SPS SPS FM Pure (SPS FM Pure) 2023 November

The complex number \(z\) satisfies the equation \(z^2 - 4iz^* + 11 = 0\). Given that \(\text{Re}(z) > 0\), find \(z\) in the form \(a + bi\), where \(a\) and \(b\) are real numbers. [4]

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. \includegraphics{figure_2}

1. Write down the polar coordinates of the points A and B. [2]
2. Explain why the curve is symmetrical about the initial line. [2]
3. In this question you must show detailed reasoning. Find in terms of \(a\) the exact area of the region enclosed by the curve. [4]

In this question you must show detailed reasoning. The roots of the equation \(2x^3 - 5x + 7 = 0\) are \(\alpha\), \(\beta\) and \(\gamma\).

1. Find \(\frac{1}{\alpha} + \frac{1}{\beta} + \frac{1}{\gamma}\). [4]
2. Find an equation with integer coefficients whose roots are \(2\alpha - 1\), \(2\beta - 1\) and \(2\gamma - 1\). [4]

In this question you must show detailed reasoning.

1. 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 \(B\). [3]
2. 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. [4]

1. Use a Maclaurin series to find a quadratic approximation for \(\ln(1 + 2x)\). [1]
2. Find the percentage error in using the approximation in part (a) to calculate \(\ln(1.2)\). [3]
3. Jane uses the Maclaurin series in part (a) to try to calculate an approximation for \(\ln 3\). Explain whether her method is valid. [2]

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 $$n^2(2n^2 - 1).$$ [5]

1. 1. Show on an Argand diagram the locus of points given by the values of \(z\) satisfying $$|z - 4 - 3i| = 5$$ Taking the initial line as the positive real axis with the pole at the origin and given that $$\theta \in [\alpha, \alpha + \pi], \text{ where } \alpha = -\arctan\left(\frac{4}{3}\right),$$
   2. show that this locus of points can be represented by the polar curve with equation $$r = 8\cos\theta + 6\sin\theta$$ (6) The set of points \(A\) is defined by $$A = \left\{z : 0 \leqslant \arg z \leqslant \frac{\pi}{3}\right\} \cap \{z : |z - 4 - 3i| \leqslant 5\}$$
2. 1. Show, by shading on your Argand diagram, the set of points \(A\).
   2. Find the exact area of the region defined by \(A\), giving your answer in simplest form. (7)

1. Use a hyperbolic substitution and calculus to show that $$\int \frac{x^2}{\sqrt{x^2 - 1}} dx = \frac{1}{2}\left[x\sqrt{x^2 - 1} + \arcosh x\right] + k$$ where \(k\) is an arbitrary constant. (6) \includegraphics{figure_8} Figure 1 shows a sketch of part of the curve \(C\) with equation $$y = \frac{4}{15}x \arcosh x \quad x \geqslant 1$$ The finite region \(R\), shown shaded in Figure 1, is bounded by the curve \(C\), the \(x\)-axis and the line with equation \(x = 3\)
2. Using algebraic integration and the result from part (a), show that the area of \(R\) is given by $$\frac{1}{15}\left[17\ln\left(3 + 2\sqrt{2}\right) - 6\sqrt{2}\right]$$ (5) This is the last question on the paper.