SPS SPS FM Pure (SPS FM Pure) 2023 February

1. Find \(\sum _ { r = 1 } ^ { n } \left( 2 r ^ { 2 } - 1 \right)\), expressing your answer in fully factorised form.
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2. Solve the equation \(2 z - 5 i z ^ { * } = 12\).
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\section*{3. In this question you must show detailed reasoning.} Fig. 4 shows the region bounded by the curve \(y = \sec \frac { 1 } { 2 } x\), the \(x\)-axis, the \(y\)-axis and the line \(x = \frac { 1 } { 2 } \pi\). \begin{figure}[h] \end{figure} This region is rotated through \(2 \pi\) radians about the \(x\)-axis.
Find, in exact form, the volume of the solid of revolution generated.
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4. The plane \(\Pi\) has equation $$\mathbf { r } = \left( \begin{array} { l } 3
3

4. The plane \(\Pi\) has equation $$\mathbf { r } = \left( \begin{array} { l } 3
3
2 \end{array} \right) + \lambda \left( \begin{array} { r } - 1
2
1 \end{array} \right) + \mu \left( \begin{array} { l } 2
0
1 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Show that vector \(2 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k }\) is perpendicular to \(\Pi\).
2. Hence find a Cartesian equation of \(\Pi\). The line \(l\) has equation $$\mathbf { r } = \left( \begin{array} { r } 4
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   2 \end{array} \right) + t \left( \begin{array} { r } 1
   6
   - 3 \end{array} \right)$$ where \(t\) is a scalar parameter.
   The point \(A\) lies on \(l\).
   Given that the shortest distance between \(A\) and \(\Pi\) is \(2 \sqrt { 29 }\)
3. determine the possible coordinates of \(A\).
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6
- 3 \end{array} \right)$$ where \(t\) is a scalar parameter.
The point \(A\) lies on \(l\).
Given that the shortest distance between \(A\) and \(\Pi\) is \(2 \sqrt { 29 }\)
(c) determine the possible coordinates of \(A\).
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5. Prove by induction that for all positive integers \(n\) $$f ( n ) = 3 ^ { 2 n + 4 } - 2 ^ { 2 n }$$ is divisible by 5
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6. In this question you must show detailed reasoning. Find \(\int _ { 2 } ^ { \infty } \frac { 1 } { 4 + x ^ { 2 } } \mathrm {~d} x\).
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7. (a) Prove that $$\tanh ^ { - 1 } ( x ) = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right) \quad - k < x < k$$ stating the value of the constant \(k\).
(b) Hence, or otherwise, solve the equation $$2 x = \tanh ( \ln \sqrt { 2 - 3 x } )$$ [BLANK PAGE]

8. The cubic equation $$a x ^ { 3 } + b x ^ { 2 } - 19 x - b = 0$$ where \(a\) and \(b\) are constants, has roots \(\alpha , \beta\) and \(\gamma\)
The cubic equation $$w ^ { 3 } - 9 w ^ { 2 } - 97 w + c = 0$$ where \(c\) is a constant, has roots \(( 4 \alpha - 1 ) , ( 4 \beta - 1 )\) and \(( 4 \gamma - 1 )\)
Without solving either cubic equation, determine the value of \(a\), the value of \(b\) and the value of \(c\).
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9. (a) Sketch, on the Argand diagram below, the locus of points satisfying the equation $$| z - 3 | = 2$$ \includegraphics[max width=\textwidth, alt={}, center]{0d8a4ccd-f88a-4f03-a70f-61864d2e30e2-18_1173_1209_301_516}
(b) There is a unique complex number \(w\) that satisfies both $$| w - 3 | = 2 \text { and } \arg ( w + 1 ) = \alpha$$ where \(\alpha\) is a constant such that \(0 < \alpha < \pi\).

1. Find the value of \(\alpha\).
2. Express \(w\) in the form \(r ( \cos \theta + i \sin \theta )\). Give each of \(r\) and \(\theta\) to two significant figures.
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10. (a) Find the general solution of the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } + \frac { 2 y } { x } = \frac { x + 3 } { x ( x - 1 ) \left( x ^ { 2 } + 3 \right) } \quad ( x > 1 )$$ (b) Find the particular solution for which \(y = 0\) when \(x = 3\). Give your answer in the form \(y = f ( x )\).
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11. In an Argand diagram, the points \(A , B\) and \(C\) are the vertices of an equilateral triangle with its centre at the origin. The point \(A\) represents the complex number \(6 + 2 \mathrm { i }\).

1. Find the complex numbers represented by the points \(B\) and \(C\), giving your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real and exact. The points \(D , E\) and \(F\) are the midpoints of the sides of triangle \(A B C\).
2. Find the exact area of triangle \(D E F\).
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12. $$\mathbf { M } = \left( \begin{array} { r r r } 2 & - 1 & 1
3 & k & 4
3 & 2 & - 1 \end{array} \right) \quad \text { where } k \text { is a constant }$$

1. Find the values of \(k\) for which the matrix \(\mathbf { M }\) has an inverse.
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2. Find, in terms of \(p\), the coordinates of the point where the following planes intersect $$\begin{aligned} & 2 x - y + z = p
   & 3 x - 6 y + 4 z = 1
   & 3 x + 2 y - z = 0 \end{aligned}$$
3. 1. Find the value of \(q\) for which the set of simultaneous equations $$\begin{aligned} & 2 x - y + z = 1
      & 3 x - 5 y + 4 z = q
      & 3 x + 2 y - z = 0 \end{aligned}$$ can be solved.
   2. For this value of \(q\), interpret the solution of the set of simultaneous equations geometrically.
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13. In this question you must show detailed reasoning. The diagram below shows the curve \(r = \sqrt { \sin \theta } \mathrm { e } ^ { \frac { 1 } { 3 } \cos \theta }\) for \(0 \leqslant \theta \leqslant \pi\).
\includegraphics[max width=\textwidth, alt={}, center]{0d8a4ccd-f88a-4f03-a70f-61864d2e30e2-26_686_1061_317_539}

1. Find the exact area enclosed by the curve.
2. Show that the greatest value of \(r\) on the curve is \(\sqrt { \frac { \sqrt { 3 } } { 2 } } \mathrm { e } ^ { \frac { 1 } { 6 } }\).
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14. (a) Use differentiation to find the first two non-zero terms of the Maclaurin expansion of \(\ln \left( \frac { 1 } { 2 } + \cos x \right)\).
(b) By considering the root of the equation \(\ln \left( \frac { 1 } { 2 } + \cos x \right) = 0\) deduce that \(\pi \approx 3 \sqrt { 3 \ln \left( \frac { 3 } { 2 } \right) }\).
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