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 \(\mathbf { D }\).]