| Exam Board | Edexcel |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Year | 2013 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Invariant lines and eigenvalues and vectors |
| Type | Find eigenvectors given eigenvalue |
| Difficulty | Standard +0.3 This is a structured multi-part question on eigenvalues/eigenvectors with given information that significantly simplifies the work. Part (a) is direct computation using Av=λv; part (b) involves solving simultaneous equations from the matrix multiplication; part (c) requires finding remaining eigenvalues of a 3×3 matrix. While it involves Further Maths content, the question provides substantial scaffolding and uses standard techniques throughout, making it slightly easier than average. |
| Spec | 4.03a Matrix language: terminology and notation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\begin{pmatrix} 4 & 2 & 3 \\ 2 & b & 0 \\ a & 1 & 8 \end{pmatrix}\begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix} = \begin{pmatrix} 8 \\ \ldots \\ \ldots \end{pmatrix} = \lambda\begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix}\), \(\lambda = 8\) | M1 | Multiplies the given matrix by the given eigenvector |
| Equates the \(x\) value to \(\lambda\) | M1 | |
| \(\lambda = 8\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(2 + 2b = 2\lambda\) or \(a + 2 = 0\) | M1 | |
| \(b = 7\) or \(a = -2\) | A1 | |
| \(b = 7\) and \(a = -2\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\begin{vmatrix} 4-\lambda & 2 & 3 \\ 2 & 7-\lambda & 0 \\ -2 & 1 & 8-\lambda \end{vmatrix} = 0\) | M1 | Correct attempt to establish Characteristic Equation; \(= 0\) required but may be implied by later work; allow in terms of \(a, b, c\) |
| \((4-\lambda)(7-\lambda)(8-\lambda) - 2\times2(8-\lambda) + 3(2+2(7-\lambda)) = 0\) | ||
| Attempts to factorise e.g. \((8-\lambda)(30 - 11\lambda + \lambda^2)\) or \((6-\lambda)(40 - 13\lambda + \lambda^2)\) or \((5-\lambda)(48 - 14\lambda + \lambda^2)\); NB \(240 - 118\lambda + 19\lambda^2 - \lambda^3 = 0\) | M1 A1 | M1: Identify linear factor and simplified quadratic; A1: Correct factorisation |
| Eigenvalues are \(5\) and \(6\) | M1 A1 | M1: Solves equation to obtain other eigenvalues; A1: \(5\) and \(6\) |
## Question 6:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\begin{pmatrix} 4 & 2 & 3 \\ 2 & b & 0 \\ a & 1 & 8 \end{pmatrix}\begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix} = \begin{pmatrix} 8 \\ \ldots \\ \ldots \end{pmatrix} = \lambda\begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix}$, $\lambda = 8$ | M1 | Multiplies the given matrix by the given eigenvector |
| Equates the $x$ value to $\lambda$ | M1 | |
| $\lambda = 8$ | A1 | |
**(3 marks)**
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $2 + 2b = 2\lambda$ or $a + 2 = 0$ | M1 | |
| $b = 7$ **or** $a = -2$ | A1 | |
| $b = 7$ **and** $a = -2$ | A1 | |
**(3 marks)**
### Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\begin{vmatrix} 4-\lambda & 2 & 3 \\ 2 & 7-\lambda & 0 \\ -2 & 1 & 8-\lambda \end{vmatrix} = 0$ | M1 | Correct attempt to establish Characteristic Equation; $= 0$ required but may be implied by later work; allow in terms of $a, b, c$ |
| $(4-\lambda)(7-\lambda)(8-\lambda) - 2\times2(8-\lambda) + 3(2+2(7-\lambda)) = 0$ | | |
| Attempts to factorise e.g. $(8-\lambda)(30 - 11\lambda + \lambda^2)$ or $(6-\lambda)(40 - 13\lambda + \lambda^2)$ or $(5-\lambda)(48 - 14\lambda + \lambda^2)$; NB $240 - 118\lambda + 19\lambda^2 - \lambda^3 = 0$ | M1 A1 | M1: Identify linear factor and simplified quadratic; A1: Correct factorisation |
| Eigenvalues are $5$ and $6$ | M1 A1 | M1: Solves equation to obtain other eigenvalues; A1: $5$ and $6$ |
**(5 marks) — Total 8**
---6. It is given that $\left( \begin{array} { l } 1 \\ 2 \\ 0 \end{array} \right)$ is an eigenvector of the matrix $\mathbf { A }$, where
$$\mathbf { A } = \left( \begin{array} { l l l }
4 & 2 & 3 \\
2 & b & 0 \\
a & 1 & 8
\end{array} \right)$$
and $a$ and $b$ are constants.
\begin{enumerate}[label=(\alph*)]
\item Find the eigenvalue of $\mathbf { A }$ corresponding to the eigenvector $\left( \begin{array} { l } 1 \\ 2 \\ 0 \end{array} \right)$.
\item Find the values of $a$ and $b$.
\item Find the other eigenvalues of $\mathbf { A }$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP3 2013 Q6 [11]}}