SPS SPS FM Pure (SPS FM Pure) 2024 September

1. (a) Sketch the graph with equation

$$y = | 2 x - 5 |$$ stating the coordinates of any points where the graph cuts or meets the coordinate axes.
(b) Find the values of \(x\) which satisfy $$| 2 x - 5 | > 7$$ (c) Find the values of \(x\) which satisfy $$| 2 x - 5 | > x - \frac { 5 } { 2 }$$ Write your answer in set notation.
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2. $$\mathbf { P } = \frac { 1 } { 2 } \left( \begin{array} { r r } 1 & \sqrt { 3 }
- \sqrt { 3 } & 1 \end{array} \right) \quad \mathbf { Q } = \left( \begin{array} { r r } - 1 & 0
0 & 1 \end{array} \right)$$ The matrices \(\mathbf { P }\) and \(\mathbf { Q }\) represent linear transformations, \(P\) and \(Q\) respectively, of the plane.
The linear transformation \(M\) is formed by first applying \(P\) and then applying \(Q\).

1. Find the matrix \(\mathbf { M }\) that represents the linear transformation \(M\).
2. Show that the invariant points of the linear transformation \(M\) form a line in the plane, stating the equation of this line.
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3. (a) Sketch, on an Argand diagram, the set of points $$X = \{ z \in \mathbb { C } : | z - 4 - 2 i | < 3 \} \cap \left\{ z \in \mathbb { C } : 0 \leqslant \arg ( z ) \leqslant \frac { \pi } { 4 } \right\}$$ On your diagram

• shade the part of the diagram that is included in the set
• use solid lines to show the parts of the boundary that are included in the set, and use dashed lines to show the parts of the boundary that are not included in the set
  (b) Show that the complex number \(z = 5 + 4 \mathrm { i }\) is in the set \(X\).
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1. (a) Prove by induction that, for all \(n \in \mathbb { Z } ^ { + }\)

$$\mathrm { f } ( n ) = n ^ { 5 } + 4 n$$ is divisible by 5
(b) Show that \(\mathrm { f } ( - x ) = - \mathrm { f } ( x )\) for all \(x \in \mathbb { R }\)
(c) Hence prove that \(\mathrm { f } ( n )\) is divisible by 5 for all \(n \in \mathbb { Z }\)
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5. (a) Show that the binomial expansion of $$( 4 + 5 x ) ^ { \frac { 1 } { 2 } }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\) is $$2 + \frac { 5 } { 4 } x + k x ^ { 2 }$$ giving the value of the constant \(k\) as a simplified fraction.
(b) (i) Use the expansion from part (a), with \(x = \frac { 1 } { 10 }\), to find an approximate value for \(\sqrt { 2 }\) Give your answer in the form \(\frac { p } { q }\) where \(p\) and \(q\) are integers.
(ii) Explain why substituting \(x = \frac { 1 } { 10 }\) into this binomial expansion leads to a valid approximation.
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6. $$\mathrm { f } ( z ) = 8 z ^ { 3 } + 12 z ^ { 2 } + 6 z + 65$$ Given that \(\frac { 1 } { 2 } - \mathrm { i } \sqrt { 3 }\) is a root of the equation \(\mathrm { f } ( z ) = 0\)

1. write down the other complex root of the equation,
2. use algebra to solve the equation \(\mathrm { f } ( z ) = 0\) completely.
3. Show the roots of \(\mathrm { f } ( z )\) on a single Argand diagram.
4. Show that the roots of \(\mathrm { f } ( z )\) form the vertices of an equilateral triangle in the complex plane.
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7. The region bounded by the curve with equation \(y = 3 + \sqrt { x }\), the \(x\)-axis and the lines \(x = 1\) and \(x = 4\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Use integration to show that the volume generated is \(\frac { 125 \pi } { 2 }\)
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8. (a) Express \(2 \sin \theta - 1.5 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) State the value of \(R\) and give the value of \(\alpha\) to 4 decimal places. Tom models the depth of water, \(D\) metres, at Southview harbour on 18th October 2017 by the formula $$D = 6 + 2 \sin \left( \frac { 4 \pi t } { 25 } \right) - 1.5 \cos \left( \frac { 4 \pi t } { 25 } \right) , \quad 0 \leqslant t \leqslant 24$$ where \(t\) is the time, in hours, after 00:00 hours on 18th October 2017.
Use Tom's model to
(b) find the depth of water at 00:00 hours on 18th October 2017,
(c) find the maximum depth of water,
(d) find the time, in the afternoon, when the maximum depth of water occurs. Give your answer to the nearest minute. Tom's model is supported by measurements of \(D\) taken at regular intervals on 18th October 2017. Jolene attempts to use a similar model in order to model the depth of water at Southview harbour on 19th October 2017. Jolene models the depth of water, \(H\) metres, at Southview harbour on 19th October 2017 by the formula $$H = 6 + 2 \sin \left( \frac { 4 \pi x } { 25 } \right) - 1.5 \cos \left( \frac { 4 \pi x } { 25 } \right) , \quad 0 \leqslant x \leqslant 24$$ where \(x\) is the time, in hours, after 00:00 hours on 19th October 2017.
By considering the depth of water at 00:00 hours on 19th October 2017 for both models,
(e) (i) explain why Jolene's model is not correct,
(ii) hence find a suitable model for \(H\) in terms of \(x\).
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9. In a chemical reaction, compound B is formed from compound A and other compounds. The mass of B at time \(t\) minutes is \(x \mathrm {~kg}\). The total mass of A and B is always 1 kg . Sadiq formulates a simple model for the reaction in which the rate at which the mass of \(B\) increases is proportional to the product of the masses of \(A\) and \(B\).

1. Show that the model can be written as \(\frac { \mathrm { d } x } { \mathrm {~d} t } = k x ( 1 - x )\), where \(k\) is a constant. Initially, the mass of B is 0.2 kg .
2. Solve the differential equation, expressing \(x\) in terms of \(k\) and \(t\). After 15 minutes, the mass of B is measured to be 0.9 kg .
3. Find the value of \(k\), correct to 3 significant figures.
4. Find the mass of B after 30 minutes.
5. Explain what the model predicts for the mass of A remaining for large values of \(t\).
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10.

1. Find \(\int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 3 } \pi } 2 \tan x \mathrm {~d} x\) giving your answer in the form \(\ln p\).
2. Show that \(\int _ { 0 } ^ { \frac { 1 } { 2 } \pi } 2 \tan x \mathrm {~d} x\) is undefined explaining your reasoning.
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