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Figure 1 shows the curve C, with equation $y = 6 \, ext{cos} \, x + 2.5 \, ext{sin} \, x$ for $0 \leq x \leq 2\pi$ - Edexcel - A-Level Maths Pure - Question 1 - 2013 - Paper 6
Question 1
Figure 1 shows the curve C, with equation $y = 6 \, ext{cos} \, x + 2.5 \, ext{sin} \, x$ for $0 \leq x \leq 2\pi$. (a) Express $6 \text{cos} \, x + 2.5 \text{sin... show full transcript
Worked Solution & Example Answer:Figure 1 shows the curve C, with equation $y = 6 \, ext{cos} \, x + 2.5 \, ext{sin} \, x$ for $0 \leq x \leq 2\pi$ - Edexcel - A-Level Maths Pure - Question 1 - 2013 - Paper 6
Step 1
Express $6 \text{cos} \, x + 2.5 \text{sin} \, x$ in the form $R \text{cos}(x - \alpha)$
Answer
To express the equation in the form ( R \text{cos}(x - \alpha) ), we first calculate:
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Calculate ( R ): [ R = \sqrt{(6)^2 + (2.5)^2} = \sqrt{36 + 6.25} = \sqrt{42.25} = 6.5 ]
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Calculate ( \alpha ): [ \tan(\alpha) = \frac{\text{sin} \alpha}{\text{cos} \alpha} = \frac{2.5}{6} ] Thus, ( \alpha = \tan^{-1}\left(\frac{2.5}{6}\right) \approx 0.395 , \text{radians} \text{ (to 3 decimal places)}.\n ]
Step 2
Find the coordinates of the points on the graph where the curve C crosses the coordinate axes
Answer
To find where the curve crosses the axes:
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X-axis crossings (): Solve ( 6 \text{cos} , x + 2.5 \text{sin} , x = 0 ). This occurs at points such as ( (0.6, 0) ) and ( (1.97, 0) ).
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Y-axis crossing (): ( y = 6 \text{cos}(0) + 2.5 \text{sin}(0) = 6. ) The coordinates are ( (0, 6) ).
Step 3
Use this function to find the maximum and minimum values of H predicted by the model
Answer
The function is given as:
[
H = 12 + 6 \text{cos}\left(\frac{2\pi t}{52}\right) + 2.5 \text{sin}\left(\frac{2\pi t}{52}\right)
]
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Maximum value of H: This occurs when ( \text{cos} , \text{and} , \text{sin} , \text{are at their maximum} [ H_{max} = 12 + 6(1) + 2.5(1) = 18.5 ]
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Minimum value of H: This occurs when ( \text{cos} , \text{and} , \text{sin} , \text{are at their minimum} ) [ H_{min} = 12 + 6(-1) + 2.5(-1) = 5.5 ]
Step 4
the value of $t$ when H = 16
Answer
Set the equation to 16: [ 16 = 12 + 6 \text{cos}\left(\frac{2\pi t}{52}\right) + 2.5 \text{sin}\left(\frac{2\pi t}{52}\right) ]
This simplifies to: [ \text{cos}\left(\frac{2\pi t}{52}\right) + \text{sin}\left(\frac{2\pi t}{52}\right) = \frac{16 - 12}{6} = \frac{2}{6} = \frac{1}{3} ]
To solve for ( t ), we find:
- First calculate the angle from the equation
- Then convert this value of angle back to weeks since the first recording.
The final value, when rounded to the nearest whole number gives us ( t \approx 4 ).