SPS SPS FM Pure (SPS FM Pure) 2025 June

1. The complex number \(z\) satisfies the equation \(z + 2 \mathrm { i } z ^ { * } = 12 + 9 \mathrm { i }\). Find \(z\), giving your answer in the form \(x + \mathrm { i } y\).
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2. (a) Use binomial expansions to show that \(\sqrt { \frac { 1 + 4 x } { 1 - x } } \approx 1 + \frac { 5 } { 2 } x - \frac { 5 } { 8 } x ^ { 2 }\)

A student substitutes \(x = \frac { 1 } { 2 }\) into both sides of the approximation shown in part (a) in an attempt to find an approximation to \(\sqrt { 6 }\)
(b) Give a reason why the student should not use \(x = \frac { 1 } { 2 }\)
(c) Substitute \(x = \frac { 1 } { 11 }\) into $$\sqrt { \frac { 1 + 4 x } { 1 - x } } = 1 + \frac { 5 } { 2 } x - \frac { 5 } { 8 } x ^ { 2 }$$ to obtain an approximation to \(\sqrt { 6 }\). Give your answer as a fraction in its simplest form.
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3. Describe a sequence of transformations which maps the graph of $$y = | 2 x - 5 |$$ onto the graph of $$y = | x |$$ [BLANK PAGE]

4. Given that $$y = \frac { 3 \sin \theta } { 2 \sin \theta + 2 \cos \theta } \quad - \frac { \pi } { 4 } < \theta < \frac { 3 \pi } { 4 }$$ show that $$\frac { \mathrm { d } y } { \mathrm {~d} \theta } = \frac { A } { 1 + \sin 2 \theta } \quad - \frac { \pi } { 4 } < \theta < \frac { 3 \pi } { 4 }$$ where \(A\) is a rational constant to be found.
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5. Two matrices \(\mathbf { A }\) and \(\mathbf { B }\) satisfy the equation $$\mathbf { A B } = \boldsymbol { I } + 2 \mathbf { A }$$ where \(\boldsymbol { I }\) is the identity matrix and \(\mathbf { B } = \left[ \begin{array} { c c } 3 & - 2
- 4 & 8 \end{array} \right]\) \section*{Find \(\mathbf { A }\).} [BLANK PAGE]

6. (a) Prove that $$1 - \cos 2 \theta \equiv \tan \theta \sin 2 \theta , \quad \theta \neq \frac { ( 2 n + 1 ) \pi } { 2 } , \quad n \in \mathbb { Z }$$ (b) Hence solve, for \(- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }\), the equation $$\left( \sec ^ { 2 } x - 5 \right) ( 1 - \cos 2 x ) = 3 \tan ^ { 2 } x \sin 2 x$$ Give any non-exact answer to 3 decimal places where appropriate.
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7. Fig. 10 shows the graph of \(x ^ { 3 } + y ^ { 3 } = x y\). \begin{figure}[h] \end{figure}

1. Find an expression for \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(x\) and \(y\).
2. P is the maximum point on the curve. The parabola \(y = k x ^ { 2 }\) intersects the curve at P . Find the value of the constant \(k\).
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8. (a) Sketch, on the Argand diagram below, the locus of points satisfying the equation $$| z - 3 | = 2$$ \includegraphics[max width=\textwidth, alt={}, center]{14f14bf3-88ee-413c-a62d-0914f41a485d-18_1339_1383_370_402}
(b) There is a unique complex number \(w\) that satisfies both $$| w - 3 | = 2 \quad \text { and } \quad \arg ( w + 1 ) = \alpha$$ where \(\alpha\) is a constant such that \(0 < \alpha < \pi\)
(b) (i) Find the value of \(\alpha\).
(b) (ii) Express \(w\) in the form \(r ( \cos \theta + \mathrm { i } \sin \theta )\).
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9. \begin{figure}[h] \end{figure} Figure 2 shows a sketch of part of the curve \(C\) with equation \(y = x \ln x , \quad x > 0\) The line \(l\) is the normal to \(C\) at the point \(P ( \mathrm { e } , \mathrm { e } )\) The region \(R\), shown shaded in Figure 2, is bounded by the curve \(C\), the line \(l\) and the \(x\)-axis. Show that the exact area of \(R\) is \(A \mathrm { e } ^ { 2 } + B\) where \(A\) and \(B\) are rational numbers to be found.
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10. Prove by induction that \(f ( n ) = 2 ^ { 4 n } + 5 ^ { 2 n } + 7 ^ { n }\) is divisible by 3 for all positive integers \(n\).
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11. Fig. 15 shows the graph of \(\mathrm { f } ( x ) = 2 x + \frac { 1 } { x } + \ln x - 4\). \begin{figure}[h] \end{figure}

