3 The equation
$$x ^ { 3 } + 5 x ^ { 2 } - 3 x - 15 = 0$$
has roots \(\alpha , \beta , \gamma\). Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\).
Hence show that the matrix \(\left( \begin{array} { c c c } 1 & \alpha & \beta \\ \alpha & 1 & \gamma \\ \beta & \gamma & 1 \end{array} \right)\) is singular.
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Question 3:
Answer Marks
Guidance
Working/Answer Marks
Guidance
\(\sum\alpha = -5\), \(\quad \sum\alpha\beta = -3\) B1
Uses known results
\(\sum\alpha^2 = (\sum\alpha)^2 - 2\sum\alpha\beta = (-5)^2 - 2\times(-3) = 31\) M1A1
Part marks: 3
\(\det\begin{pmatrix}1&\alpha&\beta\\\alpha&1&\gamma\\\beta&\gamma&1\end{pmatrix} = 1-(\alpha^2+\beta^2+\gamma^2)+2\alpha\beta\gamma\) M1A1
Evaluates determinant
\(\alpha\beta\gamma = -(-15) = 15\) \(\Rightarrow 1 - 31 + 2\times15\) M1
\(= 0 \Rightarrow\) matrix is singular A1
Part marks: 4; Total: [7]
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## Question 3:
| Working/Answer | Marks | Guidance |
|---|---|---|
| $\sum\alpha = -5$, $\quad \sum\alpha\beta = -3$ | B1 | Uses known results |
| $\sum\alpha^2 = (\sum\alpha)^2 - 2\sum\alpha\beta = (-5)^2 - 2\times(-3) = 31$ | M1A1 | Part marks: 3 |
| $\det\begin{pmatrix}1&\alpha&\beta\\\alpha&1&\gamma\\\beta&\gamma&1\end{pmatrix} = 1-(\alpha^2+\beta^2+\gamma^2)+2\alpha\beta\gamma$ | M1A1 | Evaluates determinant |
| $\alpha\beta\gamma = -(-15) = 15$ | | |
| $\Rightarrow 1 - 31 + 2\times15$ | M1 | |
| $= 0 \Rightarrow$ matrix is singular | A1 | Part marks: 4; **Total: [7]** |
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3 The equation
$$x ^ { 3 } + 5 x ^ { 2 } - 3 x - 15 = 0$$
has roots $\alpha , \beta , \gamma$. Find the value of $\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }$.
Hence show that the matrix $\left( \begin{array} { c c c } 1 & \alpha & \beta \\ \alpha & 1 & \gamma \\ \beta & \gamma & 1 \end{array} \right)$ is singular.
\hfill \mbox{\textit{CAIE FP1 2011 Q3 [7]}}