Multiple choice

If a, b and c are roots of the equation 6x3 + 11x2 – 5 = 0, then the equation whose roots are (9a – 2b – 3c), (2a -3b – 9c) and (-9a + 7b + 14c) is

  1. 3x3 + 11x2 – 10 = 0

  2. 3x3 – 11x2 – 10 = 0

  3. x3 + 11x2 – 10 = 0

  4. 3x3 – 11x2 + 10 = 0

  5. x3 + 11x2 – 2x – 10 = 0
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AI explanation

The sum of the roots a, b and c is -11/6. Let the new roots be r1, r2 and r3. Their sum r1 + r2 + r3 equals (9a - 2b - 3c) + (2a - 3b - 9c) + (-9a + 7b + 14c), which simplifies to 2a + 2b + 2c. This equals 2(a + b + c) = 2(-11/6) = -11/3. For a cubic equation ux^3 + vx^2 + wx + t = 0, the sum of the roots is -v/u. Setting -v/u = -11/3 gives the ratio of the x^2 coefficient to the x^3 coefficient as 11/3. Only the equation 3x^3 + 11x^2 - 10 = 0 matches this required ratio.

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