CAIE FP1 (Further Pure Mathematics 1) 2010 June

1 Given that 5 is an eigenvalue of the matrix $$\mathbf { A } = \left( \begin{array} { r r r } 5 & - 3 & 0
1 & 2 & 1
- 1 & 3 & 4 \end{array} \right)$$ find a corresponding eigenvector. Hence find an eigenvalue and a corresponding eigenvector of the matrix \(\mathbf { A } + \mathbf { A } ^ { 2 }\).

2 By considering the identity $$\cos [ ( 2 n - 1 ) \alpha ] - \cos [ ( 2 n + 1 ) \alpha ] \equiv 2 \sin \alpha \sin 2 n \alpha$$ show that if \(\alpha\) is not an integer multiple of \(\pi\) then $$\sum _ { n = 1 } ^ { N } \sin ( 2 n \alpha ) = \frac { 1 } { 2 } \cot \alpha - \frac { 1 } { 2 } \operatorname { cosec } \alpha \cos [ ( 2 N + 1 ) \alpha ]$$ Deduce that the infinite series $$\sum _ { n = 1 } ^ { \infty } \sin \left( \frac { 2 } { 3 } n \pi \right)$$ does not converge.

3 The sequence \(x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots\) is such that \(x _ { 1 } = 3\) and $$x _ { n + 1 } = \frac { 2 x _ { n } ^ { 2 } + 4 x _ { n } - 2 } { 2 x _ { n } + 3 }$$ for \(n = 1,2,3 , \ldots\). Prove by induction that \(x _ { n } > 2\) for all \(n\).

4 The parametric equations of a curve are $$x = \cos t + t \sin t , \quad y = \sin t - t \cos t$$ The arc of the curve joining the point where \(t = 0\) to the point where \(t = \frac { 1 } { 2 } \pi\) is rotated about the \(x\)-axis through one complete revolution. Find the area of the surface generated, leaving your result in terms of \(\pi\).

5 Use de Moivre's theorem to show that $$\sin 5 \theta = 16 \sin ^ { 5 } \theta - 20 \sin ^ { 3 } \theta + 5 \sin \theta$$ Hence find all the roots of the equation $$32 x ^ { 5 } - 40 x ^ { 3 } + 10 x + 1 = 0$$ in the form \(\sin ( q \pi )\), where \(q\) is a positive rational number.

6 The curve \(C\) has equation $$y = \frac { x ^ { 2 } - 3 x - 7 } { x + 1 }$$

1. Obtain the equations of the asymptotes of \(C\).
2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } > 1\) at all points of \(C\).
3. Draw a sketch of \(C\).

7 It is given that $$x = t ^ { 2 } \mathrm { e } ^ { - t ^ { 2 } } \quad \text { and } \quad y = t \mathrm { e } ^ { - t ^ { 2 } }$$

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 - 2 t ^ { 2 } } { 2 t - 2 t ^ { 3 } }$$
2. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) in terms of \(t\).

8 Obtain the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 5 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 y = 10 \sin 3 x - 20 \cos 3 x$$ Show that, for large positive \(x\) and independently of the initial conditions, $$y \approx R \sin ( 3 x + \phi )$$ where the constants \(R\) and \(\phi\), such that \(R > 0\) and \(0 < \phi < 2 \pi\), are to be determined correct to 2 decimal places.

9 Let $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } \sin ^ { n } \theta \mathrm {~d} \theta$$ where \(n\) is a non-negative integer. Show that \(I _ { n + 2 } = \frac { n + 1 } { n + 2 } I _ { n }\). The region \(R\) of the \(x - y\) plane is bounded by the \(x\)-axis, the line \(x = \frac { \pi } { 2 m }\) and the curve whose equation is \(y = \sin ^ { 4 } m x\), where \(m > 0\). Find the \(y\)-coordinate of the centroid of \(R\).

10 The equation $$x ^ { 4 } + x ^ { 3 } + c x ^ { 2 } + 4 x - 2 = 0$$ where \(c\) is a constant, has roots \(\alpha , \beta , \gamma , \delta\).

