| Exam Board | OCR MEI |
|---|---|
| Module | Further Extra Pure (Further Extra Pure) |
| Year | 2020 |
| Session | November |
| Marks | 8 |
| 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 guided multi-part question on eigenvectors with significant scaffolding. Part (a) requires routine verification that a given vector is an eigenvector, (b) is immediate from (a), while (c) and (d) test conceptual understanding of reflections (eigenvectors with eigenvalue 1 lie in the plane, eigenvalue -1 is perpendicular). The geometric interpretation is standard Further Maths content, making this slightly easier than average overall. |
| Spec | 4.03a Matrix language: terminology and notation4.03b Matrix operations: addition, multiplication, scalar4.03c Matrix multiplication: properties (associative, not commutative)4.03d Linear transformations 2D: reflection, rotation, enlargement, shear |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (b) | λ = 1 |
| f | B1 | 2.2a |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (c) | Since the e-val of any vector f is 1 then f must be |
| Answer | Marks |
|---|---|
| to the mirror plane. | B1 |
| Answer | Marks |
|---|---|
| [2] | 2.4 |
| 2.4 | SC1 using the word line |
| Answer | Marks | Guidance |
|---|---|---|
| 5 | (d) | Since e is the normal to the mirror plane and O must be |
| Answer | Marks |
|---|---|
| 1 0 1 | B1 |
| [1] | 3.1a |
Question 5:
5 | (b) | λ = 1
f | B1 | 2.2a
Completing and correct
conclusion
5 | (c) | Since the e-val of any vector f is 1 then f must be
parallel to (or lie in) the mirror plane.
Since the e-val of e is –1 then e must be perpendicular
to the mirror plane. | B1
B1
[2] | 2.4
2.4 | SC1 using the word line
instead of plane
Needs more than invariant
line/line of invariant points
5 | (d) | Since e is the normal to the mirror plane and O must be
in the plane the equation is
1 0 1
r. 1 = 0 . 1 =0⇒x+ y+z=0
1 0 1 | B1
[1] | 3.1a\begin{enumerate}[label=(\alph*)]
\item Show that $\mathbf { f }$ is also an eigenvector of $\mathbf { A }$.
\item State the eigenvalue associated with $\mathbf { f }$.
You are now given that $\mathbf { A }$ represents a reflection in 3-D space.
\item Explain the significance of $\mathbf { e }$ and $\mathbf { f }$ in relation to the transformation that $\mathbf { A }$ represents.
\item State the cartesian equation of the plane of reflection of the transformation represented by $\mathbf { A }$.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Extra Pure 2020 Q5 [8]}}