SPS SPS FM (SPS FM) 2022 November

1.
\includegraphics[max width=\textwidth, alt={}, center]{657b12c4-cab7-4fc1-9481-94131aeeb6b9-05_1031_938_262_529} The Argand diagram above shows a half-line \(l\) and a circle \(C\). The circle has centre 3 i and passes through the origin.

1. Write down, in complex number form, the equations of \(l\) and \(C\).
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2. Write down inequalities that define the region shaded in the diagram. [The shaded region includes the boundaries.]

2.

1. Show that \(\frac { 1 } { \sqrt { r + 2 } + \sqrt { r } } \equiv \frac { \sqrt { r + 2 } - \sqrt { r } } { 2 }\).
2. Hence find an expression, in terms of \(n\), for $$\sum _ { r = 1 } ^ { n } \frac { 1 } { \sqrt { r + 2 } + \sqrt { r } }$$
3. State, giving a brief reason, whether the series \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { \sqrt { r + 2 } + \sqrt { r } }\) converges.
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3. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable. Given that $$\int _ { 0 } ^ { 1 } \frac { 1 } { \sqrt { 16 + 9 x ^ { 2 } } } \mathrm {~d} x + \int _ { 0 } ^ { 2 } \frac { 1 } { \sqrt { 9 + 4 x ^ { 2 } } } \mathrm {~d} x = \ln a$$ find the exact value of \(a\).
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4. The cubic equation \(3 x ^ { 3 } - 9 x ^ { 2 } + 6 x + 2 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).

1. Write down the values of \(\alpha + \beta + \gamma , \alpha \beta + \beta \gamma + \gamma \alpha\) and \(\alpha \beta \gamma\). The cubic equation \(x ^ { 3 } + a x ^ { 2 } + b x + c = 0\) has roots \(\alpha ^ { 2 } , \beta ^ { 2 }\) and \(\gamma ^ { 2 }\).
2. Show that \(c = - \frac { 4 } { 9 }\) and find the values of \(a\) and \(b\).
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5. \begin{figure}[h] \end{figure} The curves in Figure 1 have equations $$y = 6 \cosh ( x ) \text { and } y = 9 - 2 \sinh ( x )$$

1. Find exact values for the \(x\)-coordinates of the two points where the curves intersect. The finite region between the two curves is shaded in Figure 1.
2. Using calculus, find the area of the shaded region, giving your answer in the form \(a \ln ( b ) + c\), where \(a , b\) and \(c\) are integers.
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6. It is given that \(\mathrm { f } ( x ) = \tanh ^ { - 1 } x\).

1. Show that \(\mathrm { f } ^ { \prime \prime \prime } ( x ) = \frac { 2 \left( 1 + 3 x ^ { 2 } \right) } { \left( 1 - x ^ { 2 } \right) ^ { 3 } }\).
2. Hence find the Maclaurin series for \(\mathrm { f } ( x )\), up to and including the term in \(x ^ { 3 }\).
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7.
\(f ( z ) = z ^ { 3 } + z ^ { 2 } + p z + q\), where \(p\) and \(q\) are real constants.
The equation \(f ( z ) = 0\) has roots \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\).
When plotted on an Argand diagram, the points representing \(z _ { 1 } , z _ { 2 }\) and \(z _ { 3 }\) form the vertices of a triangle of area 35. Given that \(z _ { 1 } = 3\), find the values of \(p\) and \(q\).
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8. The diagram is a sketch of the two curves \(C _ { 1 }\) and \(C _ { 2 }\) with polar equations
\(C _ { 1 } : r = 3 a ( 1 - \cos \theta ) , - \pi \leq \theta < \pi\)
\(\mathrm { C } _ { 2 } : r = a ( 1 + \cos \theta ) , - \pi \leq \theta < \pi\).
\includegraphics[max width=\textwidth, alt={}, center]{657b12c4-cab7-4fc1-9481-94131aeeb6b9-18_346_840_210_1107} The curves meet at the pole \(O\), and at the points \(A\) and \(B\).

1. Find, in terms of \(a\), the polar coordinates of the points \(A\) and \(B\).
2. Show that the length of the line \(A B\) is \(\frac { 3 \sqrt { 3 } } { 2 } a\). The region inside \(C _ { 2 }\) and outside \(C _ { 1 }\) is shown shaded in the diagram above.
3. Find, in terms of \(a\), the area of this region. A badge is designed which has the shape of the shaded region.
   Given that the length of the line \(A B\) is 4.5 cm ,
4. calculate the area of this badge, giving your answer to three significant figures.
   (Total 16 marks)
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