CAIE FP1 (Further Pure Mathematics 1) 2005 November

1 Write down the fifth roots of unity. Hence, or otherwise, find all the roots of the equation $$z ^ { 5 } = - 16 + ( 16 \sqrt { } 3 ) i$$ giving each root in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\).

2 The sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is such that \(u _ { 1 } = 1\) and $$u _ { n + 1 } = - 1 + \sqrt { } \left( u _ { n } + 7 \right)$$

1. Prove by induction that \(u _ { n } < 2\) for all \(n \geqslant 1\).
2. Show that if \(u _ { n } = 2 - \varepsilon\), where \(\varepsilon\) is small, then $$u _ { n + 1 } \approx 2 - \frac { 1 } { 6 } \varepsilon$$

3 The curve \(C\) has equation $$y = \frac { x ^ { 2 } } { x + \lambda }$$ where \(\lambda\) is a non-zero constant. Obtain the equations of the asymptotes of \(C\). In separate diagrams, sketch \(C\) for the cases where

1. \(\lambda > 0\),
2. \(\lambda < 0\).

4 Solve the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 3 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 2 y = 24 \mathrm { e } ^ { 2 x }$$ given that \(y = 1\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 9\) when \(x = 0\).

5 In the equation $$x ^ { 3 } + a x ^ { 2 } + b x + c = 0$$ the coefficients \(a , b\) and \(c\) are real. It is given that all the roots are real and greater than 1 .

1. Prove that \(a < - 3\).
2. By considering the sum of the squares of the roots, prove that \(a ^ { 2 } > 2 b + 3\).
3. By considering the sum of the cubes of the roots, prove that \(a ^ { 3 } < - 9 b - 3 c - 3\).

6 Let $$I _ { n } = \int _ { 0 } ^ { 1 } \left( 1 + x ^ { 2 } \right) ^ { - n } \mathrm {~d} x$$ where \(n \geqslant 1\). By considering \(\frac { \mathrm { d } } { \mathrm { d } x } \left( x \left( 1 + x ^ { 2 } \right) ^ { - n } \right)\), or otherwise, prove that $$2 n I _ { n + 1 } = ( 2 n - 1 ) I _ { n } + 2 ^ { - n }$$ Deduce that \(I _ { 3 } = \frac { 3 } { 32 } \pi + \frac { 1 } { 4 }\).
\(7 \quad\) Write down an expression in terms of \(z\) and \(N\) for the sum of the series $$\sum _ { n = 1 } ^ { N } 2 ^ { - n } z ^ { n }$$ Use de Moivre's theorem to deduce that $$\sum _ { n = 1 } ^ { 10 } 2 ^ { - n } \sin \left( \frac { 1 } { 10 } n \pi \right) = \frac { 1025 \sin \left( \frac { 1 } { 10 } \pi \right) } { 2560 - 2048 \cos \left( \frac { 1 } { 10 } \pi \right) }$$

8 Find the coordinates of the centroid of the finite region bounded by the \(x\)-axis and the curve whose equation is $$y = x ^ { 2 } ( 1 - x )$$ Deduce the coordinates of the centroid of the finite region bounded by the \(x\)-axis and the curve whose equation is $$y = x ( 1 - x ) ^ { 2 }$$

9 The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) have vector equations $$\mathbf { r } = \lambda _ { 1 } ( \mathbf { i } + \mathbf { j } - \mathbf { k } ) + \mu _ { 1 } ( 2 \mathbf { i } - \mathbf { j } + \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = \lambda _ { 2 } ( \mathbf { i } + 2 \mathbf { j } + \mathbf { k } ) + \mu _ { 2 } ( 3 \mathbf { i } + \mathbf { j } - \mathbf { k } )$$ respectively. The line \(l\) passes through the point with position vector \(4 \mathbf { i } + 5 \mathbf { j } + 6 \mathbf { k }\) and is parallel to both \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\). Find a vector equation for \(l\). Find also the shortest distance between \(l\) and the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).

