OCR MEI Further Pure Core AS 2018 June — Question 5 7 marks

Exam BoardOCR MEI
ModuleFurther Pure Core AS (Further Pure Core AS)
Year2018
SessionJune
Marks7
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Mark schemeDownload PDF ↗
TopicLinear transformations
TypeFind image coordinates under transformation
DifficultyStandard +0.3 This is a straightforward matrix transformation question requiring matrix multiplication and solving a trigonometric equation. Part (i) is routine calculation, while part (ii) involves setting up equations from the given condition and using the identity sin²θ + cos²θ = 1, which is standard technique for Further Pure AS level with no novel insight required.
Spec4.03d Linear transformations 2D: reflection, rotation, enlargement, shear4.03e Successive transformations: matrix products
A transformation of the \(x\)-\(y\) plane is represented by the matrix \(\begin{pmatrix} \cos \theta & 2 \sin \theta \\ 2 \sin \theta & -\cos \theta \end{pmatrix}\), where \(\theta\) is a positive acute angle.
  1. Write down the image of the point \((2, 3)\) under this transformation. [2]
  2. You are given that this image is the point \((a, 0)\). Find the value of \(a\). [5]
Question 5:
AnswerMarks Guidance
5(i) (2cos6sin, 4sin3cos)
[2]1.1,1.1 Accept in vector form
5(ii) 4sin3cos0
 tan 3
4
  = 36.9 or 0.644 rad
AnswerMarks
a2cos6sin5.2M1
M1
A1
M1
A1
AnswerMarks
[5]3.1a
1.1
1.1
1.1
AnswerMarks
1.1caotheir 4sin3cos0
tan  = sin  / cos  used
 = 36.9 or 0.644 rad or better
substituting their  into their
2cos  + 6 sin 
AnswerMarks
5.2or sin2 + cos2 = 1 used
sin3,cos4
or
5 5
or sin  and cos 
Question 5:
5 | (i) | (2cos6sin, 4sin3cos) | B1B1
[2] | 1.1,1.1 | Accept in vector form
5 | (ii) | 4sin3cos0
 tan 3
4
  = 36.9 or 0.644 rad
a2cos6sin5.2 | M1
M1
A1
M1
A1
[5] | 3.1a
1.1
1.1
1.1
1.1cao | their 4sin3cos0
tan  = sin  / cos  used
 = 36.9 or 0.644 rad or better
substituting their  into their
2cos  + 6 sin 
5.2 | or sin2 + cos2 = 1 used
sin3,cos4
or
5 5
or sin  and cos 
A transformation of the $x$-$y$ plane is represented by the matrix $\begin{pmatrix} \cos \theta & 2 \sin \theta \\ 2 \sin \theta & -\cos \theta \end{pmatrix}$, where $\theta$ is a positive acute angle.

\begin{enumerate}[label=(\roman*)]
\item Write down the image of the point $(2, 3)$ under this transformation. [2]
\item You are given that this image is the point $(a, 0)$. Find the value of $a$. [5]
\end{enumerate}

\hfill \mbox{\textit{OCR MEI Further Pure Core AS 2018 Q5 [7]}}
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