1. Use the substitution \(y = \frac { 1 } { x }\) to find an equation which has roots \(\frac { 1 } { \alpha } , \frac { 1 } { \beta } , \frac { 1 } { \gamma } , \frac { 1 } { \delta }\).
2. Find, in terms of \(c\), the values of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 }\) and \(\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } } + \frac { 1 } { \delta ^ { 2 } }\).
3. Hence find $$\left( \alpha - \frac { 1 } { \alpha } \right) ^ { 2 } + \left( \beta - \frac { 1 } { \beta } \right) ^ { 2 } + \left( \gamma - \frac { 1 } { \gamma } \right) ^ { 2 } + \left( \delta - \frac { 1 } { \delta } \right) ^ { 2 }$$ in terms of \(c\).
4. Deduce that when \(c = - 3\) the roots of the given equation are not all real.

11 The curve \(C\) has polar equation $$r = \frac { a } { 1 + \theta }$$ where \(a\) is a positive constant and \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).

1. Show that \(r\) decreases as \(\theta\) increases.
2. The point \(P\) of \(C\) is further from the initial line than any other point of \(C\). Show that, at \(P\), $$\tan \theta = 1 + \theta$$ and verify that this equation has a root between 1.1 and 1.2.
3. Draw a sketch of \(C\).
4. Find the area of the region bounded by the initial line, the line \(\theta = \frac { 1 } { 2 } \pi\) and \(C\), leaving your answer in terms of \(\pi\) and \(a\).

The line \(l _ { 1 }\) passes through the point \(A\) whose position vector is \(3 \mathbf { i } + \mathbf { j } + 2 \mathbf { k }\) and is parallel to the vector \(\mathbf { i } + \mathbf { j }\). The line \(l _ { 2 }\) passes through the point \(B\) whose position vector is \(- \mathbf { i } - \mathbf { k }\) and is parallel to the vector \(\mathbf { j } + 2 \mathbf { k }\). The point \(P\) is on \(l _ { 1 }\) and the point \(Q\) is on \(l _ { 2 }\) and \(P Q\) is perpendicular to both \(l _ { 1 }\) and \(l _ { 2 }\).

1. Find the length of \(P Q\).
2. Find the position vector of \(Q\).
3. Show that the perpendicular distance from \(Q\) to the plane containing \(A B\) and the line \(l _ { 1 }\) is \(\sqrt { } 3\).

The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { M } = \left( \begin{array} { r r r r } 1 & 1 & 5 & 7
3 & 9 & 17 & 25
1 & 7 & 7 & 11
3 & 6 & 16 & 23 \end{array} \right)\).

1. In either order,
   (a) show that the dimension of \(R\), the range space of T , is equal to 2 ,
   (b) obtain a basis for \(R\).
2. Show that the vector \(\left( \begin{array} { r } 1
   - 15
   - 17
   - 6 \end{array} \right)\) belongs to \(R\).
3. It is given that \(\left\{ \mathbf { e } _ { 1 } , \mathbf { e } _ { 2 } \right\}\) is a basis for the null space of T , where \(\mathbf { e } _ { 1 } = \left( \begin{array} { r } 14
   1
   - 3
   0 \end{array} \right)\) and \(\mathbf { e } _ { 2 } = \left( \begin{array} { r } 19
   2
   0
   - 3 \end{array} \right)\). Show that, for all \(\lambda\) and \(\mu\), $$\mathbf { x } = \left( \begin{array} { r } 4
   - 3
   0
   0 \end{array} \right) + \lambda \mathbf { e } _ { 1 } + \mu \mathbf { e } _ { 2 }$$ is a solution of $$\mathbf { M x } = \left( \begin{array} { r } 1
   - 15
   - 17
   - 6 \end{array} \right)$$
4. Hence find a solution of \(( * )\) of the form \(\left( \begin{array} { c } \alpha
   0
   \gamma
   \delta \end{array} \right)\).