SPS SPS FM Pure (SPS FM Pure) 2023 November

1. The complex number \(z\) satisfies the equation \(z ^ { 2 } - 4 \mathrm { i } z ^ { * } + 11 = 0\).

Given that \(\operatorname { Re } ( z ) > 0\), find \(z\) in the form \(a + b \mathrm { i }\), where \(a\) and \(b\) are real numbers.

2. Fig. 5 shows the curve with polar equation \(r = a ( 3 + 2 \cos \theta )\) for \(- \pi \leqslant \theta \leqslant \pi\), where \(a\) is a constant. \begin{figure}[h] \end{figure}

1. Write down the polar coordinates of the points A and B .
2. Explain why the curve is symmetrical about the initial line.
3. In this question you must show detailed reasoning. Find in terms of \(a\) the exact area of the region enclosed by the curve.
   [0pt] [BLANK PAGE] \section*{3. In this question you must show detailed reasoning.} The roots of the equation \(2 x ^ { 3 } - 5 x + 7 = 0\) are \(\alpha , \beta\) and \(\gamma\).
4. Find \(\frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma }\).
5. Find an equation with integer coefficients whose roots are \(2 \alpha - 1,2 \beta - 1\) and \(2 \gamma - 1\).
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6. Given that $$\frac { 1 } { r ( r + 1 ) ( r + 2 ) } = \frac { A } { r ( r + 1 ) } + \frac { B } { ( r + 1 ) ( r + 2 ) }$$ show that \(A = \frac { 1 } { 2 }\) and find the value of \(B\).
7. Use the method of differences to find $$\sum _ { r = 10 } ^ { 98 } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }$$ giving your answer as a rational number.
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5. (a) Use a Maclaurin series to find a quadratic approximation for \(\ln ( 1 + 2 x )\).
(b) Find the percentage error in using the approximation in part (a) to calculate \(\ln ( 1.2 )\).
(c) Jane uses the Maclaurin series in part (a) to try to calculate an approximation for \(\ln 3\). Explain whether her method is valid.
[0pt] [BLANK PAGE] \section*{6. In this question you must show detailed reasoning.} In this question you may assume the results for $$\sum _ { r = 1 } ^ { n } r ^ { 3 } , \quad \sum _ { r = 1 } ^ { n } r ^ { 2 } \quad \text { and } \quad \sum _ { r = 1 } ^ { n } r$$ Show that the sum of the cubes of the first \(n\) positive odd numbers is $$n ^ { 2 } \left( 2 n ^ { 2 } - 1 \right)$$ [BLANK PAGE]

7. (a) (i) Show on an Argand diagram the locus of points given by the values of \(z\) satisfying $$| z - 4 - 3 \mathbf { i } | = 5$$ Taking the initial line as the positive real axis with the pole at the origin and given that \(\theta \in [ \alpha , \alpha + \pi ]\), where \(\alpha = - \arctan \left( \frac { 4 } { 3 } \right)\),
(ii) show that this locus of points can be represented by the polar curve with equation $$r = 8 \cos \theta + 6 \sin \theta$$ The set of points \(A\) is defined by $$A = \left\{ z : 0 \leqslant \arg z \leqslant \frac { \pi } { 3 } \right\} \cap \{ z : | z - 4 - 3 \mathbf { i } | \leqslant 5 \}$$ (b) (i) Show, by shading on your Argand diagram, the set of points \(A\).
(ii) Find the exact area of the region defined by \(A\), giving your answer in simplest form.
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8. (a) Use a hyperbolic substitution and calculus to show that $$\int \frac { x ^ { 2 } } { \sqrt { x ^ { 2 } - 1 } } \mathrm {~d} x = \frac { 1 } { 2 } \left[ x \sqrt { x ^ { 2 } - 1 } + \operatorname { arcosh } x \right] + k$$ where \(k\) is an arbitrary constant. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of part of the curve \(C\) with equation $$y = \frac { 4 } { 15 } x \operatorname { arcosh } x \quad x \geqslant 1$$ The finite region \(R\), shown shaded in Figure 1, is bounded by the curve \(C\), the \(x\)-axis and the line with equation \(x = 3\)
(b) Using algebraic integration and the result from part (a), show that the area of \(R\) is given by $$\frac { 1 } { 15 } [ 17 \ln ( 3 + 2 \sqrt { 2 } ) - 6 \sqrt { 2 } ]$$ [BLANK PAGE]
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