Pre-U Pre-U 9794/2 2018 June — Question 9 13 marks

Exam BoardPre-U
ModulePre-U 9794/2 (Pre-U Mathematics Paper 2)
Year2018
SessionJune
Marks13
TopicHarmonic Form
TypeTransformations of trigonometric graphs
DifficultyStandard +0.8 This is a comprehensive harmonic form question requiring multiple techniques: converting to R cos(θ - α) form (involving both expansion and working with sin/cos combinations), describing a complex sequence of transformations (horizontal stretch, translation, and vertical stretch), and solving a trigonometric equation. While each individual step uses standard A-level techniques, the combination of all three parts, particularly getting the transformation sequence correct with proper ordering, elevates this above a routine question. The Pre-U context and multi-part nature with 'full details' requirement makes it moderately challenging but still within reach of strong A-level students.
Spec1.02w Graph transformations: simple transformations of f(x)1.05l Double angle formulae: and compound angle formulae1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.05o Trigonometric equations: solve in given intervals
9 In this question, \(x\) denotes an angle measured in degrees.
  1. Express \(4 \sin \left( 2 x + 30 ^ { \circ } \right) + 3 \cos 2 x\) in the form \(R \cos ( 2 x - \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
  2. Give full details of the sequence of transformations which maps the graph of \(y = \cos x\) onto the graph of \(y = 4 \sin \left( 2 x + 30 ^ { \circ } \right) + 3 \cos 2 x\).
  3. Find the smallest positive value of \(x\) that satisfies the equation \(4 \sin \left( 2 x + 30 ^ { \circ } \right) + 3 \cos 2 x = 6\).
Question 9(i), 9(ii), 9(iii)
(i)
- \(4\sin(2x+30°) = 2\sqrt{3}\sin 2x + 2\cos 2x\) B1 Correct expansion of first term
- \(R\cos\alpha = 5\), \(R\sin\alpha = 2\sqrt{3}\); \(R^2 = 25+12\), so \(R = \sqrt{37}\) M1 Attempt correct process to find \(R\), from expression of the form \(a\sin 2x + b\cos 2x\)
- \(\tan\alpha = \frac{2\sqrt{3}}{5}\), so \(\alpha = 34.7°\) M1 Attempt correct process to find \(\alpha\), from expression of the form \(a\sin 2x + b\cos 2x\)
- \(f(x) = \sqrt{37}\cos(2x - 34.7°)\) A1 Obtain \(\sqrt{37}\cos(2x-34.7°)\). Allow 6.08, or better, for \(\sqrt{37}\)
(ii)
- Stretch in the \(y\) direction by a scale factor of \(\sqrt{37}\) B1 FT: stretch in the \(y\) direction by a scale factor of their \(R\). Must use 'factor' or 'scale factor'. Must be using their \(R\cos(2x-\alpha)\)
- Translation in the \(x\) direction by \(34.7°\); Stretch in the \(x\) direction by sf \(\frac{1}{2}\) M1 Translation by \(\pm\)(their \(\alpha\) or \(\frac{1}{2}\alpha\)) and stretch sf \(\frac{1}{2}\) or 2, both in \(x\) direction. Allow informal language for 'translation', 'stretch' and direction (for the M1 only). Must be using their \(R\cos(2x-\alpha)\)
OR: Stretch in the \(x\) direction by sf \(\frac{1}{2}\); Translation in the \(x\) direction by \(17.4°\)
- A1 FT: Translation in the \(x\) direction by their \(\alpha\) or \(\frac{1}{2}\alpha\)
- A1 Stretch in the \(x\) direction by sf \(\frac{1}{2}\) (must follow translation) or stretch followed by correct translation. Must use 'factor' or 'scale factor'
(iii)
- \(\cos(2x-34.7°) = 0.98639\ldots\); \(2x-34.7° = -9.46°\) or \(9.46°\ldots\) M1 Attempt correct process to find at least one numerical value for \(2x - \alpha\)
- \(2x = 25.24°\) or \(44.16°\ldots\); \(x = 12.6°\) or \(22.1°\ldots\) M1 Dep on first M mark. Attempt correct process to find at least one positive numerical value for \(x\)
- \(x = 12.6°\) M1 Attempt to find smallest value of \(x\), i.e. use \(-\cos^{-1}\!\left(\frac{6}{R}\right)\)
- A1 Obtain \(x = 12.6°\) only, or better
**Question 9(i), 9(ii), 9(iii)**

