SPS SPS SM (SPS SM) 2022 January

1.

1. Express \(\frac { 21 } { \sqrt { 7 } }\) in the form \(k \sqrt { 7 }\).
2. Express \(8 ^ { - \frac { 1 } { 3 } }\) as an exact fraction in its simplest form.

2. A curve has equation \(y = 16 x + \frac { 1 } { x ^ { 2 } }\). Find
(A) \(\frac { \mathrm { d } y } { \mathrm {~d} x }\),
(B) \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\).
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3. Triangle ABC is shown in Fig. 1. \begin{figure}[h] \end{figure} Find the perimeter of triangle ABC .
(3)
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4. Find $$\int \frac { 2 x ^ { 2 } + 6 x - 5 } { 3 \sqrt { x ^ { 3 } } } d x$$ simplifying your answer.
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5. Prove, from first principles, that the derivative of \(x ^ { 3 }\) is \(3 x ^ { 2 }\)
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6. \begin{figure}[h] \end{figure} Figure 1 shows the graph of \(y = \mathrm { f } ( x )\).

1. Write down the number of solutions that exist for the equation
   1. \(\mathrm { f } ( x ) = 1\),
   2. \(\mathrm { f } ( x ) = - x\).
2. Labelling the axes in a similar way, sketch on separate diagrams in the space provided the graphs of
   1. \(\quad y = \mathrm { f } ( x - 2 )\),
   2. \(y = \mathrm { f } ( 2 x )\).
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7. Prove by contradiction that \(\sqrt [ 3 ] { 2 }\) is an irrational number.
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8. (a) Show that the equation $$4 \cos \theta - 1 = 2 \sin \theta \tan \theta$$ can be written in the form $$6 \cos ^ { 2 } \theta - \cos \theta - 2 = 0$$ (b) Hence solve, for \(0 \leqslant x < 90 ^ { \circ }\) $$4 \cos 3 x - 1 = 2 \sin 3 x \tan 3 x$$ giving your answers, where appropriate, to one decimal place.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
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9. $$\mathrm { f } ( x ) = x ^ { 3 } - 9 x ^ { 2 } + 24 x - 16$$

1. Evaluate \(\mathrm { f } ( 1 )\) and hence state a linear factor of \(\mathrm { f } ( x )\).
2. Show that \(\mathrm { f } ( x )\) can be expressed in the form $$\mathrm { f } ( x ) = ( x + p ) ( x + q ) ^ { 2 } ,$$ where \(p\) and \(q\) are integers to be found.
3. Sketch the curve \(y = \mathrm { f } ( x )\) in the space provided.
4. Using integration, find the area of the region enclosed by the curve \(y = \mathrm { f } ( x )\) and the \(x\)-axis.
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10. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of part of the curve \(C\) with equation $$y = \frac { 32 } { x ^ { 2 } } + 3 x - 8 , \quad x > 0$$ The point \(P ( 4,6 )\) lies on \(C\).
The line \(l\) is the normal to \(C\) at the point \(P\).
The region \(R\), shown shaded in Figure 4, is bounded by the line \(l\), the curve \(C\), the line with equation \(x = 2\) and the \(x\)-axis. Show that the area of \(R\) is 46
(Solutions based entirely on graphical or numerical methods are not acceptable.)
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