| Exam Board | Edexcel |
|---|---|
| Module | F3 (Further Pure Mathematics 3) |
| Year | 2023 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | 3x3 Matrices |
| Type | Eigenvalues and eigenvectors |
| Difficulty | Challenging +1.2 This is a Further Maths question requiring students to find conditions for repeated eigenvalues by setting the discriminant of the characteristic equation to zero, then solving for k and finding eigenvalues. While it involves multiple steps (characteristic polynomial, discriminant condition, solving cubic/quadratic), the techniques are standard for FM students and the algebraic manipulation, though somewhat involved, follows a well-practiced procedure without requiring novel insight. |
| Spec | 4.03l Singular/non-singular matrices |
| Answer | Marks | Guidance |
|---|---|---|
| \(\det(\mathbf{M} - \lambda\mathbf{I})\) expanded fully | M1 | Recognisable complete attempt at \(\det(\mathbf{M}-\lambda\mathbf{I})\); may use other rows/columns; allow \(\pm\) and slips including \(+2\) for first \(-2\) |
| \(= (1-\lambda)(\lambda^2+11\lambda) - 2\lambda + 2k\lambda = -\lambda^3 - 10\lambda^2 + 9\lambda + 2k\lambda = \lambda(-\lambda^2-10\lambda+9+2k)\) | M1 A1 | M1: obtains \(\{\lambda\}(a\lambda^2+b\lambda+c+dk)\), \(a,b,c,d\neq 0\); A1: correct expression, allow \(\pm\{\lambda\}(-\lambda^2-10\lambda+9+2k)\) or equivalent unsimplified form |
| For part (i): One eigenvalue zero; if repeated then \(9+2k=0 \Rightarrow k=\ldots\) | M1 | Attempts to set \(c+dk=0\) and solves for \(k\), OR considers quadratic having repeated root and uses valid strategy |
| OR \(b^2-4ac=0 \Rightarrow 100-4(-1)(9+2k)=0\) or \(100-4(1)(-9-2k)=0 \Rightarrow k=\ldots\) | M1 | |
| \(k = -\frac{9}{2}\) or \(k=-17\) | A1 | One correct value of \(k\) |
| Answer | Marks | Guidance |
|---|---|---|
| Both conditions applied simultaneously | M1 | Attempts both \(c+dk=0\) AND repeated root condition |
| \(k=-\frac{9}{2}\) with eigenvalue \(-10\) (and \(0\) repeated) | A1 | Both correct values of \(k\) with associated non-zero eigenvalues clearly assigned; no additional eigenvalues or values for \(k\) |
| \(k=-17\) with eigenvalue \(-5\) (repeated and \(0\)) |
## Question 5 (i) & (ii):
$\det(\mathbf{M} - \lambda\mathbf{I})$ expanded fully | M1 | Recognisable complete attempt at $\det(\mathbf{M}-\lambda\mathbf{I})$; may use other rows/columns; allow $\pm$ and slips including $+2$ for first $-2$
$= (1-\lambda)(\lambda^2+11\lambda) - 2\lambda + 2k\lambda = -\lambda^3 - 10\lambda^2 + 9\lambda + 2k\lambda = \lambda(-\lambda^2-10\lambda+9+2k)$ | M1 A1 | M1: obtains $\{\lambda\}(a\lambda^2+b\lambda+c+dk)$, $a,b,c,d\neq 0$; A1: correct expression, allow $\pm\{\lambda\}(-\lambda^2-10\lambda+9+2k)$ or equivalent unsimplified form
**For part (i):** One eigenvalue zero; if repeated then $9+2k=0 \Rightarrow k=\ldots$ | M1 | Attempts to set $c+dk=0$ and solves for $k$, OR considers quadratic having repeated root and uses valid strategy
**OR** $b^2-4ac=0 \Rightarrow 100-4(-1)(9+2k)=0$ or $100-4(1)(-9-2k)=0 \Rightarrow k=\ldots$ | M1 |
$k = -\frac{9}{2}$ or $k=-17$ | A1 | One correct value of $k$
**For part (ii):**
Both conditions applied simultaneously | M1 | Attempts both $c+dk=0$ AND repeated root condition
$k=-\frac{9}{2}$ with eigenvalue $-10$ (and $0$ repeated) | A1 | Both correct values of $k$ with associated non-zero eigenvalues clearly assigned; no additional eigenvalues or values for $k$
$k=-17$ with eigenvalue $-5$ (repeated and $0$) | |
**Total: 7 marks**5.
$$\mathbf { M } = \left( \begin{array} { r r r }
1 & 2 & k \\
- 1 & - 3 & 4 \\
2 & 6 & - 8
\end{array} \right) \quad \text { where } k \text { is a constant }$$
Given that $\mathbf { M }$ has a repeated eigenvalue, determine\\
(i) the possible values of $k$,\\
(ii) all corresponding eigenvalues of $\mathbf { M }$ for each value of $k$.
\hfill \mbox{\textit{Edexcel F3 2023 Q5 [7]}}