CAIE FP1 (Further Pure Mathematics 1) 2003 November

1
\includegraphics[max width=\textwidth, alt={}, center]{653d57aa-7775-4063-a8c9-11c8bc964fae-2_566_606_264_772} The curve \(C\) has polar equation $$r = \theta ^ { \frac { 1 } { 2 } } \mathrm { e } ^ { \theta ^ { 2 } / \pi }$$ where \(0 \leqslant \theta \leqslant \pi\). The area of the finite region bounded by \(C\) and the line \(\theta = \beta\) is \(\pi\) (see diagram). Show that $$\beta = ( \pi \ln 3 ) ^ { \frac { 1 } { 2 } }$$

2 Given that $$u _ { n } = \frac { 1 } { n ^ { 2 } - n + 1 } - \frac { 1 } { n ^ { 2 } + n + 1 } ,$$ find \(S _ { N } = \sum _ { n = N + 1 } ^ { 2 N } u _ { n }\) in terms of \(N\). Find a number \(M\) such that \(S _ { N } < 10 ^ { - 20 }\) for all \(N > M\).

3 Three \(n \times 1\) column vectors are denoted by \(\mathbf { x } _ { 1 } , \mathbf { x } _ { 2 } , \mathbf { x } _ { 3 }\), and \(\mathbf { M }\) is an \(n \times n\) matrix. Show that if \(\mathbf { x } _ { 1 } , \mathbf { x } _ { 2 } , \mathbf { x } _ { 3 }\) are linearly dependent then the vectors \(\mathbf { M x } _ { 1 } , \mathbf { M x } _ { 2 } , \mathbf { M x } _ { 3 }\) are also linearly dependent. The vectors \(\mathbf { y } _ { 1 } , \mathbf { y } _ { 2 } , \mathbf { y } _ { 3 }\) and the matrix \(\mathbf { P }\) are defined as follows: $$\begin{gathered} \mathbf { y } _ { 1 } = \left( \begin{array} { l } 1
5
7 \end{array} \right) , \quad \mathbf { y } _ { 2 } = \left( \begin{array} { r } 2
- 3

5
7 \end{array} \right) , \quad \mathbf { y } _ { 2 } = \left( \begin{array} { r } 2
- 3
4 \end{array} \right) , \quad \mathbf { y } _ { 3 } = \left( \begin{array} { r } 5
51
55 \end{array} \right)
\mathbf { P } = \left( \begin{array} { r r r } 1 & - 4 & 3
0 & 2 & 5
0 & 0 & - 7 \end{array} \right) . \end{gathered}$$

1. Show that \(\mathbf { y } _ { 1 } , \mathbf { y } _ { 2 } , \mathbf { y } _ { 3 }\) are linearly dependent.
2. Find a basis for the linear space spanned by the vectors \(\mathbf { P y } _ { 1 } , \mathbf { P y } _ { 2 } , \mathbf { P y } _ { 3 }\). 4 Given that \(y = x \sin x\), find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and \(\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }\), simplifying your results as far as possible, and show that $$\frac { \mathrm { d } ^ { 6 } y } { \mathrm {~d} x ^ { 6 } } = - x \sin x + 6 \cos x$$ Use induction to establish an expression for \(\frac { \mathrm { d } ^ { 2 n } y } { \mathrm {~d} x ^ { 2 n } }\), where \(n\) is a positive integer. 5 The integral \(I _ { n }\) is defined by $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sec ^ { n } x \mathrm {~d} x$$ By considering \(\frac { \mathrm { d } } { \mathrm { d } x } \left( \tan x \sec ^ { n } x \right)\), or otherwise, show that $$( n + 1 ) I _ { n + 2 } = 2 ^ { \frac { 1 } { 2 } n } + n I _ { n }$$ Find the value of \(I _ { 6 }\).

