A-Level AQA Maths Pure Basic Trigonometry: A curve C has equation $$x^3 \s

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

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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

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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.

Substitute x = \sqrt{3} and y = \frac{\pi}{6} into the equation: $x^3 \sin y + \cos y = Ax$

This gives:

$\left(\sqrt{3}\right)^3 \sin\left(\frac{\pi}{6}\right) + \cos\left(\frac{\pi}{6}\right) = A \sqrt{3}$
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})
Substitute these values back into the equation: $\left(\sqrt{3}\right)^3 \cdot \frac{1}{2} + \frac{\sqrt{3}}{2} = A \sqrt{3}$
Calculate the left-hand side: $\frac{3\sqrt{3}}{2} + \frac{\sqrt{3}}{2} = A \sqrt{3}$ $\frac{4\sqrt{3}}{2} = A \sqrt{3}$
Simplifying gives: $2\sqrt{3} = A \sqrt{3}$
Dividing both sides by (\sqrt{3}) leads to: $A = 2$ .

Step 2

Show that $$\frac{dy}{dx} = \frac{2 - 3x^2 \sin y}{x^3 \cos y - \sin y}$$

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Answer

To show the required differentiation, we start with the equation:

$x^3 \sin y + \cos y = 2x$

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
This gives: $3x^2 \sin y + (x^3 \cos y - \sin y) \frac{dy}{dx} = 2$
Rearranging for (\frac{dy}{dx}) we have: $(x^3 \cos y - \sin y) \frac{dy}{dx} = 2 - 3x^2 \sin y$
Isolating (\frac{dy}{dx}): $\frac{dy}{dx} = \frac{2 - 3x^2 \sin y}{x^3 \cos y - \sin y}$
This confirms the required expression.

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

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

Show that A = 2

Show that $$\frac{dy}{dx} = \frac{2 - 3x^2 \sin y}{x^3 \cos y - \sin y}$$

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