CAIE FP1 (Further Pure Mathematics 1) 2019 June

1 A curve \(C\) has equation \(\cos y = x\), for \(- \pi < x < \pi\).

1. Use implicit differentiation to show that $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = - \cot y \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 }$$
2. Hence find the exact value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at the point \(\left( \frac { 1 } { 2 } , \frac { 1 } { 3 } \pi \right)\) on \(C\).

2 Let \(u _ { n } = \frac { 4 \sin \left( n - \frac { 1 } { 2 } \right) \sin \frac { 1 } { 2 } } { \cos ( 2 n - 1 ) + \cos 1 }\).

1. Using the formulae for \(\cos P \pm \cos Q\) given in the List of Formulae MF10, show that $$u _ { n } = \frac { 1 } { \cos n } - \frac { 1 } { \cos ( n - 1 ) }$$
2. Use the method of differences to find \(\sum _ { n = 1 } ^ { N } u _ { n }\).
3. Explain why the infinite series \(u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots\) does not converge.

3 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\mathbf { r } = 6 \mathbf { i } + 2 \mathbf { j } + 7 \mathbf { k } + \lambda ( \mathbf { i } + \mathbf { j } )\) and \(\mathbf { r } = 4 \mathbf { i } + 4 \mathbf { j } + \mu ( - 6 \mathbf { j } + \mathbf { k } )\) respectively. The point \(P\) on \(l _ { 1 }\) and the point \(Q\) on \(l _ { 2 }\) are such that \(P Q\) is perpendicular to both \(l _ { 1 }\) and \(l _ { 2 }\). Find the position vectors of \(P\) and \(Q\).

4 It is given that, for \(n \geqslant 0\), $$I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } \mathrm { e } ^ { x ^ { 3 } } \mathrm {~d} x$$

1. Show that \(I _ { 2 } = \frac { 1 } { 3 } ( \mathrm { e } - 1 )\).
2. Show that, for \(n \geqslant 3\), $$3 I _ { n } = \mathrm { e } - ( n - 2 ) I _ { n - 3 }$$
3. Hence find the exact value of \(I _ { 8 }\).

5 A curve \(C\) is defined parametrically by $$x = \frac { 2 } { \mathrm { e } ^ { t } + \mathrm { e } ^ { - t } } \quad \text { and } \quad y = \frac { \mathrm { e } ^ { t } - \mathrm { e } ^ { - t } } { \mathrm { e } ^ { t } + \mathrm { e } ^ { - t } }$$ for \(0 \leqslant t \leqslant 1\). The area of the surface generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis is denoted by \(S\).

1. Show that \(S = 4 \pi \int _ { 0 } ^ { 1 } \frac { \mathrm { e } ^ { t } - \mathrm { e } ^ { - t } } { \left( \mathrm { e } ^ { t } + \mathrm { e } ^ { - t } \right) ^ { 2 } } \mathrm {~d} t\).
2. Using the substitution \(u = \mathrm { e } ^ { t } + \mathrm { e } ^ { - t }\), or otherwise, find \(S\) in terms of \(\pi\) and e .

6 The equation $$x ^ { 3 } - x + 1 = 0$$ has roots \(\alpha , \beta , \gamma\).

1. Use the relation \(x = y ^ { \frac { 1 } { 3 } }\) to show that the equation $$y ^ { 3 } + 3 y ^ { 2 } + 2 y + 1 = 0$$ has roots \(\alpha ^ { 3 } , \beta ^ { 3 } , \gamma ^ { 3 }\). Hence write down the value of \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }\).
   Let \(S _ { n } = \alpha ^ { n } + \beta ^ { n } + \gamma ^ { n }\).
2. Find the value of \(S _ { - 3 }\).
3. Show that \(S _ { 6 } = 5\) and find the value of \(S _ { 9 }\).

7 Find the particular solution of the differential equation $$10 \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 3 \frac { \mathrm {~d} x } { \mathrm {~d} t } - x = t + 2$$ given that when \(t = 0 , x = 0\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 0\).

