CAIE P3 2005 November — Question 5 7 marks

Exam BoardCAIE
ModuleP3 (Pure Mathematics 3)
Year2005
SessionNovember
Marks7
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicHarmonic Form
TypeExpress and solve equation
DifficultyModerate -0.3 This is a standard harmonic form question requiring routine application of the R sin(θ - α) method followed by straightforward equation solving. While it involves multiple steps (finding R and α, then solving the resulting equation in two quadrants), these are well-practiced techniques with no novel insight required, making it slightly easier than average.
Spec1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc1.05o Trigonometric equations: solve in given intervals
5 By expressing \(8 \sin \theta - 6 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), solve the equation $$8 \sin \theta - 6 \cos \theta = 7$$ for \(0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }\).
AnswerMarks
State or imply \(R = 10\) or \(R = -10\)B1
Use trig formula to find \(\alpha\)M1
Obtain \(\alpha = 36.9°\) if \(R = 10\) or \(\alpha = 216.9°\) if \(R = -10\), with no errors seenA1
Carry out evaluation of \(\sin^{-1}(\frac{7}{10}) (\approx 44.427...°)\)M1
Obtain answer \(81.3°\)A1
Carry out correct method for second answerM1
Obtain answer \(172.4°\) and no others in the range [Ignore answers outside the given range.]A1
Total: [7]
State or imply $R = 10$ or $R = -10$ | B1 |
Use trig formula to find $\alpha$ | M1 |
Obtain $\alpha = 36.9°$ if $R = 10$ or $\alpha = 216.9°$ if $R = -10$, with no errors seen | A1 |
Carry out evaluation of $\sin^{-1}(\frac{7}{10}) (\approx 44.427...°)$ | M1 |
Obtain answer $81.3°$ | A1 |
Carry out correct method for second answer | M1 |
Obtain answer $172.4°$ and no others in the range [Ignore answers outside the given range.] | A1 |

**Total: [7]**

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5 By expressing $8 \sin \theta - 6 \cos \theta$ in the form $R \sin ( \theta - \alpha )$, solve the equation

$$8 \sin \theta - 6 \cos \theta = 7$$

for $0 ^ { \circ } \leqslant \theta \leqslant 360 ^ { \circ }$.

\hfill \mbox{\textit{CAIE P3 2005 Q5 [7]}}
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