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CAT 2008 — QA Question 12

Higher Degree PolynomialsEasy
Passage / Data

Directions for next 2 questions:

The figure below shows the plan of a town. The streets are at right angles to each other. A rectangular park (P) is situated inside the town with a diagonal road running through it. There is also a prohibited region (D) in the town.

If the roots of the equation x3− ax2 + bx – c = 0 are three consecutive integers, then what is the smallest possible value of b?

Answer & solution

  • A

    -13

  • -1

  • C

    0

  • D

    1

  • E

    13

Solution

If p, q and r are the roots of a cubic equation ax3 + bx2 + cx + d = 0,

then pq + pr + qr = c/a

If a is 1, then pq + qr + pr = c

Comparing the equation ax3 + bx2 + cx + d = 0 with the equation in the question x3− ax2 + bx – c = 0, we get

pq + qr + pr = b

Let the three roots of the given equation be (n – 1), n and (n + 1).

∴ (n – 1)n + n(n + 1) + (n – 1)(n + 1) = b

∴ n2 – n + n2 + n + n2 – 1 = b

∴ 3n2 – 1 = b

∵ n2 ≥ 0, minimum value of b occurs at n = 0

∴ Minimum value of b = –1

Hence, option (b).

CAT 2008 QA Q12: If the roots of the equation x 3 − ax 2 + bx – c = 0 are three consecutive integers, then what is — Solution | TheCATExam