CAIE FP1 (Further Pure Mathematics 1) 2013 November

1 Express \(\frac { 1 } { r ( r + 1 ) ( r - 1 ) }\) in partial fractions. Find $$\sum _ { r = 2 } ^ { n } \frac { 1 } { r ( r + 1 ) ( r - 1 ) }$$ State the value of $$\sum _ { r = 2 } ^ { \infty } \frac { 1 } { r ( r + 1 ) ( r - 1 ) }$$

2 Show that the matrix \(\left( \begin{array} { r r r } 1 & 4 & 2
3 & 0 & - 2
3 & - 3 & - 4 \end{array} \right)\) has no inverse. Solve the system of equations $$\begin{array} { r } x + 4 y + 2 z = 0
3 x - 2 z = 4
3 x - 3 y - 4 z = 5 \end{array}$$

3 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y = 4 x ^ { 2 } + 8$$

4 A curve has parametric equations $$x = 2 \theta - \sin 2 \theta , \quad y = 1 - \cos 2 \theta , \quad \text { for } - 3 \pi \leqslant \theta \leqslant 3 \pi$$ Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \cot \theta$$ except for certain values of \(\theta\), which should be stated. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) when \(\theta = \frac { 1 } { 4 } \pi\).

5 The equation $$8 x ^ { 3 } + 36 x ^ { 2 } + k x - 21 = 0$$ where \(k\) is a constant, has roots \(a - d , a , a + d\). Find the numerical values of the roots and determine the value of \(k\).

6 [In this question you may use, without proof, the formula \(\int \sec x \mathrm {~d} x = \ln ( \sec x + \tan x ) + \operatorname { const }\).]

1. Let \(y = \sec x\). Find the mean value of \(y\) with respect to \(x\) over the interval \(\frac { 1 } { 6 } \pi \leqslant x \leqslant \frac { 1 } { 3 } \pi\).
2. The curve \(C\) has equation \(y = - \ln ( \cos x )\), for \(0 \leqslant x \leqslant \frac { 1 } { 3 } \pi\). Find the arc length of \(C\).

7 The curve \(C\) has equation $$y = \frac { 2 x ^ { 2 } + 5 x - 1 } { x + 2 }$$ Find the equations of the asymptotes of \(C\). Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } > 2\) at all points on \(C\). Sketch C.

8 The points \(A , B , C\) have position vectors $$4 \mathbf { i } + 5 \mathbf { j } + 6 \mathbf { k } , \quad 5 \mathbf { i } + 7 \mathbf { j } + 8 \mathbf { k } , \quad 2 \mathbf { i } + 6 \mathbf { j } + 4 \mathbf { k }$$ respectively, relative to the origin \(O\). Find a cartesian equation of the plane \(A B C\). The point \(D\) has position vector \(6 \mathbf { i } + 3 \mathbf { j } + 6 \mathbf { k }\). Find the coordinates of \(E\), the point of intersection of the line \(O D\) with the plane \(A B C\). Find the acute angle between the line \(E D\) and the plane \(A B C\).

9 Prove by mathematical induction that, for every positive integer \(n\), $$( \cos \theta + i \sin \theta ) ^ { n } = \cos n \theta + i \sin n \theta$$ Express \(\sin ^ { 5 } \theta\) in the form \(p \sin 5 \theta + q \sin 3 \theta + r \sin \theta\), where \(p , q\) and \(r\) are rational numbers to be determined.

10 The curve \(C\) has polar equation \(r = 2 \sin \theta ( 1 - \cos \theta )\), for \(0 \leqslant \theta \leqslant \pi\). Find \(\frac { \mathrm { d } r } { \mathrm {~d} \theta }\) and hence find the polar coordinates of the point of \(C\) that is furthest from the pole. Sketch \(C\). Find the exact area of the sector from \(\theta = 0\) to \(\theta = \frac { 1 } { 4 } \pi\).

Let \(I _ { n } = \int _ { 0 } ^ { 1 } \left( 1 + x ^ { 2 } \right) ^ { n } \mathrm {~d} x\). Show that, for all integers \(n\), $$( 2 n + 1 ) I _ { n } = 2 n I _ { n - 1 } + 2 ^ { n }$$ Evaluate \(I _ { 0 }\) and hence find \(I _ { 3 }\). Given that \(I _ { - 1 } = \frac { 1 } { 4 } \pi\), find \(I _ { - 3 }\).

The vector \(\mathbf { e }\) is an eigenvector of each of the \(3 \times 3\) matrices \(\mathbf { A }\) and \(\mathbf { B }\), with corresponding eigenvalues \(\lambda\) and \(\mu\) respectively. Justifying your answer, state an eigenvalue of \(\mathbf { A } + \mathbf { B }\). The matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r } 6 & - 1 & - 6
1 & 0 & - 2
3 & - 1 & - 3 \end{array} \right)$$ has eigenvectors \(\left( \begin{array} { l } 1
1
1 \end{array} \right) , \left( \begin{array} { r } 1
- 1
1 \end{array} \right) , \left( \begin{array} { l } 2
0
1 \end{array} \right)\). Find the corresponding eigenvalues. The matrix \(\mathbf { B }\), where $$\mathbf { B } = \left( \begin{array} { r r r } 8 & - 2 & - 8
2 & 0 & - 4
4 & - 2 & - 4 \end{array} \right) ,$$ also has eigenvectors \(\left( \begin{array} { l } 1
1
1 \end{array} \right) , \left( \begin{array} { r } 1
- 1
1 \end{array} \right) , \left( \begin{array} { l } 2
0
1 \end{array} \right)\), for which \(- 2,2,4\), respectively, are corresponding eigenvalues. The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \mathbf { A } + \mathbf { B } - 5 \mathbf { I }\), where \(\mathbf { I }\) is the \(3 \times 3\) identity matrix. State the eigenvalues of \(\mathbf { M }\). Find matrices \(\mathbf { R }\) and \(\mathbf { S }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { M } ^ { 5 } = \mathbf { R D S }\).
[0pt] [You should show clearly all the elements of the matrices \(\mathbf { R } , \mathbf { S }\) and \(\mathbf { D }\).] \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }