TS EAMCET · Maths · Quadratic Equation

If $α, β, γ$ are the roots of the equation $4 x^{3} + 12 x^{2} - 7 x + 165 = 0$ and $α + 5, β + 5, γ + 5$ are the roots of the equation $a x^{3} + b x^{2} + c x + d = 0$ then the product of the roots of the second equation is

A $27$
B $0$
C $- 3$
D $3 \sqrt{5} + 4$

Verified Solution

Answer & Solution

Correct Answer

(B) $0$

Step-by-step Solution

Detailed explanation

It is given that α,β and γ are the roots at the equation 4x3+12x2-7x+165=0, therefore α+β+γ=-3 ⋯i αβ+βγ+γα=-74 ⋯ii…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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If α,β,γ are the roots of the equation 4x3+12x2-7x+165=0 and α+5,β+5,γ+5 are the roots of the equation ax3+bx2+cx+d=0 then the product of the roots of the second equation is

Step-by-step Solution

If $α, β, γ$ are the roots of the equation $4 x^{3} + 12 x^{2} - 7 x + 165 = 0$ and $α + 5, β + 5, γ + 5$ are the roots of the equation $a x^{3} + b x^{2} + c x + d = 0$ then the product of the roots of the second equation is