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A curve C has equation $$x^3 \sin y + \cos y = Ax$$ where A is a constant - AQA - A-Level Maths: Pure - Question 12 - 2020 - Paper 1
Question 12
A curve C has equation $$x^3 \sin y + \cos y = Ax$$ where A is a constant. C passes through the point P (\sqrt{3}, \frac{\pi}{6}) 12 (a) Show that A = 2 12 (b) ... show full transcript
Worked Solution & Example Answer:A curve C has equation $$x^3 \sin y + \cos y = Ax$$ where A is a constant - AQA - A-Level Maths: Pure - Question 12 - 2020 - Paper 1
Step 1
Show that A = 2
Answer
To find the value of A, we start by substituting the coordinates of point P (\sqrt{3}, \frac{\pi}{6}) into the equation of the curve.
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Substitute x = \sqrt{3} and y = \frac{\pi}{6} into the equation:
This gives:
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Calculate the sine and cosine values:
- (\sin\left(\frac{\pi}{6}\right) = \frac{1}{2})
- (\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2})
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Substitute these values back into the equation:
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Calculate the left-hand side:
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Simplifying gives:
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Dividing both sides by (\sqrt{3}) leads to: .
Step 2
Show that $$\frac{dy}{dx} = \frac{2 - 3x^2 \sin y}{x^3 \cos y - \sin y}$$
Answer
To show the required differentiation, we start with the equation:
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Differentiate both sides with respect to x using implicit differentiation:
- For the left side: (3x^2 \sin y + x^3 \cos y \frac{dy}{dx} - \sin y \frac{dy}{dx}
- For the right side: 2
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This gives:
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Rearranging for (\frac{dy}{dx}) we have:
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Isolating (\frac{dy}{dx}):
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This confirms the required expression.