99 Percentile Qs Bank for JEE MainMathematicsDeterminants
If a ,   b and c are the roots of x 3 - x 2 - 2004 = 0 Then, the value of a + c a + b b + c a b + c b c + a c a + b a 2 + 1 b 2 + 1 c 2 + 1 is equal to
Options
- A- 2 a c 3 + 2 a c - 4 a 2 c + 2 a b 3 - 2 a b + 4 a b c
- B2 a c 3 + 2 a c - 4 a 2 c + 2 a b 3 - 2 a b + 4 a b c
- C- 2 a c 3 - a c - 4 a 2 c + 2 a b 3 + 2 a b + 4 a b c
- D- 2 a c 3 - a c - 4 a 2 c - 2 a b 3 - 2 a b - 4 a b c
Correct answer
A. - 2 a c 3 + 2 a c - 4 a 2 c + 2 a b 3 - 2 a b + 4 a b c
Step-by-step solution
Since, x 3 - x 2 - 2004 = 0 has three roots a ,   b ,   c ∴ a + b + c = 1       ⋯ i a b + b c + c a = 0       ⋯ ii a b c = 2004       … iii Now, given determinant, a + c a + b b + c a b + c b c + a c a + b a 2 + 1 b 2 + 1 c 2 + 1 = 1 - b 1 - c 1 - a - b c - c a - a b a 2 + 1 b 2 + 1 c 2 + 1 On applying C 1 → C 1 + C 2 + C 3 , we get 2 1 - c 1 - a 0 - c a - a b 4 b 2 + 1 c 2 + 1 = - 2 a 1 1 - c 1 - a 0 c b 0 b 2 + 2 c - 1 c 2 + 2 a