8

1. Prove by mathematical induction that, for \(z \neq 1\) and all positive integers \(n\), $$1 + z + z ^ { 2 } + \ldots + z ^ { n - 1 } = \frac { z ^ { n } - 1 } { z - 1 }$$
2. By letting \(z = \frac { 1 } { 2 } ( \cos \theta + \mathrm { i } \sin \theta )\), use de Moivre's theorem to deduce that $$\sum _ { m = 1 } ^ { \infty } \left( \frac { 1 } { 2 } \right) ^ { m } \sin m \theta = \frac { 2 \sin \theta } { 5 - 4 \cos \theta }$$

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

1. Show that \(\mathbf { e }\) is an eigenvector of \(\mathbf { A } ^ { 2 }\), with corresponding eigenvalue \(\lambda ^ { 2 }\).
   The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by $$\mathbf { A } = \left( \begin{array} { c c c } n & 1 & 3
   0 & 2 n & 0
   0 & 0 & 3 n \end{array} \right) \quad \text { and } \quad \mathbf { B } = ( \mathbf { A } + n \mathbf { I } ) ^ { 2 }$$ where \(\mathbf { I }\) is the \(3 \times 3\) identity matrix and \(n\) is a non-zero integer.
2. Find, in terms of \(n\), a non-singular matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { B } = \mathbf { P D P } \mathbf { P } ^ { - 1 }\).

10 The curves \(C _ { 1 }\) and \(C _ { 2 }\) have equations $$y = \frac { a x } { x + 5 } \quad \text { and } \quad y = \frac { x ^ { 2 } + ( a + 10 ) x + 5 a + 26 } { x + 5 }$$ respectively, where \(a\) is a constant and \(a > 2\).

1. Find the equations of the asymptotes of \(C _ { 1 }\).
2. Find the equation of the oblique asymptote of \(C _ { 2 }\).
3. Show that \(C _ { 1 }\) and \(C _ { 2 }\) do not intersect.
4. Find the coordinates of the stationary points of \(C _ { 2 }\).
5. Sketch \(C _ { 1 }\) and \(C _ { 2 }\) on a single diagram. [You do not need to calculate the coordinates of any points where \(C _ { 2 }\) crosses the axes.]

The curve \(C _ { 1 }\) has polar equation \(r ^ { 2 } = 2 \theta\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).

1. The point on \(C _ { 1 }\) furthest from the line \(\theta = \frac { 1 } { 2 } \pi\) is denoted by \(P\). Show that, at \(P\), $$2 \theta \tan \theta = 1$$ and verify that this equation has a root between 0.6 and 0.7 .
   The curve \(C _ { 2 }\) has polar equation \(r ^ { 2 } = \theta \sec ^ { 2 } \theta\), for \(0 \leqslant \theta < \frac { 1 } { 2 } \pi\). The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the pole, denoted by \(O\), and at another point \(Q\).
2. Find the exact value of \(\theta\) at \(Q\).
3. The diagram below shows the curve \(C _ { 2 }\). Sketch \(C _ { 1 }\) on this diagram.
4. Find, in exact form, the area of the region \(O P Q\) enclosed by \(C _ { 1 }\) and \(C _ { 2 }\).

The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix $$\mathbf { M } = \left( \begin{array} { r r r r } - 1 & 2 & 3 & 4
1 & 0 & 1 & - 1
1 & - 2 & - 3 & a
1 & 2 & 5 & 2 \end{array} \right) .$$

1. For \(a \neq - 4\), the range space of T is denoted by \(V\).
   (a) Find the dimension of \(V\) and show that $$\left( \begin{array} { r } - 1
   1
   1
   1 \end{array} \right) , \quad \left( \begin{array} { r } 2
   0
   - 2
   2 \end{array} \right) \quad \text { and } \quad \left( \begin{array} { r } 4
   - 1
   a
   2 \end{array} \right)$$ form a basis for \(V\).
   (b) Show that if \(\left( \begin{array} { l } x
   y
   z
   t \end{array} \right)\) belongs to \(V\) then \(x + 2 y = t\).
2. For \(a = - 4\), find the general solution of $$\mathbf { M } \mathbf { x } = \left( \begin{array} { r } - 1
   1
   1
   1 \end{array} \right)$$ If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.