CAIE FP1 (Further Pure Mathematics 1) 2012 June

1 Find the sum of the first \(n\) terms of the series $$\frac { 1 } { 1 \times 3 } + \frac { 1 } { 2 \times 4 } + \frac { 1 } { 3 \times 5 } + \ldots$$ and deduce the sum to infinity.

2 For the sequence \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\), it is given that \(u _ { 1 } = 1\) and \(u _ { r + 1 } = \frac { 3 u _ { r } - 2 } { 4 }\) for all \(r\). Prove by mathematical induction that \(u _ { n } = 4 \left( \frac { 3 } { 4 } \right) ^ { n } - 2\), for all positive integers \(n\).

3 The curve \(C\) has equation $$x y + ( x + y ) ^ { 3 } = 1$$ Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 3 } { 4 }\) at the point \(A ( 1,0 )\) on \(C\). Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at \(A\).

4 Let $$I _ { n } = \int _ { 1 } ^ { \mathrm { e } } x ^ { 2 } ( \ln x ) ^ { n } \mathrm {~d} x$$ for \(n \geqslant 0\). Show that, for all \(n \geqslant 1\), $$I _ { n } = \frac { 1 } { 3 } \mathrm { e } ^ { 3 } - \frac { 1 } { 3 } n I _ { n - 1 }$$ Find the exact value of \(I _ { 3 }\).

5 The matrix \(\mathbf { A }\) has an eigenvalue \(\lambda\) with corresponding eigenvector \(\mathbf { e }\). Prove that the matrix \(( \mathbf { A } + k \mathbf { I } )\), where \(k\) is a real constant and \(\mathbf { I }\) is the identity matrix, has an eigenvalue ( \(\lambda + k\) ) with corresponding eigenvector \(\mathbf { e }\). The matrix \(\mathbf { B }\) is given by $$\mathbf { B } = \left( \begin{array} { r r r } 2 & 2 & - 3
2 & 2 & 3
- 3 & 3 & 3 \end{array} \right) .$$ Two of the eigenvalues of \(\mathbf { B }\) are - 3 and 4 . Find corresponding eigenvectors. Given that \(\left( \begin{array} { r } 1
- 1
- 2 \end{array} \right)\) is an eigenvector of \(\mathbf { B }\), find the corresponding eigenvalue. Hence find the eigenvalues of \(\mathbf { C }\), where $$\mathbf { C } = \left( \begin{array} { r r r } - 1 & 2 & - 3
2 & - 1 & 3
- 3 & 3 & 0 \end{array} \right) ,$$ and state corresponding eigenvectors.

6 The curve \(C\) has equation \(y = \frac { x ^ { 2 } } { x - 2 }\). Find the equations of the asymptotes of \(C\). Find the coordinates of the turning points on \(C\). Draw a sketch of \(C\).

7 Expand \(\left( z + \frac { 1 } { z } \right) ^ { 4 } \left( z - \frac { 1 } { z } \right) ^ { 2 }\) and, by substituting \(z = \cos \theta + \mathrm { i } \sin \theta\), find integers \(p , q , r , s\) such that $$64 \sin ^ { 2 } \theta \cos ^ { 4 } \theta = p + q \cos 2 \theta + r \cos 4 \theta + s \cos 6 \theta$$ Using the substitution \(x = 2 \cos \theta\), show that $$\int _ { 1 } ^ { 2 } x ^ { 4 } \sqrt { } \left( 4 - x ^ { 2 } \right) \mathrm { d } x = \frac { 4 } { 3 } \pi + \sqrt { } 3$$

8 The cubic equation \(x ^ { 3 } - x ^ { 2 } - 3 x - 10 = 0\) has roots \(\alpha , \beta , \gamma\).

1. Let \(u = - \alpha + \beta + \gamma\). Show that \(u + 2 \alpha = 1\), and hence find a cubic equation having roots \(- \alpha + \beta + \gamma\), \(\alpha - \beta + \gamma , \alpha + \beta - \gamma\).
2. State the value of \(\alpha \beta \gamma\) and hence find a cubic equation having roots \(\frac { 1 } { \beta \gamma } , \frac { 1 } { \gamma \alpha } , \frac { 1 } { \alpha \beta }\).

9 The plane \(\Pi _ { 1 }\) has parametric equation $$\mathbf { r } = 2 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } - \mathbf { k } ) + \mu ( \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k } )$$ Find a cartesian equation of \(\Pi _ { 1 }\). The plane \(\Pi _ { 2 }\) has cartesian equation \(3 x - 2 y - 3 z = 4\). Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\). Find a vector equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).

10 Find the set of values of \(a\) for which the system of equations $$\begin{aligned} x - 2 y - 2 z & = - 7
2 x + ( a - 9 ) y - 10 z & = - 11
3 x - 6 y + 2 a z & = - 29 \end{aligned}$$ has a unique solution. Show that the system has no solution in the case \(a = - 3\). Given that \(a = 5\),

1. show that the number of solutions is infinite,
2. find the solution for which \(x + y + z = 2\).

The curve \(C\) has cartesian equation $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = a ^ { 2 } \left( x ^ { 2 } - y ^ { 2 } \right)$$ where \(a\) is a positive constant. Show that \(C\) has polar equation $$r ^ { 2 } = a ^ { 2 } \cos 2 \theta$$ Sketch \(C\) for \(- \pi < \theta \leqslant \pi\). Find the area of the sector between \(\theta = - \frac { 1 } { 4 } \pi\) and \(\theta = \frac { 1 } { 4 } \pi\). Find the polar coordinates of all points of \(C\) where the tangent is parallel to the initial line.

Show that the substitution \(y = x z\) reduces the differential equation $$\frac { 1 } { x } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + \left( \frac { 6 } { x } - \frac { 2 } { x ^ { 2 } } \right) \frac { \mathrm { d } y } { \mathrm {~d} x } + \left( \frac { 9 } { x } - \frac { 6 } { x ^ { 2 } } + \frac { 2 } { x ^ { 3 } } \right) y = 169 \sin 2 x$$ to the differential equation $$\frac { \mathrm { d } ^ { 2 } z } { \mathrm {~d} x ^ { 2 } } + 6 \frac { \mathrm {~d} z } { \mathrm {~d} x } + 9 z = 169 \sin 2 x$$ Find the particular solution for \(y\) in terms of \(x\), given that when \(x = 0 , z = - 10\) and \(\frac { \mathrm { d } z } { \mathrm {~d} x } = 5\).