AP EAMCET · Maths · Quadratic Equation
If \(\alpha, \beta, \gamma\) are the roots of the equation \(\mathrm{x}^3+\mathrm{px}^2+\mathrm{qx}+\mathrm{r}=0\), then \(\alpha^3+\beta^3+\gamma^3=\)
- A \(p^3-3 p q+r\)
- B \(\mathrm{p}^2-2 \mathrm{pq}+\mathrm{r}\)
- C \(3 p q-3 r-p^3\)
- D \(3 p q+3 r+p^3\)
Answer & Solution
Correct Answer
(C) \(3 p q-3 r-p^3\)
Step-by-step Solution
Detailed explanation
\(\sum\alpha = -p\) \(\sum\alpha\beta = q\) \(\sum\alpha^2 = (\sum\alpha)^2 - 2\sum\alpha\beta = (-p)^2 - 2q = p^2 - 2q\) \(\alpha^3 = -p\alpha^2 - q\alpha - r\) \(\sum\alpha^3 = -p\sum\alpha^2 - q\sum\alpha - 3r\) \(\sum\alpha^3 = -p(p^2-2q) - q(-p) - 3r\)…
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