10 It is given that the eigenvalues of the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r } 4 & 1 & - 1
- 4 & - 1 & 4
0 & - 1 & 5 \end{array} \right)$$ are \(1,3,4\). Find a set of corresponding eigenvectors. Write down a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { M } ^ { n } = \mathbf { P } \mathbf { D } \mathbf { P } ^ { - 1 }$$ where \(n\) is a positive integer. Find \(\mathbf { P } ^ { - 1 }\) and deduce that $$\lim _ { n \rightarrow \infty } 4 ^ { - n } \mathbf { M } ^ { n } = \left( \begin{array} { r r r } - \frac { 1 } { 3 } & 0 & - \frac { 1 } { 3 }
\frac { 4 } { 3 } & 0 & \frac { 4 } { 3 }
\frac { 4 } { 3 } & 0 & \frac { 4 } { 3 } \end{array} \right)$$

11 Find the rank of the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r r } 1 & 1 & 2 & 3
4 & 3 & 5 & 16
6 & 6 & 13 & 13
14 & 12 & 23 & 45 \end{array} \right)$$ Find vectors \(\mathbf { x } _ { 0 }\) and \(\mathbf { e }\) such that any solution of the equation $$\mathbf { A x } = \left( \begin{array} { r } 0
2
- 1
3 \end{array} \right)$$ can be expressed in the form \(\mathbf { x } _ { 0 } + \lambda \mathbf { e }\), where \(\lambda \in \mathbb { R }\). Hence show that there is no vector which satisfies (*) and has all its elements positive.

Show that \(\left( n + \frac { 1 } { 2 } \right) ^ { 3 } - \left( n - \frac { 1 } { 2 } \right) ^ { 3 } \equiv 3 n ^ { 2 } + \frac { 1 } { 4 }\). Use this result to prove that \(\sum _ { n = 1 } ^ { N } n ^ { 2 } = \frac { 1 } { 6 } N ( N + 1 ) ( 2 N + 1 )\). The sums \(S , T\) and \(U\) are defined as follows: $$\begin{aligned} & S = 1 ^ { 2 } + 2 ^ { 2 } + 3 ^ { 2 } + 4 ^ { 2 } + \ldots + ( 2 N ) ^ { 2 } + ( 2 N + 1 ) ^ { 2 } ,
& T = 1 ^ { 2 } + 3 ^ { 2 } + 5 ^ { 2 } + 7 ^ { 2 } + \ldots + ( 2 N - 1 ) ^ { 2 } + ( 2 N + 1 ) ^ { 2 } ,
& U = 1 ^ { 2 } - 2 ^ { 2 } + 3 ^ { 2 } - 4 ^ { 2 } + \ldots - ( 2 N ) ^ { 2 } + ( 2 N + 1 ) ^ { 2 } . \end{aligned}$$ Find and simplify expressions in terms of \(N\) for each of \(S , T\) and \(U\). Hence

1. describe the behaviour of \(\frac { S } { T }\) as \(N \rightarrow \infty\),
2. prove that if \(\frac { S } { U }\) is an integer then \(\frac { T } { U }\) is an integer.

The curves \(C _ { 1 }\) and \(C _ { 2 }\) have polar equations $$r = 4 \cos \theta \quad \text { and } \quad r = 1 + \cos \theta$$ respectively, where \(- \frac { 1 } { 2 } \pi \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).

1. Show that \(C _ { 1 }\) and \(C _ { 2 }\) meet at the points \(A \left( \frac { 4 } { 3 } , \alpha \right)\) and \(B \left( \frac { 4 } { 3 } , - \alpha \right)\), where \(\alpha\) is the acute angle such that \(\cos \alpha = \frac { 1 } { 3 }\).
2. In a single diagram, draw sketch graphs of \(C _ { 1 }\) and \(C _ { 2 }\).
3. Show that the area of the region bounded by the arcs \(O A\) and \(O B\) of \(C _ { 1 }\), and the \(\operatorname { arc } A B\) of \(C _ { 2 }\), is $$4 \pi - \frac { 1 } { 3 } \sqrt { } 2 - \frac { 13 } { 2 } \alpha .$$