| Exam Board | OCR |
|---|---|
| Module | Further Pure Core 2 (Further Pure Core 2) |
| Year | 2022 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Parallel and perpendicular planes |
| Difficulty | Challenging +1.8 This is a challenging Further Maths question requiring multiple sophisticated steps: finding normal vectors with constraints (x=y component), using the distance formula between parallel planes, determining two solution planes, and calculating the angle between them. It demands strong 3D vector manipulation and systematic problem-solving beyond standard A-level, but the individual techniques are well-established for FM students. |
| Spec | 4.04b Plane equations: cartesian and vector forms4.04d Angles: between planes and between line and plane |
| Answer | Marks |
|---|---|
| 10 | 1 0 |
| Answer | Marks |
|---|---|
| acute angle is awrt 12.4 | *M1 |
| Answer | Marks |
|---|---|
| (8) | Finding AB or an expression for |
| Answer | Marks |
|---|---|
| or awrt 0.217 rads | from r.n = p = 3a – 2a – c and |
Question 10:
10 | 1 0
A B = 1 2
1 0
A B .n
2 A B c o s = = with AB from above
n
used
a 1
n = a or n = 1 oe
c s
1 1 9 a 2 + 1 1 0 a c + 2 4 c 2 = 0
( 7 a + 4 c ) ( 1 7 a + 6 c ) = 0
4
e .g . a = 4 , c = − 7 n = 4
1
− 7
6
e .g . a = 6 , c = − 1 7 n = 6
2
− 1 7
4 6
4 . 6 = 1 6 7
− 7 − 1 7
167 167
cos= =
919 171
acute angle is awrt 12.4 | *M1
dep*M
1
*M1
A1
dep*M
1
A1
Dep*M
1
A1
(8) | Finding AB or an expression for
p – p or d – d
B A B A
Use of dot product to express correct
AB
shortest distance in terms of
and the normal to the planes or
seeing an appropriate difference
between plane constants
a
a used consistently
c
Quadratic formed
Using one solution of quadratic
to obtain a normal of one of the
solution planes.
Both correct.
Finding the dot product of their
solution normals.
or awrt 0.217 rads | from r.n = p = 3a – 2a – c and
A
r.n = p = 13a + 10a + 9c
B
or r .nˆ = d and r .nˆ = d
A B
ie 22a + 10c
or d – d = 2 or
B A
p p
B − A =2
a2 +b2 +c2 a2 +b2 +c2
allow omission of
Or 171a2∓ 22a – 24 = 0
171c2∓ 20c – 119 = 0
24s2 + 110s + 119 = 0
Dependent on all previous M marks
Dependent on all previous M marks
PMT
Need to get in touch?
If you ever have any questions about OCR qualifications or services (including administration, logistics and teaching) please feel free to get in
touch with our customer support centre.
Call us on
01223 553998
Alternatively, you can email us on
support@ocr.org.uk
For more information visit
ocr.org.uk/qualifications/resource-finder
ocr.org.uk
Twitter/ocrexams
/ocrexams
/company/ocr
/ocrexams
OCR is part of Cambridge University Press & Assessment, a department of the University of Cambridge.
For staff training purposes and as part of our quality assurance programme your call may be recorded or monitored. © OCR
2022 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 operates academic and vocational qualifications regulated by Ofqual, Qualifications Wales and CCEA as listed in their
qualifications registers including A Levels, GCSEs, Cambridge Technicals and Cambridge Nationals.
OCR provides resources to help you deliver our qualifications. These resources do not represent any particular teaching method
we expect you to use. We update our resources regularly and aim to make sure content is accurate but please check the OCR
website so that you have the most up-to-date version. OCR cannot be held responsible for any errors or omissions in these
resources.
Though we make every effort to check our resources, there may be contradictions between published support and the
specification, so it is important that you always use information in the latest specification. We indicate any specification changes
within the document itself, change the version number and provide a summary of the changes. If you do notice a discrepancy
between the specification and a resource, please contact us.
Whether you already offer OCR qualifications, are new to OCR or are thinking about switching, you can request more
information using our Expression of Interest form.
Please get in touch if you want to discuss the accessibility of resources we offer to support you in delivering our qualifications.10 The coordinates of the points $A$ and $B$ are ( $3 , - 2 , - 1$ ) and ( $13,10,9$ ) respectively.
\begin{itemize}
\item The plane $\Pi _ { A }$ contains $A$ and the plane $\Pi _ { B }$ contains $B$.
\item The planes $\Pi _ { A }$ and $\Pi _ { B }$ are parallel.
\item The $x$ and $y$ components of any normal to plane $\Pi _ { A }$ are equal.
\item The shortest distance between $\Pi _ { A }$ and $\Pi _ { B }$ is 2 .
\end{itemize}
There are two possible solution planes for $\Pi _ { A }$ which satisfy the above conditions.\\
Determine the acute angle between these two possible solution planes.
\hfill \mbox{\textit{OCR Further Pure Core 2 2022 Q10 [8]}}