| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2017 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Invariant lines and eigenvalues and vectors |
| Type | Find eigenvalues/vectors of matrix combination |
| Difficulty | Standard +0.3 Part (i) requires standard eigenvector calculation for three given eigenvalues (routine substitution into (A-λI)v=0), while part (ii) tests understanding that subtracting 2I shifts eigenvalues by -2 but preserves eigenvectors. This is a straightforward Further Maths question requiring mechanical computation and basic eigenvalue theory, slightly above average due to 3×3 matrices and the conceptual element in part (ii). |
| Spec | 4.03a Matrix language: terminology and notation4.03b Matrix operations: addition, multiplication, scalar4.03c Matrix multiplication: properties (associative, not commutative) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Eigenvectors are \(\mathbf{i}+2\mathbf{j}+2\mathbf{k}\), \(\mathbf{i}+\mathbf{k}\) and \(\mathbf{j}+\mathbf{k}\) (OE) for \(\lambda = 1, -1\) and \(-2\) respectively | M1A1 | |
| (Award M1A1 for any one correct and A1 for each of the other two.) | A1A1 | |
| Total: 4 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Eigenvalues for B are \(-1, -3\) and \(-4\) | B1 | |
| Eigenvectors are the same as for A respectively | B1 FT | |
| Total: 2 |
## Question 5:
**Part (i):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Eigenvectors are $\mathbf{i}+2\mathbf{j}+2\mathbf{k}$, $\mathbf{i}+\mathbf{k}$ and $\mathbf{j}+\mathbf{k}$ (OE) for $\lambda = 1, -1$ and $-2$ respectively | M1A1 | |
| (Award M1A1 for any one correct and A1 for each of the other two.) | A1A1 | |
| **Total: 4** | | |
**Part (ii):**
| Answer | Marks | Guidance |
|--------|-------|----------|
| Eigenvalues for **B** are $-1, -3$ and $-4$ | B1 | |
| Eigenvectors are the same as for **A** respectively | B1 FT | |
| **Total: 2** | | |5 The matrix $\mathbf { A }$, given by
$$\mathbf { A } = \left( \begin{array} { l l l }
1 & 2 & - 2 \\
6 & 4 & - 6 \\
6 & 5 & - 7
\end{array} \right)$$
has eigenvalues $1 , - 1$ and - 2 .\\
(i) Find a set of corresponding eigenvectors.\\
(ii) The matrix $\mathbf { B }$ is given by $\mathbf { B } = \mathbf { A } - 2 \mathbf { I }$, where $\mathbf { I }$ is the $3 \times 3$ identity matrix. Write down the eigenvalues of $\mathbf { B }$, and state a set of corresponding eigenvectors.\\
\hfill \mbox{\textit{CAIE FP1 2017 Q5 [6]}}