CAIE FP1 (Further Pure Mathematics 1) 2011 June

1 Express \(\frac { 1 } { ( 2 r + 1 ) ( 2 r + 3 ) }\) in partial fractions and hence use the method of differences to find $$\sum _ { r = 1 } ^ { n } \frac { 1 } { ( 2 r + 1 ) ( 2 r + 3 ) }$$ Deduce the value of $$\sum _ { r = 1 } ^ { \infty } \frac { 1 } { ( 2 r + 1 ) ( 2 r + 3 ) }$$

2 The roots of the equation $$x ^ { 3 } + p x ^ { 2 } + q x + r = 0$$ are \(\frac { \beta } { k } , \beta , k \beta\), where \(p , q , r , k\) and \(\beta\) are non-zero real constants. Show that \(\beta = - \frac { q } { p }\). Deduce that \(r p ^ { 3 } = q ^ { 3 }\).

3 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { M } = \left( \begin{array} { r r r r } 1 & 3 & - 2 & 4
5 & 15 & - 9 & 19
- 2 & - 6 & 3 & - 7
3 & 9 & - 5 & 11 \end{array} \right)\).

1. Find the rank of \(\mathbf { M }\).
2. Obtain a basis for the null space of T .

4 It is given that \(\mathrm { f } ( n ) = 3 ^ { 3 n } + 6 ^ { n - 1 }\).

1. Show that \(\mathrm { f } ( n + 1 ) + \mathrm { f } ( n ) = 28 \left( 3 ^ { 3 n } \right) + 7 \left( 6 ^ { n - 1 } \right)\).
2. Hence, or otherwise, prove by mathematical induction that \(\mathrm { f } ( n )\) is divisible by 7 for every positive integer \(n\).

5 The curve \(C\) has polar equation \(r = 2 \cos 2 \theta\). Sketch the curve for \(0 \leqslant \theta < 2 \pi\). Find the exact area of one loop of the curve.

6 The line \(l _ { 1 }\) passes through the point with position vector \(8 \mathbf { i } + 8 \mathbf { j } - 7 \mathbf { k }\) and is parallel to the vector \(4 \mathbf { i } + 3 \mathbf { j }\). The line \(l _ { 2 }\) passes through the point with position vector \(7 \mathbf { i } - 2 \mathbf { j } + 4 \mathbf { k }\) and is parallel to the vector \(4 \mathbf { i } - \mathbf { k }\). 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 \(l _ { 2 }\). In either order,

1. show that \(P Q = 13\),
2. find the position vectors of \(P\) and \(Q\).

7 The variables \(x\) and \(y\) are related by the differential equation $$y ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 y ^ { 2 } \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } - 5 y ^ { 3 } = 8 \mathrm { e } ^ { - x }$$ Given that \(v = y ^ { 3 }\), show that $$\frac { \mathrm { d } ^ { 2 } v } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} v } { \mathrm {~d} x } - 15 v = 24 \mathrm { e } ^ { - x }$$ Hence find the general solution for \(y\) in terms of \(x\).

8 Find the eigenvalues and corresponding eigenvectors of the matrix \(\mathbf { A } = \left( \begin{array} { r r r } 4 & - 1 & 1
- 1 & 0 & - 3
1 & - 3 & 0 \end{array} \right)\). Find a non-singular matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } ^ { 5 } = \mathbf { P D P } ^ { - 1 }\).

9 The curve \(C\) has equation \(y = x ^ { \frac { 3 } { 2 } }\). Find the coordinates of the centroid of the region bounded by \(C\), the lines \(x = 1 , x = 4\) and the \(x\)-axis. Show that the length of the arc of \(C\) from the point where \(x = 5\) to the point where \(x = 28\) is 139 .

10 Let $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \cos ^ { n } x \mathrm {~d} x$$ where \(n \geqslant 0\). Show that, for all \(n \geqslant 2\), $$I _ { n } = \frac { n - 1 } { n } I _ { n - 2 }$$ A curve has parametric equations \(x = a \sin ^ { 3 } t\) and \(y = a \cos ^ { 3 } t\), where \(a\) is a constant and \(0 \leqslant t \leqslant \frac { 1 } { 2 } \pi\). Show that the mean value \(m\) of \(y\) over the interval \(0 \leqslant x \leqslant a\) is given by $$m = 3 a \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \left( \cos ^ { 4 } t - \cos ^ { 6 } t \right) \mathrm { d } t$$ Find the exact value of \(m\), in terms of \(a\).

Use de Moivre's theorem to prove that $$\tan 3 \theta = \frac { 3 \tan \theta - \tan ^ { 3 } \theta } { 1 - 3 \tan ^ { 2 } \theta }$$ State the exact values of \(\theta\), between 0 and \(\pi\), that satisfy \(\tan 3 \theta = 1\). Express each root of the equation \(t ^ { 3 } - 3 t ^ { 2 } - 3 t + 1 = 0\) in the form \(\tan ( k \pi )\), where \(k\) is a positive rational number. For each of these values of \(k\), find the exact value of \(\tan ( k \pi )\).

The curve \(C\) has equation $$y = \frac { x ^ { 2 } + \lambda x - 6 \lambda ^ { 2 } } { x + 3 }$$ where \(\lambda\) is a constant such that \(\lambda \neq 1\) and \(\lambda \neq - \frac { 3 } { 2 }\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and deduce that if \(C\) has two stationary points then \(- \frac { 3 } { 2 } < \lambda < 1\).
2. Find the equations of the asymptotes of \(C\).
3. Draw a sketch of \(C\) for the case \(0 < \lambda < 1\).
4. Draw a sketch of \(C\) for the case \(\lambda > 3\).