| Exam Board | OCR MEI |
|---|---|
| Module | Further Pure Core AS (Further Pure Core AS) |
| Year | 2024 |
| Session | June |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Parallel and perpendicular planes |
| Difficulty | Challenging +1.2 This is a Further Maths question requiring students to find a vector satisfying three simultaneous conditions: unit magnitude, parallel to one plane, and making a specific angle with another plane's normal. While it involves multiple constraints and requires systematic algebraic manipulation (setting up equations from dot products and solving), the techniques are standard for Further Maths vectors. The problem is more involved than typical A-level questions but follows a clear methodical approach without requiring exceptional insight. |
| Spec | 1.10c Magnitude and direction: of vectors4.04c Scalar product: calculate and use for angles4.04d Angles: between planes and between line and plane |
| Answer | Marks |
|---|---|
| 9 | DR |
| Answer | Marks |
|---|---|
| a = 13 , b = − 23 , c = 23 v = 13 i − 23 j + 23 k | M1 |
| Answer | Marks |
|---|---|
| [8] | 3.1a |
| Answer | Marks |
|---|---|
| 3.2a | define vector v |
Question 9:
9 | DR
Let v = a i + b j + c k
a 2 + b 2 + c 2 = 1
(ai+bj+ck).(2i+2j+k)=0
2 a + 2 b + c = 0
(ai+bj+ck).(i+k)
=()cos45
a2 +b2 +c2. 2
a+c= a2+b2+c2
a + c = 1
c=1−a, b=−1(a+1) a2+1(a+1)2+(1−a)2 =1
2 4
9a2−6a+1=0(3a−1)2 =0
a = 13 , b = − 23 , c = 23 v = 13 i − 23 j + 23 k | M1
B1
M1
A1
M1
A1
M1
A1
[8] | 3.1a
1.1
3.1a
1.1
3.1a
1.1
2.1
3.2a | define vector v
oe
scalar product with normal to plane = 0
use of formula for angle between vector and normal to
x + z = 5
Complete method for obtaining an equation in one unknown
dep both previous M1 marks and B1 obtained
( v = − 13 i + 23 j − 23 k is also a valid solution, following use of
the negative sign from the optional ± above)
PMT
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Please get in touch if you want to discuss the accessibility of resources we offer to support you in delivering our qualifications.9 In this question you must show detailed reasoning.
Find a vector $\mathbf { v }$ which has the following properties.
\begin{itemize}
\item It is a unit vector.
\item It is parallel to the plane $2 x + 2 y + z = 10$.
\item It makes an angle of $45 ^ { \circ }$ with the normal to the plane $\mathrm { x } + \mathrm { z } = 5$.
\end{itemize}
\section*{END OF QUESTION PAPER}
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\hfill \mbox{\textit{OCR MEI Further Pure Core AS 2024 Q9 [8]}}