SPS SPS FM Pure (SPS FM Pure) 2023 September

1. Show that \(\int _ { 5 } ^ { \infty } ( x - 1 ) ^ { - \frac { 3 } { 2 } } \mathrm {~d} x = 1\).
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2. (i) Sketch the graph of \(y = | 2 x - 7 a |\), where \(a\) is a positive constant. State the coordinates of the points where the graph meets each axis.
(ii) Solve the inequality \(| 2 x - 7 a | < 4 a\).
(iii) Deduce the largest integer \(N\) satisfying the inequality \(| 2 \ln N - 10.5 | < 6\).
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3. Solve each of the following equations, for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).

1. \(\sin \frac { 1 } { 2 } x = 0.8\)
2. \(\sin x = 3 \cos x\)
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4. Find, in exact form, the area of the region on an Argand diagram which represents the locus of points for which \(| z - 5 - 2 \mathrm { i } | \leqslant \sqrt { 32 }\) and \(\operatorname { Re } ( z ) \geq 9\).
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5.
\includegraphics[max width=\textwidth, alt={}, center]{c9751c50-bab1-43fa-b580-909e1ce06a9d-12_423_743_123_731} The diagram shows the curve \(y = \mathrm { f } ( x )\), where f is the function defined for all real values of \(x\) by $$f ( x ) = 3 + 4 e ^ { - x }$$

1. State the range of f.
2. Find an expression for \(f ^ { - 1 } ( x )\), and state the domain and range of \(f ^ { - 1 }\).
3. The straight line \(y = x\) meets the curve \(y = \mathrm { f } ( x )\) at the point \(P\). By using an iterative process based on the equation \(x = \mathrm { f } ( x )\), with a starting value of 3 , find the coordinates of the point \(P\). Show all your working and give each coordinate correct to 3 decimal places.
4. How is the point \(P\) related to the curves \(y = \mathrm { f } ( x )\) and \(y = \mathrm { f } ^ { - 1 } ( x )\) ?
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6. The matrix \(\left( \begin{array} { l l } 1 & 3
0 & 1 \end{array} \right)\) represents a transformation \(P\).

1. Describe fully the transformation \(P\). The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { r r } - 3 & - 1
   - 1 & 0 \end{array} \right)\).
2. Given that \(M\) represents transformation \(Q\) followed by transformation \(P\), find the matrix that represents the transformation Q and describe fully the transformation Q .
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7. The complex number \(2 - \mathrm { i }\) is denoted by \(z\).

1. Find \(| z |\) and arg \(z\).
2. Given that \(a z + b z ^ { * } = 4 - 8 \mathrm { i }\), find the values of the real constants \(a\) and \(b\).
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8. A curve has equation \(y = 2 + \mathrm { e } ^ { \frac { 1 } { 2 } x }\). The region \(R\) is bounded by the curve and by the straight lines \(x = 0 , x = 4\) and \(y = 0\). Find the exact volume of the solid obtained when \(R\) is rotated completely about the x-axis.
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9. The equations of two lines are $$\mathbf { r } = \left( \begin{array} { l } 3
0
2 \end{array} \right) + \lambda \left( \begin{array} { l } 1
1
3 \end{array} \right) \text { and } \mathbf { r } = \left( \begin{array} { r } - 1
8
2 \end{array} \right) + \mu \left( \begin{array} { c } - 3
1
- 5 \end{array} \right)$$ Find the coordinates of the point where these lines intersect.
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10. The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l } 2 & 2
0 & 1 \end{array} \right)\). Prove by induction that, for \(n \geq 1\), $$\mathbf { M } ^ { n } = \left( \begin{array} { c c } 2 ^ { n } & 2 ^ { n + 1 } - 2
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11. A curve has parametric equations \(x = \frac { 1 } { t } - 1\) and \(y = 2 t + \frac { 1 } { t ^ { 2 } }\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\), simplifying your answer.
2. Find the coordinates of the stationary point and, by considering the gradient of the curve on either side of this point, determine its nature.
3. Find a cartesian equation of the curve.
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12.

1. Express \(\frac { 16 + 5 x - 2 x ^ { 2 } } { ( x + 1 ) ^ { 2 } ( x + 4 ) }\) in partial fractions.
2. It is given that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { \left( 16 + 5 x - 2 x ^ { 2 } \right) y } { ( x + 1 ) ^ { 2 } ( x + 4 ) }$$ and that \(y = \frac { 1 } { 256 }\) when \(x = 0\). Find the exact value of \(y\) when \(x = 2\). Give your answer in the form \(A e ^ { n }\).
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13. The matrix \(\mathbf { M }\) is given by $$\mathbf { M } = \left( \begin{array} { r r } - \frac { 3 } { 5 } & \frac { 4 } { 5 }
\frac { 4 } { 5 } & \frac { 3 } { 5 } \end{array} \right)$$

1. The diagram below shows the unit square \(O A B C\). The image of the unit square under the transformation represented by M is \(O A ^ { \prime } B ^ { \prime } C ^ { \prime }\). Draw and clearly label \(O A ^ { \prime } B ^ { \prime } C ^ { \prime }\).
   \includegraphics[max width=\textwidth, alt={}, center]{c9751c50-bab1-43fa-b580-909e1ce06a9d-28_778_1095_513_557}
2. Find the equation of the line of invariant points of this transformation.
3. 1. Find the determinant of M .
   2. Describe briefly how this value relates to the transformation represented by M .
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14. Use the substitution \(u = 1 + \ln x\) to find \(\int \frac { \ln x } { x ( 1 + \ln x ) ^ { 2 } } \mathrm {~d} x\).
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