SPS SPS SM Pure (SPS SM Pure) 2020 October

1. a) Find

$$\int \frac { x } { x ^ { 2 } + 1 } d x$$ b) Find. $$\int 2 \pi ( 4 x + 3 ) ^ { 10 } d x$$ c) Find. $$\int \frac { 2 } { e ^ { 4 x } } d x$$

1. a) Find \(\frac { d y } { d x }\) if \(y = 4 \ln ( 3 x )\)
   b) Differentiate \(\frac { 2 x } { \sqrt { 3 x + 1 } }\) giving your answer in the form \(\frac { 3 x + c } { \sqrt { ( 3 x + 1 ) ^ { p } } }\), where \(c , p \in \mathbb { N }\)
2. Expand \(( x - 2 y ) ^ { 5 }\).
3. What transformations could be used, and in which order, to transform the curve \(y = \sin x\) into the curve \(y = 2 \sin ( 3 x + 30 )\) ?

Find the equation of the tangent to the curve $$y = 3 x ^ { 2 } ( x + 2 ) ^ { 6 }$$ at the point \(( - 1,3 )\), giving your answer in the form \(y = m x + c\).

6.

1. Express \(\frac { x } { ( x + 1 ) ( x + 2 ) }\) in partial fractions.
2. Hence find \(\int \frac { x } { ( x + 1 ) ( x + 2 ) } \mathrm { d } x\).

7.

1. Express \(24 \sin \theta + 7 \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R > 0\) and \(0 ^ { \circ } < \alpha < 90 ^ { \circ }\).
2. Hence solve the equation \(24 \sin \theta + 7 \cos \theta = 12\) for \(0 ^ { \circ } < \theta < 360 ^ { \circ }\).

8.

1. 1. Sketch the graph of \(y = \operatorname { cosec } x\) for \(0 < x < 4 \pi\).
   2. It is given that \(\operatorname { cosec } \alpha = \operatorname { cosec } \beta\), where \(\frac { 1 } { 2 } \pi < \alpha < \pi\) and \(2 \pi < \beta < \frac { 5 } { 2 } \pi\). By using your sketch, or otherwise, express \(\beta\) in terms of \(\alpha\).
2. 1. Write down the identity giving \(\tan 2 \theta\) in terms of \(\tan \theta\).
   2. Given that \(\cot \phi = 4\), find the exact value of \(\tan \phi \cot 2 \phi \tan 4 \phi\), showing all your working.

9. Earth is being added to a pile so that, when the height of the pile is \(h\) metres, its volume is \(V\) cubic metres, where $$V = \left( h ^ { 6 } + 16 \right) ^ { \frac { 1 } { 2 } } - 4$$

1. Find the value of \(\frac { \mathrm { d } V } { \mathrm {~d} h }\) when \(h = 2\).
2. The volume of the pile is increasing at a constant rate of 8 cubic metres per hour. Find the rate, in metres per hour, at which the height of the pile is increasing at the instant when \(h = 2\). Give your answer correct to 2 significant figures.

10.

1. Prove that $$\cos ^ { 2 } \left( \theta + 45 ^ { \circ } \right) - \frac { 1 } { 2 } ( \cos 2 \theta - \sin 2 \theta ) \equiv \sin ^ { 2 } \theta .$$
2. Hence solve the equation $$6 \cos ^ { 2 } \left( \frac { 1 } { 2 } \theta + 45 ^ { \circ } \right) - 3 ( \cos \theta - \sin \theta ) = 2$$ for \(- 90 ^ { \circ } < \theta < 90 ^ { \circ }\).