| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2018 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Invariant lines and eigenvalues and vectors |
| Type | Find eigenvectors given eigenvalue |
| Difficulty | Moderate -0.3 This is a straightforward Further Maths question testing basic linear algebra concepts. Part (i) requires computing a 3×3 determinant to verify linear independence (routine calculation), and part (ii) involves solving a 3×3 system of linear equations. Both are standard textbook exercises with no conceptual difficulty or novel insight required, though the calculations are slightly more involved than typical A-level questions due to the Further Maths context. |
| Spec | 4.04b Plane equations: cartesian and vector forms |
| Answer | Marks |
|---|---|
| 1(i) | EITHER: |
| Answer | Marks | Guidance |
|---|---|---|
| 1 0 4 | M1 A1 | Calculates determinant |
| Answer | Marks | Guidance |
|---|---|---|
| ⇒β=0⇒α=γ=0 | (M1A1) | Solves homogeneous system of equations. |
| Answer | Marks | Guidance |
|---|---|---|
| R3. | A1 | States either that vectors are linearly independent or that |
| Answer | Marks |
|---|---|
| 1(ii) | 1 2 3 0 |
| Answer | Marks | Guidance |
|---|---|---|
| ⇒l=m=−1 and n=1. | M1 | Sets up system of equations. |
| ⇒d=c−b−a | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Question | Answer | Marks |
Question 1:
--- 1(i) ---
1(i) | EITHER:
1 2 3
2 9 3 =36−10−3×9=−1≠0
1 0 4 | M1 A1 | Calculates determinant
OR:
α +2β+3γ =0
2α+9β+3γ =0 ⇒2β−γ=0
α +4γ=0
9β−5γ=0
⇒β=0⇒α=γ=0 | (M1A1) | Solves homogeneous system of equations.
Eliminates one variable.
Therefore a,b,c are linearly independent (and span R3) so form a basis for
R3. | A1 | States either that vectors are linearly independent or that
vectors span R3.
3
--- 1(ii) ---
1(ii) | 1 2 3 0
l 2 +m 9 +n 3 = −8
1 0 4 3
⇒l=m=−1 and n=1. | M1 | Sets up system of equations.
⇒d=c−b−a | A1
2
Question | Answer | Marks | GuidanceThe vectors $\mathbf{a}$, $\mathbf{b}$, $\mathbf{c}$ and $\mathbf{d}$ in $\mathbb{R}^3$ are given by
$$\mathbf{a} = \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}, \quad \mathbf{b} = \begin{pmatrix} 2 \\ 9 \\ 0 \end{pmatrix}, \quad \mathbf{c} = \begin{pmatrix} 3 \\ 5 \\ 4 \end{pmatrix} \quad \text{and} \quad \mathbf{d} = \begin{pmatrix} 0 \\ -8 \\ -3 \end{pmatrix}.$$
\begin{enumerate}[label=(\roman*)]
\item Show that $\{\mathbf{a}, \mathbf{b}, \mathbf{c}\}$ is a basis for $\mathbb{R}^3$. [3]
\item Express $\mathbf{d}$ in terms of $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$. [2]
\end{enumerate}
\hfill \mbox{\textit{CAIE FP1 2018 Q1 [5]}}