10 The plane P has normal vector \(2 \mathbf { i } + a \mathbf { j } - \mathbf { k }\), where \(a\) is a positive constant, and the point ( \(3 , - 1,1\) ) lies in P . The plane \(\mathrm { x } - \mathrm { z } = 3\) makes an angle of \(45 ^ { \circ }\) with P .
Find the cartesian equation of P .
\section*{END OF QUESTION PAPER}
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Question 10:
Answer Marks
10 Normal to x − z = 3 is vector i − k
(i−k).(2i+aj−k)
cos45=
2+5
2 a
1 3
=
2 2 a 2 + 5
a 2+5 =3
a = 2
Plane equation is 2x + ay – z = k
(3, –1, 1) lies in plane 6 – a – 1 = k
Answer Marks
k = 3 so plane equation is 2x + 2y – z = 3 B1
M1
A1
A1
M1
M1
A1
1.1a
1.1
1.1
1.1
3.1a
Answer Marks
3.2a may be implied by i − k seen (or column vector)
allow 1 slip
must have 1/2 for cos 45
or with their a
substituting (3, −1, 1) into plane equation (k = 5 − their a)
PMT
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Copy
Question 10:
10 | Normal to x − z = 3 is vector i − k
(i−k).(2i+aj−k)
cos45=
2+5
2 a
1 3
=
2 2 a 2 + 5
a 2+5 =3
a = 2
Plane equation is 2x + ay – z = k
(3, –1, 1) lies in plane 6 – a – 1 = k
k = 3 so plane equation is 2x + 2y – z = 3 | B1
M1
A1
A1
M1
M1
A1
[7] | 3.1a
1.1a
1.1
1.1
1.1
3.1a
3.2a | may be implied by i − k seen (or column vector)
allow 1 slip
must have 1/2 for cos 45
or with their a
substituting (3, −1, 1) into plane equation (k = 5 − their a)
Response | Mark
PMT
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touch with our customer support centre.
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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
2023 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
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10 The plane P has normal vector $2 \mathbf { i } + a \mathbf { j } - \mathbf { k }$, where $a$ is a positive constant, and the point ( $3 , - 1,1$ ) lies in P . The plane $\mathrm { x } - \mathrm { z } = 3$ makes an angle of $45 ^ { \circ }$ with P .
Find the cartesian equation of P .
\section*{END OF QUESTION PAPER}
\hfill \mbox{\textit{OCR MEI Further Pure Core AS 2023 Q10 [7]}}