Quantitative Ability-Question of the Day

Question:-
If the roots of the equation x^3 + 2x^2 – 3x + 1 = 0 are A, B and C, and the roots of the equation x^3 + px^2 + qx + r = 0 are (A + B), (B + C) and (C + A), what is the value of the product pr?

OPTIONS
1) 28
2) −5
3) 1
4) 5
5) –28

Solution
From observation of the coefficients of the second equation, we can see that the product of its roots is –r.
Hence, (A + B)(B + C)(C + A) = −r, or (A + B + C − C)(A + B + C − A)(A + B + C – B) = −r
On observing the coefficients of the first equation, we can see that the sum of its roots is −2.
∴  A + B + C = −2
Hence, (−2 – C)(−2 – A)(−2 – B) = −r, or (2 + A)(2 + B)(2 + C) = r
Thus, (2 + A)(4 + 2B + 2C + BC) = r
∴ 8 + 4(A + B + C) + 2(AB + BC + AC) + ABC = r
From the first equation, AB + BC + AC = –3
∴ 8 + 4(−2) + 2(–3) – 1 = r
∴ r = –7
From the second equation, we can write –p = A + B + B + C + C + A = 2(A + B + C) = 2(−2) = –4
∴ p = 4
∴ pr = 4 × (–7) = –28
Hence, option 5.


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