| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 1 (Further Pure Core 1) |
| Year | 2021 |
| Session | November |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Invariant lines and eigenvalues and vectors |
| Type | Verify invariant line property |
| Difficulty | Standard +0.3 This is a standard Further Maths FP1 question on invariant lines under matrix transformations. Part (a) requires setting up the condition that points on y=kx map back to the same line, leading to a straightforward eigenvalue-type calculation. Part (b) is a simple check of whether points are fixed. The technique is well-practiced in FP1 with no novel insight required, making it slightly easier than average overall. |
| Spec | 4.03g Invariant points and lines |
| Answer | Marks | Guidance |
|---|---|---|
| 9 | (a) | 2 1 x 2x+kx |
| Answer | Marks |
|---|---|
| ( i.e. y =−x ) | M1 |
| Answer | Marks |
|---|---|
| A1 | 3.1a |
| Answer | Marks |
|---|---|
| 1.1 | Value of k can be implied by the correct equation |
| Answer | Marks |
|---|---|
| (b) | 2 1 x 2x−x x |
| Answer | Marks | Guidance |
|---|---|---|
| to itself and it is a line of invariant points | B1 | 2.4 |
Question 9:
9 | (a) | 2 1 x 2x+kx
=
−1 0kx −x
same line⇒−x=k(2x+kx) for all x (≠0 )
⇒−1=k(2+k)⇒k2 +2k+1=0
⇒k =−1
( i.e. y =−x ) | M1
A1
M1
A1 | 3.1a
1.1
2.1
1.1 | Value of k can be implied by the correct equation
[4]
(b) | 2 1 x 2x−x x
= = so each point maps
−1 0−x −x −x
to itself and it is a line of invariant points | B1 | 2.4 | Must have a reason
e.g. it is sufficient to test one point other than (0, 0)
[1]You are given that the matrix $\begin{pmatrix} 2 & 1 \\ -1 & 0 \end{pmatrix}$ represents a transformation T.
\begin{enumerate}[label=(\alph*)]
\item You are given that the line with equation $y = kx$ is invariant under T.
Determine the value of $k$. [4]
\item Determine whether the line with equation $y = kx$ in part (a) is a line of invariant points under T. [1]
\end{enumerate}
\hfill \mbox{\textit{OCR Further Pure Core 1 2021 Q9 [5]}}