OCR MEI Further Pure Core AS 2021 November — Question 3 7 marks

Exam BoardOCR MEI
ModuleFurther Pure Core AS (Further Pure Core AS)
Year2021
SessionNovember
Marks7
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Mark schemeDownload PDF ↗
TopicVectors: Lines & Planes
TypeAngle between two planes
DifficultyStandard +0.3 Part (a) is routine matrix manipulation (writing as Ax=b and solving), while part (b) requires the standard formula for angle between planes using normal vectors (cos θ = |n₁·n₂|/(|n₁||n₂|)). Both are direct applications of learned techniques with no problem-solving insight required, making this slightly easier than average for Further Maths.
Spec4.03r Solve simultaneous equations: using inverse matrix4.04c Scalar product: calculate and use for angles4.04d Angles: between planes and between line and plane
3 Three planes have the following equations. $$\begin{aligned} 2 x - 3 y + z & = - 3 \\ x - 4 y + 2 z & = 1 \\ - 3 x - 2 y + 3 z & = 14 \end{aligned}$$
    1. Write the system of equations in matrix form.
    2. Hence find the point of intersection of the planes.
  2. In this question you must show detailed reasoning. Find the acute angle between the planes \(2 x - 3 y + z = - 3\) and \(x - 4 y + 2 z = 1\).
Question 3:
AnswerMarks Guidance
3(a) (i)
 1 −4 2y= 1 
    
AnswerMarks Guidance
−3 −2 3z 14B1
[1]2.5 allow 1 slip
3(a) (ii)
[y=] 1 −4 2  1 
     
z −3 −2 3 14
AnswerMarks
⇒x=−1, y=2, z=5M1
A1
AnswerMarks
[2]1.1
1.1−3
or M −1 1  soi
 
14
BC allow unsupported answers
AnswerMarks Guidance
3(b) DR
Normal vectors are 2i−3j+k and i−4j+2k
2×1+(−3)×(−4)+1×2
cosθ=
22+(−3)2+12 × 12+(−4)2+22
16
=
14 21
AnswerMarks
θ=21.1°M1
M1
A1
A1
AnswerMarks
[4]1.1a
1.1
1.1
AnswerMarks
1.1soi
(0.9331…)
or 0.368 rad , 21° or 0.37 rad or better
Question 3:
3 | (a) | (i) |  2 −3 1x −3
 1 −4 2y= 1 
    
−3 −2 3z 14 | B1
[1] | 2.5 | allow 1 slip
3 | (a) | (ii) | x  2 −3 1 −1 −3
[y=] 1 −4 2  1 
     
z −3 −2 3 14
⇒x=−1, y=2, z=5 | M1
A1
[2] | 1.1
1.1 | −3
or M −1 1  soi
 
14
BC allow unsupported answers
3 | (b) | DR
Normal vectors are 2i−3j+k and i−4j+2k
2×1+(−3)×(−4)+1×2
cosθ=
22+(−3)2+12 × 12+(−4)2+22
16
=
14 21
θ=21.1° | M1
M1
A1
A1
[4] | 1.1a
1.1
1.1
1.1 | soi
(0.9331…)
or 0.368 rad , 21° or 0.37 rad or better
3 Three planes have the following equations.

$$\begin{aligned}
2 x - 3 y + z & = - 3 \\
x - 4 y + 2 z & = 1 \\
- 3 x - 2 y + 3 z & = 14
\end{aligned}$$
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Write the system of equations in matrix form.
\item Hence find the point of intersection of the planes.
\end{enumerate}\item In this question you must show detailed reasoning.

Find the acute angle between the planes $2 x - 3 y + z = - 3$ and $x - 4 y + 2 z = 1$.
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Pure Core AS 2021 Q3 [7]}}
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