1. Show that the equation $$2 x + \frac { 1 } { x } + \ln x - 4 = 0$$ has a root, \(\alpha\), such that \(0.1 < \alpha < 0.9\).
2. Obtain the following Newton-Raphson iteration for the equation in part (i). $$x _ { r + 1 } = x _ { r } - \frac { 2 x _ { r } ^ { 3 } + x _ { r } + x _ { r } ^ { 2 } \left( \ln x _ { r } - 4 \right) } { 2 x _ { r } ^ { 2 } - 1 + x _ { r } }$$
3. Explain why this iteration fails to find \(\alpha\) using each of the following starting values.
   (A) \(x _ { 0 } = 0.4\)
   (B) \(x _ { 0 } = 0.5\)
   (C) \(x _ { 0 } = 0.6\)
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   \includegraphics[max width=\textwidth, alt={}]{14f14bf3-88ee-413c-a62d-0914f41a485d-26_819_589_173_826}
   The curve \(C\) has parametric equations $$x = \frac { 1 } { \sqrt { 2 + t } } , \quad y = \ln ( 1 + t ) , \quad 2 \leq t < \infty$$ The point \(P\) on curve \(C\) has \(x\)-coordinate \(\frac { 1 } { 2 }\).
   (a) Find the exact \(y\)-coordinate of \(P\). The tangent to \(C\) at \(P\) meets the \(y\)-axis at point \(Y\).
   (b) Determine the exact coordinates of \(Y\). The curve \(C\) and the line segment \(P Y\) are rotated \(2 \pi\) radians about the \(y\)-axis.
   (c) Determine the exact volume of the solid generated. Give your answer in the form \(\pi ( \ln p + q )\), where \(p\) and \(q\) are rational numbers.
   [0pt] [You are given that the volume of a cone with radius \(r\) and height \(h\) is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\) ]
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13.

1. Using a suitable substitution, find $$\int \sqrt { 1 - x ^ { 2 } } d x$$
2. Show that the differential equation $$\frac { d y } { d x } = 2 \sqrt { 1 - x ^ { 2 } - y ^ { 2 } + x ^ { 2 } y ^ { 2 } }$$ given that \(y = 0\) when \(x = 0 , | x | < 1\) and \(| y | < 1\), has the solution $$y = x \cos \left( x \sqrt { 1 - x ^ { 2 } } \right) + \sqrt { 1 - x ^ { 2 } } \sin \left( x \sqrt { 1 - x ^ { 2 } } \right) .$$ [BLANK PAGE]

14. The three dimensional non-zero vector \(\boldsymbol { u }\) has the following properties:

• The angle \(\theta\) between \(\boldsymbol { u }\) and the vector \(\left( \begin{array} { l } 1
  5
  9 \end{array} \right)\) is acute.
• The (non-reflex) angle between \(\boldsymbol { u }\) and the vector \(\left( \begin{array} { l } 9
  5
  1 \end{array} \right)\) is \(2 \theta\).
• \(\boldsymbol { u }\) is perpendicular to the vector \(\left( \begin{array} { l } 1
  1
  1 \end{array} \right)\).

Find the angle \(\theta\).
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