SPS SPS FM Pure (SPS FM Pure) 2025 February

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.
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2. Prove by mathematical induction that \(\sum _ { r = 1 } ^ { n } ( r \times r ! ) = ( n + 1 ) ! - 1\) for all positive integers \(n\).
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3. The curve \(C\) has equation $$y = 31 \sinh x - 2 \sinh 2 x \quad x \in \mathbb { R }$$ Determine, in terms of natural logarithms, the exact \(x\) coordinates of the stationary points of \(C\).
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4. The plane \(\Pi _ { 1 }\) has equation $$\mathbf { r } = 2 \mathbf { i } + 4 \mathbf { j } - \mathbf { k } + \lambda ( \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k } ) + \mu ( - \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Find a Cartesian equation for \(\Pi _ { 1 }\) The line \(l\) has equation $$\frac { x - 1 } { 5 } = \frac { y - 3 } { - 3 } = \frac { z + 2 } { 4 }$$
2. Find the coordinates of the point of intersection of \(l\) with \(\Pi _ { 1 }\) The plane \(\Pi _ { 2 }\) has equation $$\mathbf { r } \cdot ( 2 \mathbf { i } - \mathbf { j } + 3 \mathbf { k } ) = 5$$
3. Find, to the nearest degree, the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\)
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7. The equation of a curve, in polar coordinates, is $$r = \sec \theta + \tan \theta , \quad \text { for } 0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi$$

1. Sketch the curve.
2. Find the exact area of the region bounded by the curve and the lines \(\theta = 0\) and \(\theta = \frac { 1 } { 3 } \pi\).
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8. (a) Solve the equation \(z ^ { 3 } = \sqrt { 2 } - \sqrt { 6 } \mathrm { i }\), giving your answers in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\) where \(r > 0\) and \(0 \leq \theta < 2 \pi\)
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(b) The transformation represented by the matrix \(\mathbf { M } = \left[ \begin{array} { l l } 5 & 1
1 & 3 \end{array} \right]\) acts on the points on an Argand Diagram which represent the roots of the equation in part (a). Find the exact area of the shape formed by joining the transformed points.
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9. In this question, you must show detailed reasoning. Find the sum of all the integers from 1 to 999 inclusive that are not square or cube numbers.
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10. Three planes have equations $$\begin{array} { r } 4 x - 5 y + z = 8
3 x + 2 y - k z = 6
( k - 2 ) x + k y - 8 z = 6 \end{array}$$ where \(k\) is a real constant.
The planes do not meet at a unique point.

1. Find the possible values of \(k\).
2. For each value of \(k\) found in part (a), identify the configuration of the given planes. Fully justify your answer, stating in each case whether or not the equations of the planes form a consistent system.
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11. The infinite series C and S are defined by $$\begin{aligned} & \mathrm { C } = \cos \theta + \frac { 1 } { 2 } \cos 5 \theta + \frac { 1 } { 4 } \cos 9 \theta + \frac { 1 } { 8 } \cos 13 \theta + \ldots
& \mathrm { S } = \sin \theta + \frac { 1 } { 2 } \sin 5 \theta + \frac { 1 } { 4 } \sin 9 \theta + \frac { 1 } { 8 } \sin 13 \theta + \ldots \end{aligned}$$ Given that the series C and S are both convergent,

1. show that $$C + i S = \frac { 2 e ^ { i \theta } } { 2 - e ^ { 4 i \theta } }$$
2. Hence show that $$\mathrm { S } = \frac { 4 \sin \theta + 2 \sin 3 \theta } { 5 - 4 \cos 4 \theta }$$ [BLANK PAGE]

12. The population density \(P\), in suitable units, of a certain bacterium at time \(t\) hours is to be modelled by a differential equation. Initially, the population density is zero, and its long-term value is 5 . The model uses the differential equation $$\frac { d P } { d t } - \frac { P } { t \left( 1 + t ^ { 2 } \right) } = \frac { t e ^ { - t } } { \sqrt { 1 + t ^ { 2 } } }$$ Find \(P\) as a function of \(t\). [You may assume that \(\lim _ { t \rightarrow \infty } t e ^ { - t } = 0\) ].
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13. (a) Write down the Maclaurin series of \(\mathrm { e } ^ { x }\), in ascending power of \(x\), up to and including the term in \(x ^ { 3 }\)
(b) Hence, without differentiating, determine the Maclaurin series of $$\mathrm { e } ^ { \left( \mathrm { e } ^ { x } - 1 \right) }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 3 }\), giving each coefficient in simplest form.
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