;ax n bx n1 cx n2 dx n3 0 b a c a d a e aNote:(sum of roots, one at a time) (sum of roots, two at a time) (sum of roots, three at a time) (sum of roots, four at a time)2 2 2 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 7 5 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 7 5 2 3 2 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 7 5 2 3 2 1 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 7 5 2 5 1 4 4 4 7 4 7 2 23 2 1 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 7 5 2 5 1 4 4 4 7 4 7 2 2 27 23 2 1 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 7 5 2 5 1 4 4 4 7 4 7 2 2 27 2 1 1 1 b) 3 2 1 2 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 7 5 2 5 1 4 4 4 7 4 7 2 2 27 2 1 1 1 b) 3 2 1 2 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 7 5 2 5 1 4 4 4 7 4 7 2 2 27 2 1 1 1 b) 3 2 1 2 3 2 1 2 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 7 5 2 5 1 4 4 4 7 4 7 2 2 27 2 1 1 1 b) 3 2 1 2 3 3 2 1 2 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 7 5 2 3 2 5 1 4 4 4 7 4 7 2 2 27 2 1 1 1 b) c) 2 2 23 2 1 2 3 1 2 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 7 5 2 3 2 1 25 1 4 4 4 7 4 7 2 2 27 2 1 1 1 b) c) 2 2 2 2 2 3 2 1 2 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 7 5 2 3 2 1 25 1 4 4 4 7 4 7 2 2 27 2 1 1 1 b) c) 2 2 2 2 2 3 2 5 3 2 2 2 2 1 2 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 7 5 2 3 2 1 25 1 4 4 4 7 4 7 2 2 27 2 1 1 1 b) c) 2 2 2 2 2 3 2 5 3 2 2 2 2 1 37 2 4 3 34. 1988 Extension 1 HSC Q2c) 3 If , and are the roots of x 3 x 1 0 find: (i) 35. 1988 Extension 1 HSC Q2c) 3 If , and are the roots of x 3 x 1 0 find: (i) 0 36. 1988 Extension 1 HSC Q2c) 3 If , and are the roots of x 3 x 1 0 find: (i) 0 (ii) 37. 1988 Extension 1 HSC Q2c) 3 If , and are the roots of x 3 x 1 0 find: (i) 0 (ii) 1 38. 1988 Extension 1 HSC Q2c) 3 If , and are the roots of x 3 x 1 0 find: (i) 0 (ii) 1 (iii)111 39. 1988 Extension 1 HSC Q2c) 3 If , and are the roots of x 3 x 1 0 find: (i) 0 (ii) 1 (iii)111 111 40. 1988 Extension 1 HSC Q2c) 3 If , and are the roots of x 3 x 1 0 find: (i) 0 (ii) 1 (iii)111 1113 1 41. 1988 Extension 1 HSC Q2c) 3 If , and are the roots of x 3 x 1 0 find: (i) 0 (ii) 1 (iii)111 1113 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 are reciprocals 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 are reciprocals of each other. Find the value of k. 1 Let 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 are reciprocals of each other. Find the value of k. 1 Let 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 are reciprocals of each other. Find the value of k. 1 Let 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 are reciprocals of each other. Find the value of k. 1 Let the roots be , and 1 6 2 3P 3 0 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 are reciprocals of each other. Find the value of k. 1 Let the roots be , and 1 6 2 3P 3 0 2 3 3 k 3 6 0 32 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 are reciprocals of each other. Find the value of k. 1 Let the roots be , and 1 6 2 3P 3 0 2 3 3 k 3 6 0 32 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 are reciprocals of each other. Find the value of k. 1 Let the roots be , and 1 6 2 3P 3 0 2 3 3 k 3 6 0 32 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 are reciprocals of each other. Find the value of k. 1 Let the roots be , and 1 6 2 3P 3 0 2 3 3 k 3 6 0 32 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 real numbers, 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 real numbers, 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 real numbers, 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 real numbers, 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 real numbers, has three real zeros, 1, and (i) Find the value of r 1 r r 1 (ii) Find the value of s + t1 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 real numbers, has three real zeros, 1, and (i) Find the value of r 1 r r 1 (ii) Find the value of s + t1 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 real numbers, has three real zeros, 1, and (i) Find the value of r 1 r r 1 (ii) Find the value of s + t1 1 s s 21 t 58. 2006 Extension 1 HSC Q4a) The cubic polynomial P x x 3 rx 2 sx t , where r, s and t are real numbers, has three real zeros, 1, and (i) Find the value of r 1 r r 1 (ii) Find the value of s + t1 1 s s 21 t t 2 59. 2006 Extension 1 HSC Q4a) The cubic polynomial P x x 3 rx 2 sx t , where r, s and t are real numbers, has three real zeros, 1, and (i) Find the value of r 1 r r 1 (ii) Find the value of s + t1 1 s s 21 t t 2s t 0 60. Exercise 4F; 2, 4, 5ac, 6ac, 8, 10a, 13, 15, 16ad, 17, 18, 19</p>