OCR Further Pure Core 2 (Further Pure Core 2) 2021 November

1 Two matrices, \(\mathbf { A }\) and \(\mathbf { B }\), are given by \(\mathbf { A } = \left( \begin{array} { r r r } 1 & - 2 & - 1 \\ 2 & - 3 & 1 \\ a & 1 & 1 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { r r r } - 6 & 3 & - 4 \\ - 1 & 6 & - 4 \\ 8 & - 8 & - 1 \end{array} \right)\) where \(a\) is a constant. Find the value of \(a\) for which \(\mathbf { A B } = \mathbf { B A }\).

2 In this question you must show detailed reasoning. The complex numbers \(z _ { 1 }\) and \(z _ { 2 }\) are given by \(z _ { 1 } = 3 - 7 \mathrm { i }\) and \(z _ { 2 } = 2 + 4 \mathrm { i }\).

1. Express each of the following as exact numbers in the form \(a + b \mathrm { i }\).
   1. \(3 z _ { 1 } + 4 z _ { 2 }\)
   2. \(z _ { 1 } z _ { 2 }\)
   3. \(\frac { Z _ { 1 } } { Z _ { 2 } }\)
2. Write \(z _ { 1 }\) in modulus-argument form giving the modulus in exact form and the argument correct to \(\mathbf { 3 }\) significant figures.

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 \(\mathbf { r } \cdot \left( \begin{array} { r } 2 \\ - 5 \\ - 3 \end{array} \right) = 4\).

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.

4 In this question you must show detailed reasoning.
Determine the value of \(\sum _ { r = 1 } ^ { 100 } ( 2 r + 3 ) ^ { 2 }\).

5 In this question you must show detailed reasoning.

1. Using the definition of \(\cosh x\) in terms of exponentials, show that \(\cosh 2 x \equiv 2 \cosh ^ { 2 } x - 1\).
2. Solve the equation \(\cosh 2 x = 3 \cosh x + 1\), giving all your answers in exact logarithmic form.

6 In this question you must show detailed reasoning.
The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } 1 & 2 \\ 0 & 1 \end{array} \right)\).

1. Define the transformation represented by \(\mathbf { A }\).
2. Show that the area of any object shape is invariant under the transformation represented by \(\mathbf { A }\). The matrix \(\mathbf { B }\) is given by \(\mathbf { B } = \left( \begin{array} { r l } 7 & 2 \\ 21 & 7 \end{array} \right)\). You are given that \(\mathbf { B }\) represents the transformation which is the result of applying the following three transformations in the given order.

7 In this question you must show detailed reasoning.

1. Find the values of \(A , B\) and \(C\) for which \(\frac { x ^ { 3 } + x ^ { 2 } + 9 x - 1 } { x ^ { 3 } + x ^ { 2 } + 4 x + 4 } \equiv A + \frac { B x + C } { x ^ { 3 } + x ^ { 2 } + 4 x + 4 }\).
2. Hence express \(\frac { x ^ { 3 } + x ^ { 2 } + 9 x - 1 } { x ^ { 3 } + x ^ { 2 } + 4 x + 4 }\) using partial fractions.
3. Using your answer to part (b), determine \(\int _ { 0 } ^ { 2 } \frac { x ^ { 3 } + x ^ { 2 } + 9 x - 1 } { x ^ { 3 } + x ^ { 2 } + 4 x + 4 } \mathrm {~d} x\) expressing your answer in the form \(a + \ln b + c \pi\) where \(a\) is an integer, and \(b\) and \(c\) are both rational.

8 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, \(\frac { d v } { d t } + v = 2 e ^ { - 2 t }\).
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).
5. Determine the greatest value of \(t\) for which the model is valid.

9 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l } 2 & 3 \\ 0 & 2 \end{array} \right)\).

1. By considering \(\mathbf { A } , \mathbf { A } ^ { 2 } , \mathbf { A } ^ { 3 }\) and \(\mathbf { A } ^ { 4 }\) make a conjecture about the form of the matrix \(\mathbf { A } ^ { n }\) in terms of \(n\) for \(n \geqslant 1\).
2. Use induction to prove the conjecture made in part (a).

10 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 \(\pi ^ { \mathrm { e } }\).