SPS SPS SM Pure (SPS SM Pure) 2023 October

1. technology.
   1. Differentiate with respect to \(x\)
      1. \(x ^ { 2 } \mathrm { e } ^ { 3 x + 2 }\),
      2. \(\frac { \cos \left( 2 x ^ { 3 } \right) } { 3 x }\).
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   (i) The curve \(C\) has equation $$y = \frac { x } { 9 + x ^ { 2 } }$$ Use calculus to find the coordinates of the turning points of \(C\).
   (ii) Given that $$y = \left( 1 + \mathrm { e } ^ { 2 x } \right) ^ { \frac { 3 } { 2 } }$$ find the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(x = \frac { 1 } { 2 } \ln 3\).
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3.

1. Given that \(\cos A = \frac { 3 } { 4 }\), where \(270 ^ { \circ } < A < 360 ^ { \circ }\), find the exact value of \(\sin 2 A\).
2. 1. Show that \(\cos \left( 2 x + \frac { \pi } { 3 } \right) + \cos \left( 2 x - \frac { \pi } { 3 } \right) \equiv \cos 2 x\). Given that $$y = 3 \sin ^ { 2 } x + \cos \left( 2 x + \frac { \pi } { 3 } \right) + \cos \left( 2 x - \frac { \pi } { 3 } \right)$$
   2. show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \sin 2 x\).
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4. $$\mathrm { f } ( x ) = 12 \cos x - 4 \sin x$$ Given that \(\mathrm { f } ( x ) = R \cos ( x + \alpha )\), where \(R \geqslant 0\) and \(0 \leqslant \alpha \leqslant 90 ^ { \circ }\),

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 \(0 \leqslant x < 360 ^ { \circ }\), giving your answers to one decimal place.
3. 1. Write down the minimum value of \(12 \cos x - 4 \sin x\).
   2. Find, to 2 decimal places, the smallest positive value of \(x\) for which this minimum value occurs.
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5. The curve \(C\) has equation $$y = \frac { 3 + \sin 2 x } { 2 + \cos 2 x }$$

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 6 \sin 2 x + 4 \cos 2 x + 2 } { ( 2 + \cos 2 x ) ^ { 2 } }$$
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 \(y = a x + b\), where \(a\) and \(b\) are exact constants.
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