OCR MEI Paper 3 2022 June — Question 11 3 marks

Exam BoardOCR MEI
ModulePaper 3 (Paper 3)
Year2022
SessionJune
Marks3
PaperDownload PDF ↗
Mark schemeDownload PDF ↗
TopicSmall angle approximation
TypeVerify approximation for specific angle
DifficultyModerate -0.5 This is a straightforward verification question requiring conversion between degrees and radians (θ = 45°, x = π/4), then substituting into two given formulas and showing they yield the same numerical result. It involves careful arithmetic but no problem-solving, proof construction, or conceptual insight—purely mechanical substitution and calculation.
Spec1.05a Sine, cosine, tangent: definitions for all arguments
11 Show that, for the angle \(45 ^ { \circ }\), the formula \(\sin \theta \approx \frac { 4 \theta ( 180 - \theta ) } { 40500 - \theta ( 180 - \theta ) }\) given in line 28 gives the same approximation for the sine of the angle as the formula \(\sin x \approx \frac { 16 x ( \pi - x ) } { 5 \pi ^ { 2 } - 4 x ( \pi - x ) }\) given in line 23.
Question 11:
AnswerMarks Guidance
\(\dfrac{4 \times 45(180-45)}{40500 - 45(180-45)} = \dfrac{12}{17}\)B1 1.1
\(45° = \dfrac{\pi}{4}\)B1 1.2
\(\dfrac{16 \times \dfrac{\pi}{4}\left(\pi - \dfrac{\pi}{4}\right)}{5\pi^2 - 4 \times \dfrac{\pi}{4}\left(\pi - \dfrac{\pi}{4}\right)} = \dfrac{12}{17}\)B1 2.2a
[3 marks]
## Question 11:

$\dfrac{4 \times 45(180-45)}{40500 - 45(180-45)} = \dfrac{12}{17}$ | B1 | 1.1 | Substitution seen or implied by partial working. Condone $0.70588...$

$45° = \dfrac{\pi}{4}$ | B1 | 1.2 | Soi maybe in substitution

$\dfrac{16 \times \dfrac{\pi}{4}\left(\pi - \dfrac{\pi}{4}\right)}{5\pi^2 - 4 \times \dfrac{\pi}{4}\left(\pi - \dfrac{\pi}{4}\right)} = \dfrac{12}{17}$ | B1 | 2.2a | Substitution seen or implied by partial working. Both answers must be $\dfrac{12}{17}$

**[3 marks]**

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11 Show that, for the angle $45 ^ { \circ }$, the formula $\sin \theta \approx \frac { 4 \theta ( 180 - \theta ) } { 40500 - \theta ( 180 - \theta ) }$ given in line 28 gives the same approximation for the sine of the angle as the formula $\sin x \approx \frac { 16 x ( \pi - x ) } { 5 \pi ^ { 2 } - 4 x ( \pi - x ) }$ given in line 23.

\hfill \mbox{\textit{OCR MEI Paper 3 2022 Q11 [3]}}
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Q1 2 Q2 6 Q3 4 Q4 5 Q5 7 Q6 8 Q7 12 Q8 16 Q9 2 Q10 5 Q11 3 Q12 5