CAIE FP1 (Further Pure Mathematics 1) 2018 June

1 The curve \(C\) is defined parametrically by $$x = \mathrm { e } ^ { t } - t , \quad y = 4 \mathrm { e } ^ { \frac { 1 } { 2 } t }$$ Find the length of the arc of \(C\) from the point where \(t = 0\) to the point where \(t = 3\).

2 It is given that \(\mathrm { f } ( n ) = 2 ^ { 3 n } + 8 ^ { n - 1 }\). By simplifying \(\mathrm { f } ( k ) + \mathrm { f } ( k + 1 )\), or otherwise, prove by mathematical induction that \(\mathrm { f } ( n )\) is divisible by 9 for every positive integer \(n\).

3 The curve \(C\) has polar equation \(r = \cos 2 \theta\), for \(- \frac { 1 } { 4 } \pi \leqslant \theta \leqslant \frac { 1 } { 4 } \pi\).

1. Sketch \(C\).
2. Find the area of the region enclosed by \(C\), showing full working.
3. Find a cartesian equation of \(C\).

4 It is given that the equation $$x ^ { 3 } - 21 x ^ { 2 } + k x - 216 = 0$$ where \(k\) is a constant, has real roots \(a , a r\) and \(a r ^ { - 1 }\).

1. Find the numerical values of the roots.
2. Deduce the value of \(k\).

5 Let \(S _ { n } = \sum _ { r = 1 } ^ { n } ( - 1 ) ^ { r - 1 } r ^ { 2 }\).

1. Use the standard result for \(\sum _ { r = 1 } ^ { n } r ^ { 2 }\) given in the List of Formulae (MF10) to show that $$S _ { 2 n } = - n ( 2 n + 1 )$$
2. State the value of \(\lim _ { n \rightarrow \infty } \frac { S _ { 2 n } } { n ^ { 2 } }\) and find \(\lim _ { n \rightarrow \infty } \frac { S _ { 2 n + 1 } } { n ^ { 2 } }\).

6 The curve \(C\) has equation $$y = \frac { x ^ { 2 } + b } { x + b }$$ where \(b\) is a positive constant.

1. Find the equations of the asymptotes of \(C\).
2. Show that \(C\) does not intersect the \(x\)-axis.
3. Justifying your answer, find the number of stationary points on \(C\).
4. Sketch C. Your sketch should indicate the coordinates of any points of intersection with the \(y\)-axis. You do not need to find the coordinates of any stationary points.

7 Find the particular solution of the differential equation $$49 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 14 \frac { \mathrm {~d} y } { \mathrm {~d} x } + y = 49 x + 735$$ given that when \(x = 0 , y = 0\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\).

8 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 3 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r c r } 1 & 2 & \alpha & - 1
2 & 6 & - 3 & - 3
3 & 10 & - 6 & - 5 \end{array} \right)$$ and \(\alpha\) is a constant. When \(\alpha \neq 0\) the null space of T is denoted by \(K _ { 1 }\).

1. Find a basis for \(K _ { 1 }\).
   When \(\alpha = 0\) the null space of T is denoted by \(K _ { 2 }\).
2. Find a basis for \(K _ { 2 }\).
3. Determine, justifying your answer, whether \(K _ { 1 }\) is a subspace of \(K _ { 2 }\).

9

1. Using the substitution \(u = \tan x\), or otherwise, find \(\int \sec ^ { 2 } x \tan ^ { 2 } x \mathrm {~d} x\).
   It is given that, for \(n \geqslant 0\), $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sec ^ { n } x \tan ^ { 2 } x \mathrm {~d} x$$
2. Using the result that \(\frac { \mathrm { d } } { \mathrm { d } x } ( \sec x ) = \tan x \sec x\), show that, for \(n \geqslant 2\), $$( n + 1 ) I _ { n } = ( \sqrt { } 2 ) ^ { n - 2 } + ( n - 2 ) I _ { n - 2 }$$
3. Hence find the mean value of \(\sec ^ { 4 } x \tan ^ { 2 } x\) with respect to \(x\) over the interval \(0 \leqslant x \leqslant \frac { 1 } { 4 } \pi\), giving your answer in exact form.

10 The line \(l _ { 1 }\) is parallel to the vector \(a \mathbf { i } - \mathbf { j } + \mathbf { k }\), where \(a\) is a constant, and passes through the point whose position vector is \(9 \mathbf { j } + 2 \mathbf { k }\). The line \(l _ { 2 }\) is parallel to the vector \(- a \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k }\) and passes through the point whose position vector is \(- 6 \mathbf { i } - 5 \mathbf { j } + 10 \mathbf { k }\).

1. It is given that \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
   (a) Show that \(a = - \frac { 6 } { 13 }\).
   (b) Find a cartesian equation of the plane containing \(l _ { 1 }\) and \(l _ { 2 }\).
2. Given instead that the perpendicular distance between \(l _ { 1 }\) and \(l _ { 2 }\) is \(3 \sqrt { } ( 30 )\), find the value of \(a\).

1. Show that if \(z = \mathrm { e } ^ { \mathrm { i } \theta }\) and \(z \neq - 1\) then $$\frac { z - 1 } { z + 1 } = \mathrm { i } \tan \frac { 1 } { 2 } \theta$$
2. Hence, or otherwise, show that if \(z\) is a cube root of unity then $$\frac { z ^ { 3 } - 1 } { z ^ { 3 } + 1 } + \frac { z ^ { 2 } - 1 } { z ^ { 2 } + 1 } + \frac { z - 1 } { z + 1 } = 0$$
3. Hence write down three roots of the equation $$\left( z ^ { 3 } - 1 \right) \left( z ^ { 2 } + 1 \right) ( z + 1 ) + \left( z ^ { 2 } - 1 \right) \left( z ^ { 3 } + 1 \right) ( z + 1 ) + ( z - 1 ) \left( z ^ { 3 } + 1 \right) \left( z ^ { 2 } + 1 \right) = 0$$ and find the other three roots. Give your answers in an exact form.

It is given that \(\mathbf { e }\) is an eigenvector of the matrix \(\mathbf { A }\), with corresponding eigenvalue \(\lambda\).

1. Write down another eigenvector of \(\mathbf { A }\) corresponding to \(\lambda\).
2. Write down an eigenvector and corresponding eigenvalue of \(\mathbf { A } ^ { n }\), where \(n\) is a positive integer.
   Let \(\mathbf { A } = \left( \begin{array} { l l l } 3 & 0 & 0
   2 & 7 & 0
   4 & 8 & 1 \end{array} \right)\).
3. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { n } = \mathbf { P D P } \mathbf { P } ^ { - 1 }\).
4. Determine the set of values of the real constant \(k\) such that $$\sum _ { n = 1 } ^ { \infty } k ^ { n } \left( \mathbf { A } ^ { n } - k \mathbf { A } ^ { n + 1 } \right) = k \mathbf { A } .$$ If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.