| Exam Board | Edexcel |
|---|---|
| Module | F3 (Further Pure Mathematics 3) |
| Year | 2020 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Invariant lines and eigenvalues and vectors |
| Type | Normalized eigenvectors |
| Difficulty | Standard +0.3 This is a straightforward multi-part eigenvalue question requiring standard techniques: using det(M - λI) = 0 to find k, solving the characteristic equation for remaining eigenvalues, and normalizing an eigenvector. All steps are routine Further Maths procedures with no novel insight required, making it slightly easier than average. |
| Spec | 4.03i Determinant: area scale factor and orientation4.03j Determinant 3x3: calculation |
| VIIIV SIHI NI JIIHM ION OC | VARV SHAL NI ALIAM LON OO | VERV SIHI NI JIIIM ION OO |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Notes |
| \( | \mathbf{M}-\lambda\mathbf{I} | \) evaluated with \(\lambda=3\): \((0)+4[2-k]+k[-5+5]=0\) |
| \((0)+4[2-k]+k[-5+5]=0 \Rightarrow k=\ldots\) | M1 | Uses \( |
| \(k=2\) | A1 | cao |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Notes |
| \((3-\lambda)\left[(\lambda+2)(\lambda-5)+10\right]+4(5-\lambda-2)+2(-5+2+\lambda)=0\) | M1 | Attempts \( |
| \((\lambda+2)(\lambda-5)+12 \Rightarrow \lambda^2-3\lambda+2=0 \Rightarrow (\lambda-2)(\lambda-1)=0\) | M1 | Uses \(\lambda=3\) as a factor to obtain and solve a 3TQ to find other eigenvalues (alternatively may use calculator to solve \(\lambda^3-6\lambda^2+11\lambda-6=0\)) |
| \(\lambda = 1, 2\) | A1 | Correct values |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Notes |
| \(\begin{pmatrix}3&-4&2\\1&-2&2\\1&-5&5\end{pmatrix}\begin{pmatrix}x\\y\\z\end{pmatrix}=3\begin{pmatrix}x\\y\\z\end{pmatrix} \Rightarrow \begin{cases}3x-4y+2z=3x\\x-2y+2z=3y\\x-5y+5z=3z\end{cases}\) | M1 | Uses eigenvalue 3 and their \(k\) to form at least 2 equations in \(x\), \(y\) and \(z\) |
| \(\alpha\begin{pmatrix}1\\1\\2\end{pmatrix}\) (\(\alpha\) a constant) | A1 | Any correct eigenvector; allow any constant multiple of \(\mathbf{i}+\mathbf{j}+2\mathbf{k}\) |
| \(\frac{1}{\sqrt{6}}\begin{pmatrix}1\\1\\2\end{pmatrix}\) | A1 | Correct normalised vector |
# Question 3:
$$\mathbf{M} = \begin{pmatrix} 3 & -4 & k \\ 1 & -2 & k \\ 1 & -5 & 5 \end{pmatrix}$$
## Part (a):
| Working/Answer | Mark | Notes |
|---|---|---|
| $|\mathbf{M}-\lambda\mathbf{I}|$ evaluated with $\lambda=3$: $(0)+4[2-k]+k[-5+5]=0$ | M1 | Attempts $|\mathbf{M}-\lambda\mathbf{I}|$ using $\lambda=3$ |
| $(0)+4[2-k]+k[-5+5]=0 \Rightarrow k=\ldots$ | M1 | Uses $|\mathbf{M}-\lambda\mathbf{I}|=0$ and solves for $k$ |
| $k=2$ | A1 | cao |
## Part (b):
| Working/Answer | Mark | Notes |
|---|---|---|
| $(3-\lambda)\left[(\lambda+2)(\lambda-5)+10\right]+4(5-\lambda-2)+2(-5+2+\lambda)=0$ | M1 | Attempts $|\mathbf{M}-\lambda\mathbf{I}|=0$ using their value of $k$ |
| $(\lambda+2)(\lambda-5)+12 \Rightarrow \lambda^2-3\lambda+2=0 \Rightarrow (\lambda-2)(\lambda-1)=0$ | M1 | Uses $\lambda=3$ as a factor to obtain and solve a 3TQ to find other eigenvalues (alternatively may use calculator to solve $\lambda^3-6\lambda^2+11\lambda-6=0$) |
| $\lambda = 1, 2$ | A1 | Correct values |
## Part (c):
| Working/Answer | Mark | Notes |
|---|---|---|
| $\begin{pmatrix}3&-4&2\\1&-2&2\\1&-5&5\end{pmatrix}\begin{pmatrix}x\\y\\z\end{pmatrix}=3\begin{pmatrix}x\\y\\z\end{pmatrix} \Rightarrow \begin{cases}3x-4y+2z=3x\\x-2y+2z=3y\\x-5y+5z=3z\end{cases}$ | M1 | Uses eigenvalue 3 and their $k$ to form at least 2 equations in $x$, $y$ and $z$ |
| $\alpha\begin{pmatrix}1\\1\\2\end{pmatrix}$ ($\alpha$ a constant) | A1 | Any correct eigenvector; allow any constant multiple of $\mathbf{i}+\mathbf{j}+2\mathbf{k}$ |
| $\frac{1}{\sqrt{6}}\begin{pmatrix}1\\1\\2\end{pmatrix}$ | A1 | Correct normalised vector |
---3.
$$\mathbf { M } = \left( \begin{array} { c c c }
3 & - 4 & k \\
1 & - 2 & k \\
1 & - 5 & 5
\end{array} \right) \text { where } k \text { is a constant }$$
Given that 3 is an eigenvalue of $\mathbf { M }$,
\begin{enumerate}[label=(\alph*)]
\item find the value of $k$.
\item Hence find the other two eigenvalues of $\mathbf { M }$.
\item Find a normalised eigenvector corresponding to the eigenvalue 3\\
.
\begin{center}
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VIIIV SIHI NI JIIHM ION OC & VARV SHAL NI ALIAM LON OO & VERV SIHI NI JIIIM ION OO \\
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\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel F3 2020 Q3 [9]}}