**(i)**
- $4\sin(2x+30°) = 2\sqrt{3}\sin 2x + 2\cos 2x$ **B1** Correct expansion of first term
- $R\cos\alpha = 5$, $R\sin\alpha = 2\sqrt{3}$; $R^2 = 25+12$, so $R = \sqrt{37}$ **M1** Attempt correct process to find $R$, from expression of the form $a\sin 2x + b\cos 2x$
- $\tan\alpha = \frac{2\sqrt{3}}{5}$, so $\alpha = 34.7°$ **M1** Attempt correct process to find $\alpha$, from expression of the form $a\sin 2x + b\cos 2x$
- $f(x) = \sqrt{37}\cos(2x - 34.7°)$ **A1** Obtain $\sqrt{37}\cos(2x-34.7°)$. Allow 6.08, or better, for $\sqrt{37}$

**(ii)**
- Stretch in the $y$ direction by a scale factor of $\sqrt{37}$ **B1** FT: stretch in the $y$ direction by a scale factor of their $R$. Must use 'factor' or 'scale factor'. Must be using their $R\cos(2x-\alpha)$
- Translation in the $x$ direction by $34.7°$; Stretch in the $x$ direction by sf $\frac{1}{2}$ **M1** Translation by $\pm$(their $\alpha$ or $\frac{1}{2}\alpha$) and stretch sf $\frac{1}{2}$ or 2, both in $x$ direction. Allow informal language for 'translation', 'stretch' and direction (for the M1 only). Must be using their $R\cos(2x-\alpha)$

OR: Stretch in the $x$ direction by sf $\frac{1}{2}$; Translation in the $x$ direction by $17.4°$

- **A1** FT: Translation in the $x$ direction by their $\alpha$ or $\frac{1}{2}\alpha$
- **A1** Stretch in the $x$ direction by sf $\frac{1}{2}$ (must follow translation) or stretch followed by correct translation. Must use 'factor' or 'scale factor'

**(iii)**
- $\cos(2x-34.7°) = 0.98639\ldots$; $2x-34.7° = -9.46°$ or $9.46°\ldots$ **M1** Attempt correct process to find at least one numerical value for $2x - \alpha$
- $2x = 25.24°$ or $44.16°\ldots$; $x = 12.6°$ or $22.1°\ldots$ **M1** Dep on first M mark. Attempt correct process to find at least one positive numerical value for $x$
- $x = 12.6°$ **M1** Attempt to find smallest value of $x$, i.e. use $-\cos^{-1}\!\left(\frac{6}{R}\right)$
- **A1** Obtain $x = 12.6°$ only, or better
9 In this question, $x$ denotes an angle measured in degrees.\\
(i) Express $4 \sin \left( 2 x + 30 ^ { \circ } \right) + 3 \cos 2 x$ in the form $R \cos ( 2 x - \alpha )$, where $R > 0$ and $0 ^ { \circ } < \alpha < 90 ^ { \circ }$.\\
(ii) Give full details of the sequence of transformations which maps the graph of $y = \cos x$ onto the graph of $y = 4 \sin \left( 2 x + 30 ^ { \circ } \right) + 3 \cos 2 x$.\\
(iii) Find the smallest positive value of $x$ that satisfies the equation $4 \sin \left( 2 x + 30 ^ { \circ } \right) + 3 \cos 2 x = 6$.

\hfill \mbox{\textit{Pre-U Pre-U 9794/2 2018 Q9 [13]}}
This paper (10 questions)
View full paper
Q1 4 Q2 11 Q3 11 Q4 12 Q5 10 Q6 12 Q7 10 Q8 8 Q9 13 Q10 10