8 The cubic equation \(k x ^ { 3 } + 6 x ^ { 2 } + x - 3 = 0\), where \(k\) is a non-zero constant, has roots \(\alpha , \beta\) and \(\gamma\).
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Question 8:
Answer Marks
Guidance
Answer Marks
Guidance
*Either:* \(\sum\alpha = -\frac{6}{k},\ \sum\alpha\beta = \frac{1}{k}\) B1B1
Correct values stated or used
\(\sum\alpha\beta + 2\sum\alpha + 3\) M1
Expand brackets
A1 Obtain correct expression aef
\(3 - \frac{11}{k}\) M1
Use their values, in terms of \(k\), for \(\sum\alpha\) and \(\sum\alpha\beta\)
A1 Obtain correct answer aef
[6] *Or:* \(x = u - 1\) B1
State or use substitution \(x = u-1\)
M1 Expand and attempt to simplify coefficients
\(ku^3 + (6-3k)u^2 + (3k-11)u + 2 - k = 0\) A1A1
Obtain at least correct \(1^{st}\) and \(3^{rd}\) terms
M1 Use their \(\frac{c}{a}\)
\(3 - \frac{11}{k}\) A1
Obtain correct answer a.e.f.
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## Question 8:
| Answer | Marks | Guidance |
|--------|-------|----------|
| *Either:* | | |
| $\sum\alpha = -\frac{6}{k},\ \sum\alpha\beta = \frac{1}{k}$ | B1B1 | Correct values stated or used |
| $\sum\alpha\beta + 2\sum\alpha + 3$ | M1 | Expand brackets |
| | A1 | Obtain correct expression aef |
| $3 - \frac{11}{k}$ | M1 | Use their values, in terms of $k$, for $\sum\alpha$ and $\sum\alpha\beta$ |
| | A1 | Obtain correct answer aef |
| **[6]** | | |
| *Or:* | | |
| $x = u - 1$ | B1 | State or use substitution $x = u-1$ |
| | M1 | Expand and attempt to simplify coefficients |
| $ku^3 + (6-3k)u^2 + (3k-11)u + 2 - k = 0$ | A1A1 | Obtain at least correct $1^{st}$ and $3^{rd}$ terms |
| | M1 | Use their $\frac{c}{a}$ |
| $3 - \frac{11}{k}$ | A1 | Obtain correct answer a.e.f. |
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8 The cubic equation $k x ^ { 3 } + 6 x ^ { 2 } + x - 3 = 0$, where $k$ is a non-zero constant, has roots $\alpha , \beta$ and $\gamma$.\\
Find the value of $( \alpha + 1 ) ( \beta + 1 ) + ( \beta + 1 ) ( \gamma + 1 ) + ( \gamma + 1 ) ( \alpha + 1 )$ in terms of $k$.
\hfill \mbox{\textit{OCR FP1 2013 Q8 [6]}}