| Exam Board | OCR MEI |
|---|---|
| Module | Further Pure Core AS (Further Pure Core AS) |
| Year | 2019 |
| Session | June |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Invariant lines and eigenvalues and vectors |
| Type | Find constant from invariant line or area condition |
| Difficulty | Standard +0.8 This is a multi-part Further Maths question requiring determinant conditions, eigenvalue analysis for invariant lines, and applying the area scaling property (det M). Part (a)(i) requires completing the square on a quadratic inequality, (b) involves finding eigenvalues/eigenvectors (characteristic equation with discriminant analysis), and (c) connects determinant to area scaling. While each technique is standard, the combination across multiple concepts and the algebraic manipulation required elevates this above typical A-level questions. |
| Spec | 4.03g Invariant points and lines4.03h Determinant 2x2: calculation4.03i Determinant: area scale factor and orientation |
| Answer | Marks | Guidance |
|---|---|---|
| 6 | (a) | (i) |
| Answer | Marks |
|---|---|
| ⇒−4<k <0 | B1 |
| Answer | Marks |
|---|---|
| [3] | 1.1 |
| Answer | Marks | Guidance |
|---|---|---|
| 1.1 | attempt to find discriminant | or (λ−k/2)2 > k2/4 + k |
| 6 | (a) | (ii) |
| preserves the orientation of shapes | B1 | |
| [1] | 1.2 | condone ‘doesn’t reflect’ |
| 6 | (b) | λ −2 x λx−2y |
| Answer | Marks |
|---|---|
| for m, i.e. there are no invariant lines | B1 |
| Answer | Marks |
|---|---|
| [4] | 2.1 |
| Answer | Marks |
|---|---|
| 2.3 | oe (e.g. with y = mx [+ c]) |
| Answer | Marks |
|---|---|
| x+(λ+2)y = m(λx−2y)[+c] | x+(λ+2)(mx+c) |
| Answer | Marks | Guidance |
|---|---|---|
| 6 | (c) | detM =λ2 +2λ+2=5 |
| Answer | Marks |
|---|---|
| ⇒λ=−3 or 1 | M1 |
| Answer | Marks |
|---|---|
| [3] | 3.1a |
| Answer | Marks |
|---|---|
| 1.1 | det × area by 5 soi |
Question 6:
6 | (a) | (i) | detM =λ(λ−k)−k
detM >0⇒λ2 −kλ−k >0 for all
λ⇒k2 +4k <0
⇒−4<k <0 | B1
M1
A1cao
[3] | 1.1
3.1a
1.1 | attempt to find discriminant | or (λ−k/2)2 > k2/4 + k
6 | (a) | (ii) | The transformation represented by M always
preserves the orientation of shapes | B1
[1] | 1.2 | condone ‘doesn’t reflect’
6 | (b) | λ −2 x λx−2y
=
1 λ+2y x+(λ+2)y
invariant line if x+λmx+2mx=m(λx−2mx)
⇒2m2 +2m+1=0
discriminant is 22 −2×4=−4<0 so no real roots
for m, i.e. there are no invariant lines | B1
M1
A1
A1
[4] | 2.1
2.1
1.1
2.3 | oe (e.g. with y = mx [+ c])
subst y = mx[+c] into
x+(λ+2)y = m(λx−2y)[+c] | x+(λ+2)(mx+c)
= m(λx−2mx−2c)+c
6 | (c) | detM =λ2 +2λ+2=5
⇒λ2 +2λ−3=0
⇒λ=−3 or 1 | M1
A1
A1
[3] | 3.1a
1.1
1.1 | det × area by 5 soi
correct equation in any form
BC6 A linear transformation $T$ of the $x - y$ plane has an associated matrix $\mathbf { M }$, where $\mathbf { M } = \left( \begin{array} { c c } \lambda & k \\ 1 & \lambda - k \end{array} \right)$, and $\lambda$\\
and $k$ are real constants. and $k$ are real constants.
\begin{enumerate}[label=(\alph*)]
\item You are given that $\operatorname { det } \mathbf { M } > 0$ for all values of $\lambda$.
\begin{enumerate}[label=(\roman*)]
\item Find the range of possible values of $k$.
\item What is the significance of the condition $\operatorname { det } \mathbf { M } > 0$ for the transformation T?
For the remainder of this question, take $k = - 2$.
\end{enumerate}\item Determine whether there are any lines through the origin that are invariant lines for the transformation T.
\item The transformation T is applied to a triangle with area 3 units ${ } ^ { 2 }$. The area of the resulting image triangle is 15 units ${ } ^ { 2 }$.\\
Find the possible values of $\lambda$.
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Pure Core AS 2019 Q6 [11]}}