3 The cubic equation \(x ^ { 3 } - 2 x ^ { 2 } - 3 x + 4 = 0\) has roots \(\alpha , \beta , \gamma\). Given that \(c = \alpha + \beta + \gamma\), state the value of \(c\).
Use the substitution \(y = c - x\) to find a cubic equation whose roots are \(\alpha + \beta , \beta + \gamma , \gamma + \alpha\).
Find a cubic equation whose roots are \(\frac { 1 } { \alpha + \beta } , \frac { 1 } { \beta + \gamma } , \frac { 1 } { \gamma + \alpha }\).
Hence evaluate \(\frac { 1 } { ( \alpha + \beta ) ^ { 2 } } + \frac { 1 } { ( \beta + \gamma ) ^ { 2 } } + \frac { 1 } { ( \gamma + \alpha ) ^ { 2 } }\).
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Question 3:
Answer Marks
Guidance
Answer/Working Marks
Guidance
\(c = 2\) B1
Uses \(\sum\alpha = \frac{-b}{a}\)
\((\alpha+\beta = c-\gamma \text{ etc.}) \Rightarrow y=c-x \Rightarrow x=c-y\) M1, M1
Uses substitution
\((2-y)^3 - 2(2-y)^2 - 3(2-y)+4=0\) …(their \(c\)) \(\Rightarrow y^3 - 4y^2 + y + 2 = 0\) A1
Obtains required cubic equation
Uses \(z=y^{-1}\) to obtain \(2z^3+z^2-4z+1=0\) M1A1
Obtains equation whose roots are reciprocals of those in previous cubic
\(\sum\frac{1}{(\alpha+\beta)^2} = \left(\frac{1}{2}\right)^2 - 2(-2) = 4\frac{1}{4}\) M1A1
Uses \(\sum\alpha^2=(\sum\alpha)^2-2\sum\alpha\beta\)
Total: [8]
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## Question 3:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $c = 2$ | B1 | Uses $\sum\alpha = \frac{-b}{a}$ |
| $(\alpha+\beta = c-\gamma \text{ etc.}) \Rightarrow y=c-x \Rightarrow x=c-y$ | M1, M1 | Uses substitution |
| $(2-y)^3 - 2(2-y)^2 - 3(2-y)+4=0$ …(their $c$) | | |
| $\Rightarrow y^3 - 4y^2 + y + 2 = 0$ | A1 | Obtains required cubic equation |
| Uses $z=y^{-1}$ to obtain $2z^3+z^2-4z+1=0$ | M1A1 | Obtains equation whose roots are reciprocals of those in previous cubic |
| $\sum\frac{1}{(\alpha+\beta)^2} = \left(\frac{1}{2}\right)^2 - 2(-2) = 4\frac{1}{4}$ | M1A1 | Uses $\sum\alpha^2=(\sum\alpha)^2-2\sum\alpha\beta$ |
**Total: [8]**
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3 The cubic equation $x ^ { 3 } - 2 x ^ { 2 } - 3 x + 4 = 0$ has roots $\alpha , \beta , \gamma$. Given that $c = \alpha + \beta + \gamma$, state the value of $c$.
Use the substitution $y = c - x$ to find a cubic equation whose roots are $\alpha + \beta , \beta + \gamma , \gamma + \alpha$.
Find a cubic equation whose roots are $\frac { 1 } { \alpha + \beta } , \frac { 1 } { \beta + \gamma } , \frac { 1 } { \gamma + \alpha }$.
Hence evaluate $\frac { 1 } { ( \alpha + \beta ) ^ { 2 } } + \frac { 1 } { ( \beta + \gamma ) ^ { 2 } } + \frac { 1 } { ( \gamma + \alpha ) ^ { 2 } }$.
\hfill \mbox{\textit{CAIE FP1 2013 Q3 [8]}}