**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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