CAIE FP1 (Further Pure Mathematics 1) 2012 November

1 Show that \(\sum _ { r = n + 1 } ^ { 2 n } r ^ { 2 } = \frac { 1 } { 6 } n ( 2 n + 1 ) ( 7 n + 1 )\).

2 Find the set of values of \(a\) for which the system of equations $$\begin{aligned} a x + y + 2 z & = 0
3 x - 2 y & = 4
3 x - 4 y - 6 a z & = 14 \end{aligned}$$ has a unique solution.

3 Let \(S _ { N } = \frac { 1 } { 2 ! } + \frac { 2 } { 3 ! } + \frac { 3 } { 4 ! } + \ldots + \frac { N } { ( N + 1 ) ! }\). Prove by mathematical induction that, for all positive integers \(N\), $$S _ { N } = 1 - \frac { 1 } { ( N + 1 ) ! }$$

4 The points \(A , B\) and \(C\) have position vectors \(\mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k } , 2 \mathbf { i } + 4 \mathbf { j } + 5 \mathbf { k }\) and \(2 \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k }\) respectively. Find \(\overrightarrow { A B } \times \overrightarrow { A C }\). Deduce, in either order, the exact value of

1. the area of the triangle \(A B C\),
2. the perpendicular distance from \(C\) to \(A B\).

5 The curve \(C\) has polar equation \(r = 1 + 2 \cos \theta\). Sketch the curve for \(- \frac { 2 } { 3 } \pi \leqslant \theta < \frac { 2 } { 3 } \pi\). Find the area bounded by \(C\) and the half-lines \(\theta = - \frac { 1 } { 3 } \pi , \theta = \frac { 1 } { 3 } \pi\).

6 The curve \(C\) has parametric equations $$x = t ^ { 2 } , \quad y = \frac { 1 } { 4 } t ^ { 4 } - \ln t$$ for \(1 \leqslant t \leqslant 2\). Find the area of the surface generated when \(C\) is rotated through \(2 \pi\) radians about the \(y\)-axis.

7 A cubic equation has roots \(\alpha , \beta\) and \(\gamma\) such that $$\begin{aligned} \alpha + \beta + \gamma & = 4
\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } & = 14
\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } & = 34 \end{aligned}$$ Find the value of \(\alpha \beta + \beta \gamma + \gamma \alpha\). Show that the cubic equation is $$x ^ { 3 } - 4 x ^ { 2 } + x + 6 = 0$$ and solve this equation.

8 Let \(z = \cos \theta + \mathrm { i } \sin \theta\). Show that $$1 + z = 2 \cos \frac { 1 } { 2 } \theta \left( \cos \frac { 1 } { 2 } \theta + \mathrm { i } \sin \frac { 1 } { 2 } \theta \right)$$ By considering \(( 1 + z ) ^ { n }\), where \(n\) is a positive integer, deduce the sum of the series $$\binom { n } { 1 } \sin \theta + \binom { n } { 2 } \sin 2 \theta + \ldots + \binom { n } { n } \sin n \theta$$

9 The curve \(C\) has equation \(y = \frac { x ^ { 2 } - 3 x + 3 } { x - 2 }\). Find the equations of the asymptotes of \(C\). Show that there are no points on \(C\) for which \(- 1 < y < 3\). Find the coordinates of the turning points of \(C\). Sketch \(C\).

10 The curve \(C\) has equation \(x ^ { 3 } + y ^ { 3 } = 3 x y\), for \(x > 0\) and \(y > 0\). Find a relationship between \(x\) and \(y\) when \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\). Find the exact coordinates of the turning point of \(C\), and determine the nature of this turning point.

11 Show that \(\int x \left( 1 - x ^ { 2 } \right) ^ { \frac { 1 } { 2 } } \mathrm {~d} x = - \frac { 1 } { 3 } \left( 1 - x ^ { 2 } \right) ^ { \frac { 3 } { 2 } } + c\), where \(c\) is a constant. Given that \(I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } \left( 1 - x ^ { 2 } \right) ^ { \frac { 1 } { 2 } } \mathrm {~d} x\), prove that, for \(n \geqslant 2\), $$( n + 2 ) I _ { n } = ( n - 1 ) I _ { n - 2 }$$ Use the substitution \(x = \sin u\) to show that $$\int _ { 0 } ^ { 1 } \left( 1 - x ^ { 2 } \right) ^ { \frac { 1 } { 2 } } \mathrm {~d} x = \frac { 1 } { 4 } \pi$$ Find \(I _ { 4 }\).

The vector \(\mathbf { e }\) is an eigenvector of each of the \(n \times n\) matrices \(\mathbf { A }\) and \(\mathbf { B }\), with corresponding eigenvalues \(\lambda\) and \(\mu\) respectively. Prove that \(\mathbf { e }\) is an eigenvector of the matrix \(\mathbf { A B }\) with eigenvalue \(\lambda \mu\). It is given that the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r } 3 & 2 & 2
- 2 & - 2 & - 2
1 & 2 & 2 \end{array} \right) ,$$ has eigenvectors \(\left( \begin{array} { r } 0
1
- 1 \end{array} \right)\) and \(\left( \begin{array} { r } 1
0
- 1 \end{array} \right)\). Find the corresponding eigenvalues. Given that 2 is also an eigenvalue of \(\mathbf { A }\), find a corresponding eigenvector. The matrix \(\mathbf { B }\), where $$\mathbf { B } = \left( \begin{array} { r r r } - 1 & 2 & 2
2 & 2 & 2
- 3 & - 6 & - 6 \end{array} \right) ,$$ has the same eigenvectors as \(\mathbf { A }\). Given that \(\mathbf { A B } = \mathbf { C }\), find a non-singular matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that $$\mathbf { P } ^ { - 1 } \mathbf { C } ^ { 2 } \mathbf { P } = \mathbf { D }$$

Obtain the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 6 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 13 x = 75 \cos 2 t$$ Given that \(x = 5\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 0\) when \(t = 0\), find \(x\) in terms of \(t\). Show that, for large positive values of \(t\) and for any initial conditions, $$x \approx 5 \cos ( 2 t - \phi ) ,$$ where the constant \(\phi\) is such that \(\tan \phi = \frac { 4 } { 3 }\).