SPS SPS SM (SPS SM) 2021 November

1. Find \(\frac { d y } { d x }\) for the following functions, simplifying your answers as far as possible.
   i) \(y = \cos x - 2 \sin 2 x\)
   ii) \(y = \frac { 1 } { 2 } x ^ { 4 } + 2 x ^ { 4 } \ln x\)
   iii) \(y = \frac { 2 e ^ { 3 x } - 1 } { 3 e ^ { 3 x } - 1 }\)

a. Express \(\frac { 5 x + 7 } { ( x + 3 ) ( x + 1 ) ^ { 2 } }\) in partial fractions. In this question you must show all of your algebraic steps clearly. The function \(f ( x ) = \frac { 2 - 6 x + 5 x ^ { 2 } } { x ^ { 2 } ( 1 - 2 x ) }\) can be written in the form; $$f ( x ) = \frac { - 2 } { x } + \frac { 2 } { x ^ { 2 } } + \frac { 1 } { 1 - 2 x }$$ b. Hence find the exact value of \(\int _ { 2 } ^ { 3 } \frac { 2 - 6 x + 5 x ^ { 2 } } { x ^ { 2 } ( 1 - 2 x ) } d x\)

3. In this question you must show detailed algebraic reasoning. Find the coordinates of any stationary points on the curve below. $$y = ( 1 - 3 x ) ( 3 - x ) ^ { 3 }$$

1. Find the equation of the normal to the curve \(y = 4 \ln ( 2 x - 3 )\) at the point where the curve crosses the \(x\) axis. Give your answer in the form \(a x + b y + k = 0\) where \(a > 0\).

i) Write \(\log _ { 16 } y - \log _ { 16 } x\) as a single logarithm.
ii) Solve the simultaneous equations, giving your answers in an exact form. $$\begin{gathered} \log _ { 3 } y = \log _ { 3 } ( 9 - 6 x ) + 1
\log _ { 16 } y - \log _ { 16 } x = \frac { 1 } { 4 } \end{gathered}$$

6. a. Prove the following trigonometric identities. You must show all of your algebraic steps clearly. $$( \cos x + \sin x ) ( \operatorname { cosec } x - \sec x ) \equiv 2 \cot 2 x$$ b. Solve the following equation for \(x\) in the interval \(0 \leq x \leq \pi\). Giving your answers in terms of \(\pi\). $$\sin \left( 2 x + \frac { \pi } { 6 } \right) = \frac { 1 } { 2 } \sin \left( 2 x - \frac { \pi } { 6 } \right)$$

1. The diagram below represents the graph of the function \(y = ( 2 x - 5 ) ^ { 4 } - 1\)
   \includegraphics[max width=\textwidth, alt={}, center]{1650b28f-be4e-4600-89ca-67c2d3026c5b-10_784_657_233_694}
   a. Find the intersections of this graph with the \(x\) axis.
   b. Hence find the exact value of the area bounded by the curve and the \(x\) axis.

a. Express \(2 \sqrt { 3 } \cos 2 x - 6 \sin 2 x\) in the form \(R \cos ( 2 x + \alpha )\) where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\)
b. Hence
i. Solve the equation \(2 \sqrt { 3 } \cos 2 x - 6 \sin 2 x = 6\) for \(0 \leq x \leq 2 \pi\) Giving your answers in terms of \(\pi\).
c. It can be shown that \(y = 9 \sin 2 x + 4 \cos 2 x\) can be written as \(y = \sqrt { 97 } \sin \left( 2 x + 24.0 ^ { \circ } \right)\)
i. State the transformations in the order of occurrence which transform the curve \(y = 9 \sin 2 x + 4 \cos 2 x\) to the curve \(y = \sin x\)
ii. Find the exact maximum and minimum values of the function; $$f ( x ) = \frac { 1 } { 11 - 9 \sin 2 x - 4 \cos 2 x }$$

9. a.
i. Show that \(\cos ^ { 2 } x \equiv \frac { 1 } { 2 } + \frac { 1 } { 2 } \cos 2 x\)
ii. Hence find \(\int 2 \cos ^ { 2 } 4 x d x\)
b. Find \(\int \sin ^ { 3 } x d x\)

10.
a. The parametric equations of a curve are \(x = \theta \cos \theta\) and \(y = \sin \theta\) Find the gradient of the curve at the point for which \(\theta = \pi\)
b. A curve is defined parametrically by the equations; $$x = \cos \theta \quad y = \left( \frac { \sin \theta } { 2 } \right) \left( \sin \frac { \theta } { 2 } \right)$$ Show that the cartesian equation of the curve can be written as \(y ^ { 2 } = \frac { 1 } { 8 } ( 1 - x ) ^ { 2 } ( 1 + x )\)