SPS SPS FM Pure (SPS FM Pure) 2025 September

2. Prove by induction that \(11 \times 7 ^ { n } - 13 ^ { n } - 1\) is divisible by 3 , for all integers \(n \geq 0\).
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4. The curve \(C\) has parametric equations $$x = 2 \cos t , \quad y = \sqrt { 3 } \cos 2 t , \quad 0 \leqslant t \leqslant \pi$$

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\). The point \(P\) lies on \(C\) where \(t = \frac { 2 \pi } { 3 }\)
   The line \(l\) is the normal to \(C\) at \(P\).
2. Show that an equation for \(l\) is $$2 x - 2 \sqrt { 3 } y - 1 = 0$$ The line \(l\) intersects the curve \(C\) again at the point \(Q\).
3. Find the exact coordinates of \(Q\). You must show clearly how you obtained your answers.
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4. On the Argand diagram below, sketch the locus, \(L\), of points satisfying the equation $$\arg ( z + \mathrm { i } ) = \frac { \pi } { 6 }$$ [2 marks]
   \includegraphics[max width=\textwidth, alt={}, center]{75e73922-4324-441c-b942-c709a71d9025-12_1310_1354_513_463}
5. \(\quad z _ { 1 }\) is a point on \(L\) such that \(| z |\) is a minimum. Find the exact value of \(z _ { 1 }\) in the form \(a + b \mathrm { i }\)
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7. (a) Prove the identity \(\frac { \cos x } { \sec x + 1 } + \frac { \cos x } { \sec x - 1 } \equiv 2 \cot ^ { 2 } x\)
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(b) Hence, solve the equation $$\frac { \cos \left( 2 \theta + \frac { \pi } { 3 } \right) } { \sec \left( 2 \theta + \frac { \pi } { 3 } \right) + 1 } = \cot \left( 2 \theta + \frac { \pi } { 3 } \right) - \frac { \cos \left( 2 \theta + \frac { \pi } { 3 } \right) } { \sec \left( 2 \theta + \frac { \pi } { 3 } \right) - 1 }$$ in the interval \(0 \leq \theta \leq 2 \pi\), giving your values of \(\theta\) to three significant figures where appropriate.
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8. A population of meerkats is being studied. The population is modelled by the differential equation $$\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { 1 } { 22 } P ( 11 - 2 P ) , \quad t \geqslant 0 , \quad 0 < P < 5.5$$ where \(P\), in thousands, is the population of meerkats and \(t\) is the time measured in years since the study began. Given that there were 1000 meerkats in the population when the study began, determine the time taken, in years, for this population of meerkats to double.
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9. A curve \(C\) has equation \(y = \mathrm { f } ( x )\) where $$\mathrm { f } ( x ) = x + 2 \ln ( \mathrm { e } - x )$$

1. 1. Show that the equation of the normal to \(C\) at the point where \(C\) crosses the \(y\)-axis is given by $$y = \left( \frac { \mathrm { e } } { 2 - \mathrm { e } } \right) x + 2$$
   2. Find the exact area enclosed by the normal and the coordinate axes. Fully justify your answer.
2. The equation \(\mathrm { f } ( x ) = 0\) has one positive root, \(\alpha\).
   1. Show that \(\alpha\) lies between 2 and 3 Fully justify your answer.
   2. Show that the roots of \(\mathrm { f } ( x ) = 0\) satisfy the equation $$x = \mathrm { e } - \mathrm { e } ^ { - \frac { x } { 2 } }$$ [2 marks]
   3. Use the recurrence relation $$x _ { n + 1 } = \mathrm { e } - \mathrm { e } ^ { - \frac { x _ { n } } { 2 } }$$ with \(x _ { 1 } = 2\) to find the values of \(x _ { 2 }\) and \(x _ { 3 }\) giving your answers to three decimal places.
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   4. Figure 1 below shows a sketch of the graphs of \(y = e - e ^ { - \frac { x } { 2 } }\) and \(y = x\), and the position of \(x _ { 1 }\) On Figure 1, draw a cobweb or staircase diagram to show how convergence takes place, indicating the positions of \(x _ { 2 }\) and \(x _ { 3 }\) on the \(x\)-axis.
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      \captionsetup{labelformat=empty} \caption{Figure 1} \includegraphics[alt={},max width=\textwidth]{75e73922-4324-441c-b942-c709a71d9025-22_1236_1566_1519_360}
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