SPS SPS FM Pure (SPS FM Pure) 2022 February

1. (a) Express \(\frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) }\) in partial fractions.
   (b) Hence find \(\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r - 1 ) ( 2 r + 1 ) }\), expressing the result as a single fraction.
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$$\mathbf { A } = \left( \begin{array} { r r } 4 & - 2
5 & 3 \end{array} \right)$$ The matrix \(\mathbf { A }\) represents the linear transformation \(M\).
Prove that, for the linear transformation \(M\), there are no invariant lines.
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3. The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 1
- 3
3 \end{array} \right) + \lambda \left( \begin{array} { r } 3
2
- 2 \end{array} \right)\).
The plane \(\Pi\) has equation \(\mathbf { r } . \left( \begin{array} { r } 2
- 5
- 3 \end{array} \right) = 4\).

1. Find the position vector of the point of intersection of \(l _ { 1 }\) and \(\Pi\).
2. Find the acute angle between \(l _ { 1 }\) and \(\Pi\).
   \(A\) is the point on \(l _ { 1 }\) where \(\lambda = 1\).
   \(l _ { 2 }\) is the line with the following properties.
   • \(l _ { 2 }\) passes through \(A\)
   • \(l _ { 2 }\) is perpendicular to \(l _ { 1 }\)
   • \(l _ { 2 }\) is parallel to \(\Pi\)
   • Find, in vector form, the equation of \(l _ { 2 }\).
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1. 1. \(\mathbf { A }\) is a 2 by 2 matrix and \(\mathbf { B }\) is a 2 by 3 matrix.

Giving a reason for your answer, explain whether it is possible to evaluate

1. \(\mathbf { A B }\)
2. \(\mathbf { A } + \mathbf { B }\)
   (ii) Given that $$\left( \begin{array} { r r r } - 5 & 3 & 1
   a & 0 & 0
   b & a & b \end{array} \right) \left( \begin{array} { r r r } 0 & 5 & 0
   2 & 12 & - 1
   - 1 & - 11 & 3 \end{array} \right) = \lambda \mathbf { I }$$ where \(a , b\) and \(\lambda\) are constants,
3. determine
   • the value of \(\lambda\)
   • the value of \(a\)
   • the value of \(b\)
   • Hence deduce the inverse of the matrix \(\left( \begin{array} { r r r } - 5 & 3 & 1
     a & 0 & 0
     b & a & b \end{array} \right)\)
     (iii) Given that
   $$\mathbf { M } = \left( \begin{array} { c c c } 1 & 1 & 1
   0 & \sin \theta & \cos \theta
   0 & \cos 2 \theta & \sin 2 \theta \end{array} \right) \quad \text { where } 0 \leqslant \theta < \pi$$ determine the values of \(\theta\) for which the matrix \(\mathbf { M }\) is singular.
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5. Points \(A , B\) and \(C\) have coordinates \(( 4,2,0 ) , ( 1,5,3 )\) and \(( 1,4 , - 2 )\) respectively. The line \(l\) passes through \(A\) and \(B\).

1. Find a cartesian equation for \(l\).
   \(M\) is the point on \(l\) that is closest to \(C\).
2. Find the coordinates of \(M\).
3. Find the exact area of the triangle \(A B C\).
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   There are exactly four points on \(C\) where the tangent is perpendicular to the initial line.
4. Show that the range possible values for \(p\) is
5. Sketch the curve with equation $$r = a ( 3 + 2 \cos \theta ) \quad 0 \leqslant \theta < 2 \pi \quad \text { where } a > 0$$ John digs a hole in his garden in order to make a pond.
   The pond has a uniform horizontal cross section that is modelled by the curve with equation $$r = 20 ( 3 + 2 \cos \theta ) \quad 0 \leqslant \theta < 2 \pi$$ where \(r\) is measured in centimetres. The depth of the pond is 90 centimetres.
   Water flows through a hosepipe into the pond at a rate of 50 litres per minute.
   Given that the pond is initially empty,
6. determine how long it will take to completely fill the pond with water using the hosepipe, according to the model. Give your answer to the nearest minute.
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7. The matrix \(\mathbf { M }\) is defined by \(\mathbf { M } = \left[ \begin{array} { c c c } 3 & 2 & - 2
0 & 1 & 0
0 & 0 & 1 \end{array} \right]\)
Prove by induction that \(\mathbf { M } ^ { n } = \left[ \begin{array} { c c c } 3 ^ { n } & 3 ^ { n } - 1 & - 3 ^ { n } + 1
0 & 1 & 0
0 & 0 & 1 \end{array} \right]\) for all integers \(n \geq 1\)
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8. The complex number \(z\) satisfies the equations $$\left| z ^ { * } - 1 - 2 i \right| = | z - 3 |$$ and $$| z - a | = 3$$ where \(a\) is real.
Show that \(a\) must lie in the interval \([ 1 - s \sqrt { t } , 1 + s \sqrt { t } ]\), where \(s\) and \(t\) are prime numbers.
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9. The equation \(4 x ^ { 4 } - 4 x ^ { 3 } + p x ^ { 2 } + q x - 9 = 0\), where \(p\) and \(q\) are constants, has roots \(\alpha , - \alpha , \beta\) and \(\frac { 1 } { \beta }\).

1. Determine the exact roots of the equation.
2. Determine the values of \(p\) and \(q\).
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10. You are given that \(\mathrm { f } ( x ) = 4 \sinh x + 3 \cosh x\).

1. Show that the curve \(y = \mathrm { f } ( x )\) has no turning points.
2. Determine the exact solution of the equation \(\mathrm { f } ( x ) = 5\).
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11. A particle \(P\) of mass 2 kg can only move along the straight line segment \(O A\), where \(O A\) is on a rough horizontal surface. The particle is initially at rest at \(O\) and the distance \(O A\) is 0.9 m . When the time is \(t\) seconds the displacement of \(P\) from \(O\) is \(x \mathrm {~m}\) and the velocity of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\). \(P\) is subject to a force of magnitude \(4 \mathrm { e } ^ { - 2 t } \mathrm {~N}\) in the direction of \(A\) for any \(t \geqslant 0\). The resistance to the motion of \(P\) is modelled as being proportional to \(v\). At the instant when \(t = \ln 2 , v = 0.5\) and the resultant force on \(P\) is 0 N .

1. Show that, according to the model, \(\frac { \mathrm { d } v } { \mathrm {~d} t } + v = 2 \mathrm { e } ^ { - 2 t }\).
2. Find an expression for \(v\) in terms of \(t\) for \(t \geqslant 0\).
3. By considering the behaviour of \(v\) as \(t\) becomes large explain why, according to the model, \(P\) 's speed must reach a maximum value for some \(t > 0\).
4. Determine the maximum speed considered in part (c).
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12. In this question you must show detailed reasoning.

1. By using an appropriate Maclaurin series prove that if \(x > 0\) then \(\mathrm { e } ^ { x } > 1 + x\).
2. Hence, by using a suitable substitution, deduce that \(\mathrm { e } ^ { t } > \mathrm { e } t\) for \(t > 1\).
3. Using the inequality in part (b), and by making a suitable choice for \(t\), determine which is greater, \(\mathrm { e } ^ { \pi }\) or \(\pi ^ { \mathrm { e } }\).
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