SPS SPS FM Pure (SPS FM Pure) 2023 June

1. You are given that \(g f ( x ) = | 3 x - 1 |\) for \(x \in \mathbb { R }\).
   1. Given that \(f ( x ) = 3 x - 1\), express \(g ( x )\) in terms of \(x\).
   2. State the range of \(g f ( x )\).
   3. Solve the inequality \(| 3 x - 1 | > 1\).
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   \section*{2. In this question you must show detailed reasoning.}
2. Express \(8 \cos x + 5 \sin x\) in the form \(R \cos ( x - \alpha )\), where \(R\) and \(\alpha\) are constants with \(R > 0\) and \(0 < \alpha < \frac { 1 } { 2 } \pi\).
3. Hence solve the equation \(8 \cos x + 5 \sin x = 6\) for \(0 \leqslant x < 2 \pi\), giving your answers correct to 4 decimal places.
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3. You are given that \(f ( x ) = \ln ( 2 x - 5 ) + 2 x ^ { 2 } - 30\), for \(x > 2.5\).

1. Show that \(f ( x ) = 0\) has a root \(\alpha\) in the interval [3.5, 4]. A student takes 4 as the first approximation to \(\alpha\).
   Given \(f ( 4 ) = 3.099\) and \(f ^ { \prime } ( 4 ) = 16.67\) to 4 significant figures,
2. apply the Newton-Raphson procedure once to obtain a second approximation for \(\alpha\), giving your answer to 3 significant figures.
3. Show that \(\alpha\) is the only root of \(f ( x ) = 0\).
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4. You are given that \(\boldsymbol { M } = \left( \begin{array} { c c } 1 & - \sqrt { 3 }
\sqrt { 3 } & 1 \end{array} \right)\).

1. Show that \(\boldsymbol { M }\) is non-singular. The hexagon \(R\) is transformed to the hexagon \(S\) by the transformation represented by the matrix \(\boldsymbol { M }\). Given that the area of hexagon \(R\) is 5 square units,
2. find the area of hexagon \(S\). The matrix \(\boldsymbol { M }\) represents an enlargement, with centre \(( 0,0 )\) and scale factor \(k\), where \(k > 0\), followed by a rotation anti-clockwise through an angle \(\theta\) about \(( 0,0 )\).
3. Find the value of \(k\).
4. Find the value of \(\theta\).
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5. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of a triangle \(A B C\).
Given \(\overrightarrow { A B } = 2 \mathbf { i } + 3 \mathbf { j } + \mathbf { k }\) and \(\overrightarrow { B C } = \mathbf { i } - 9 \mathbf { j } + 3 \mathbf { k }\), show that \(\angle B A C = 105.9 ^ { \circ }\) to one decimal place.
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6. A spherical balloon is inflated so that its volume increases at a rate of \(10 \mathrm {~cm} ^ { 3 }\) per second. Find the rate of increase of the balloon's surface area when its diameter is 8 cm .
[0pt] [For a sphere of radius \(r\), surface area \(= 4 \pi r ^ { 2 }\) and volume \(= \frac { 4 } { 3 } \pi r ^ { 3 }\) ].
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7. Prove that for all \(n \in \mathbb { N }\) $$\left( \begin{array} { c c } 3 & 4 i
i & - 1 \end{array} \right) ^ { n } = \left( \begin{array} { c c } 2 n + 1 & 4 n i
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8. (i) Shade on an Argand diagram the set of points $$\{ z \in \mathbb { C } : | z - 4 \mathrm { i } | \leqslant 3 \} \cap \left\{ z \in \mathbb { C } : - \frac { \pi } { 2 } < \arg ( z + 3 - 4 \mathrm { i } ) \leqslant \frac { \pi } { 4 } \right\}$$ The complex number \(w\) satisfies \(| w - 4 i | = 3\).
(ii) Find the maximum value of \(\arg w\) in the interval \(( - \pi , \pi ]\). Give your answer in radians correct to 2 decimal places.
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9. (i) Use the binomial expansion to show that \(( 1 - 2 x ) ^ { - \frac { 1 } { 2 } } \approx 1 + x + \frac { 3 } { 2 } x ^ { 2 }\) for sufficiently small values of \(x\).
(ii) For what values of \(x\) is the expansion valid?
(iii) Find the expansion of \(\sqrt { \frac { 1 + 2 x } { 1 - 2 x } }\) in ascending powers of \(x\) as far as the term in \(x ^ { 2 }\).
(iv) Use \(x = \frac { 1 } { 20 }\) in your answer to part (iii) to find an approximate value for \(\sqrt { 11 }\).
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10. The complex number \(z\) is given by \(z = k + 3 i\), where \(k\) is a negative real number. Given that \(z + \frac { 12 } { z }\) is real, find \(k\) and express \(z\) in exact modulus-argument form.
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11. In this question you must show detailed reasoning. A curve has parametric equations $$x = \cos t - 3 t \text { and } y = 3 t - 4 \cos t - \sin 2 t , \text { for } 0 \leqslant t \leqslant \pi .$$ Show that the gradient of the curve is always negative.
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12.
\includegraphics[max width=\textwidth, alt={}, center]{368f5263-8f2e-40a9-8bc0-f074d8a98ee1-26_689_1203_182_447} The figure shows part of the graph of \(y = ( x - 3 ) \sqrt { \ln x }\). The portion of the graph below the \(x\)-axis is rotated by \(2 \pi\) radians around the \(x\)-axis to form a solid of revolution, \(S\). Determine the exact volume of \(S\).
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13. (i) Solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = y ( 1 + y ) ( 1 - x ) ,$$ given that \(y = 1\) when \(x = 1\). Give your answer in the form \(y = \mathrm { f } ( x )\), where f is a function to be determined.
(ii) By considering the sign of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) near \(( 1,1 )\), or otherwise, show that this point is a maximum point on the curve \(y = \mathrm { f } ( x )\).
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14. A curve \(C\) has equation $$x ^ { 3 } + y ^ { 3 } = 3 x y + 48$$ Prove that \(C\) has two stationary points and find their coordinates.
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15. In this question you must use detailed reasoning.

1. Show that \(\int _ { \frac { \pi } { 3 } } ^ { \frac { \pi } { 2 } } \frac { 1 + \sin 2 x } { - \cos 2 x } d x = \ln ( \sqrt { a } + b )\), where \(a\) and \(b\) are integers to be determined.
2. Show that \(\frac { \pi } { \frac { \pi } { 4 } } \frac { 1 + \sin 2 x } { - \cos 2 x } d x\) is undefined, explaining your reasoning clearly.
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