SPS SPS SM Pure (SPS SM Pure) 2023 October

In all questions you must show all stages of your working, justifying solutions and not relying solely on calculator technology.

1. Differentiate with respect to \(x\)
   1. \(x^2 e^{3x + 2}\), [4]
   2. \(\frac{\cos(2x^4)}{3x}\). [4]

1. The curve \(C\) has equation $$y = \frac{x}{9 + x^2}.$$ Use calculus to find the coordinates of the turning points of \(C\). [6]
2. Given that $$y = (1 + e^{2x})^{\frac{3}{2}},$$ find the value of \(\frac{dy}{dx}\) at \(x = \frac{1}{2} \ln 3\). [5]

1. Given that \(\cos A = \frac{3}{4}\), where \(270° < A < 360°\), find the exact value of \(\sin 2A\). [5]
2. 1. Show that \(\cos\left(2x + \frac{\pi}{3}\right) + \cos\left(2x - \frac{\pi}{3}\right) = \cos 2x\). [3] Given that $$y = 3\sin^2 x + \cos\left(2x + \frac{\pi}{3}\right) + \cos\left(2x - \frac{\pi}{3}\right),$$
   2. show that \(\frac{dy}{dx} = \sin 2x\). [4]

$$f(x) = 12 \cos x - 4 \sin x.$$ Given that \(f(x) = R \cos(x + \alpha)\), where \(R \geq 0\) and \(0 \leq \alpha \leq 90°\),

1. find the value of \(R\) and the value of \(\alpha\). [4]
2. Hence solve the equation $$12 \cos x - 4 \sin x = 7$$ for \(0 \leq x < 360°\), giving your answers to one decimal place. [5]
3. 1. Write down the minimum value of \(12 \cos x - 4 \sin x\). [1]
   2. Find, to 2 decimal places, the smallest positive value of \(x\) for which this minimum value occurs. [2]

The curve \(C\) has equation $$y = \frac{3 + \sin 2x}{2 + \cos 2x}$$

1. Show that $$\frac{dy}{dx} = \frac{6\sin 2x + 4\cos 2x + 2}{(2 + \cos 2x)^2}$$ [4]
2. Find an equation of the tangent to \(C\) at the point on \(C\) where \(x = \frac{\pi}{2}\). Write your answer in the form \(y = ax + b\), where \(a\) and \(b\) are exact constants. [4]