| Exam Board | OCR MEI |
|---|---|
| Module | Further Pure Core AS (Further Pure Core AS) |
| Year | 2018 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Perpendicularity conditions |
| Difficulty | Standard +0.3 This is a standard Further Pure question on plane arrangements requiring systematic row reduction and geometric interpretation. Part (i) involves routine Gaussian elimination to identify when planes meet at a line (a=0) versus are parallel with no common point (a≠0). Part (ii) is a straightforward check whether three vectors are coplanar by testing if they satisfy the plane equations. While it requires multiple techniques, the methods are algorithmic with no novel insight needed, making it slightly easier than average for Further Maths AS. |
| Spec | 4.03s Consistent/inconsistent: systems of equations4.03t Plane intersection: geometric interpretation4.04b Plane equations: cartesian and vector forms |
| Answer | Marks | Guidance |
|---|---|---|
| 10 | (i) | detM0 [so no unique solution] |
| Answer | Marks |
|---|---|
| as there are no solutions | B1 |
| Answer | Marks |
|---|---|
| [6] | 3.1a |
| Answer | Marks |
|---|---|
| 3.1a | or M is singular |
| Answer | Marks |
|---|---|
| allow ‘prism’ | or state no unique solution |
| Answer | Marks | Guidance |
|---|---|---|
| 10 | (ii) | These are the normals to the three planes |
| Answer | Marks |
|---|---|
| the same plane | M1 |
| A1dep | 1.1a |
| 2.3 | dep previous part correct |
| Answer | Marks |
|---|---|
| coplanar | M1 |
| Answer | Marks |
|---|---|
| [2] | showing linear dependence |
Question 10:
10 | (i) | detM0 [so no unique solution]
no planes parallel [so prism or sheaf]
when a = 0, they form a sheaf
as the system has solutions
when a ≠ 0, they form a prismatic
intersection
as there are no solutions | B1
B1
B1
B1
B1
B1
[6] | 3.1a
1.1
2.2a
2.2a
2.2a
3.1a | or M is singular
allow ‘intersect in a line’
o.e. e.g. finding solutions
allow ‘prism’ | or state no unique solution
by direct solution of
equations
10 | (ii) | These are the normals to the three planes
In either of the above cases, they must lie in
the same plane | M1
A1dep | 1.1a
2.3 | dep previous part correct
OR
(e.g) i + j = i + 2j + k + 2i j k
coplanar | M1
A1
[2] | showing linear dependence
OR
(e.g) i + j = i + 2j + k + 2i j k
coplanar
M1
A1
PPMMTT
OCR (Oxford Cambridge and RSA Examinations)
The Triangle Building
Shaftesbury Road
Cambridge
CB2 8EA
OCR Customer Contact Centre
Education and Learning
Telephone: 01223 553998
Facsimile: 01223 552627
Email: general.qualifications@ocr.org.uk
www.ocr.org.uk
For staff training purposes and as part of our quality assurance
programme your call may be recorded or monitored
Oxford Cambridge and RSA Examinations
is a Company Limited by Guarantee
Registered in England
Registered Office; The Triangle Building, Shaftesbury Road, Cambridge, CB2 8EA
Registered Company Number: 3484466
OCR is an exempt Charity
OCR (Oxford Cambridge and RSA Examinations)
Head office
Telephone: 01223 552552
Facsimile: 01223 552553
© OCR 2018Three planes have equations
\begin{align}
-x + 2y + z &= 0\\
2x - y - z &= 0\\
x + y &= a
\end{align}
where $a$ is a constant.
\begin{enumerate}[label=(\roman*)]
\item Investigate the arrangement of the planes:
\begin{itemize}
\item when $a = 0$;
\item when $a \neq 0$.
\end{itemize}
[6]
\item Chris claims that the position vectors $-\mathbf{i} + 2\mathbf{j} + \mathbf{k}$, $2\mathbf{i} - \mathbf{j} - \mathbf{k}$ and $\mathbf{i} + \mathbf{j}$ lie in a plane. Determine whether or not Chris is correct. [2]
\end{enumerate}
\hfill \mbox{\textit{OCR MEI Further Pure Core AS 2018 Q10 [8]}}