7 \end{array} \right) , \quad \mathbf { y } _ { 2 } = \left( \begin{array} { r } 2
- 3
4 \end{array} \right) , \quad \mathbf { y } _ { 3 } = \left( \begin{array} { r } 5
51
55 \end{array} \right)
\mathbf { P } = \left( \begin{array} { r r r } 1 & - 4 & 3
0 & 2 & 5
0 & 0 & - 7 \end{array} \right) . \end{gathered}$$

1. Show that \(\mathbf { y } _ { 1 } , \mathbf { y } _ { 2 } , \mathbf { y } _ { 3 }\) are linearly dependent.
2. Find a basis for the linear space spanned by the vectors \(\mathbf { P y } _ { 1 } , \mathbf { P y } _ { 2 } , \mathbf { P y } _ { 3 }\). 4 Given that \(y = x \sin x\), find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) and \(\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }\), simplifying your results as far as possible, and show that $$\frac { \mathrm { d } ^ { 6 } y } { \mathrm {~d} x ^ { 6 } } = - x \sin x + 6 \cos x$$ Use induction to establish an expression for \(\frac { \mathrm { d } ^ { 2 n } y } { \mathrm {~d} x ^ { 2 n } }\), where \(n\) is a positive integer. 5 The integral \(I _ { n }\) is defined by $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sec ^ { n } x \mathrm {~d} x$$ By considering \(\frac { \mathrm { d } } { \mathrm { d } x } \left( \tan x \sec ^ { n } x \right)\), or otherwise, show that $$( n + 1 ) I _ { n + 2 } = 2 ^ { \frac { 1 } { 2 } n } + n I _ { n }$$ Find the value of \(I _ { 6 }\). 6 Find the sum of the squares of the roots of the equation $$x ^ { 3 } + x + 12 = 0$$ and deduce that only one of the roots is real. The real root of the equation is denoted by \(\alpha\). Prove that \(- 3 < \alpha < - 2\), and hence prove that the modulus of each of the other roots lies between 2 and \(\sqrt { } 6\). 7 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 4 y = \mathrm { e } ^ { - \alpha t }$$ where \(\alpha\) is a constant and \(\alpha \neq 2\). Show that if \(\alpha < 2\) then, whatever the initial conditions, \(y \mathrm { e } ^ { \alpha t } \rightarrow \frac { 1 } { ( 2 - \alpha ) ^ { 2 } }\) as \(t \rightarrow \infty\).

8 Given that \(z = \mathrm { e } ^ { \mathrm { i } \theta }\) and \(n\) is a positive integer, show that $$z ^ { n } + \frac { 1 } { z ^ { n } } = 2 \cos n \theta \quad \text { and } \quad z ^ { n } - \frac { 1 } { z ^ { n } } = 2 \mathrm { i } \sin n \theta$$ Hence express \(\sin ^ { 6 } \theta\) in the form $$p \cos 6 \theta + q \cos 4 \theta + r \cos 2 \theta + s$$ where the constants \(p , q , r , s\) are to be determined. Hence find the mean value of \(\sin ^ { 6 } \theta\) with respect to \(\theta\) over the interval \(0 \leqslant \theta \leqslant \frac { 1 } { 4 } \pi\).

9 The line \(l _ { 1 }\) passes through the point \(A\) with position vector \(\mathbf { i } - \mathbf { j } - 2 \mathbf { k }\) and is parallel to the vector \(3 \mathbf { i } - 4 \mathbf { j } - 2 \mathbf { k }\). The variable line \(l _ { 2 }\) passes through the point \(( 1 + 5 \cos t ) \mathbf { i } - ( 1 + 5 \sin t ) \mathbf { j } - 14 \mathbf { k }\), where \(0 \leqslant t < 2 \pi\), and is parallel to the vector \(15 \mathbf { i } + 8 \mathbf { j } - 3 \mathbf { k }\). The points \(P\) and \(Q\) are on \(l _ { 1 }\) and \(l _ { 2 }\) respectively, and \(P Q\) is perpendicular to both \(l _ { 1 }\) and \(l _ { 2 }\).

1. Find the length of \(P Q\) in terms of \(t\).
2. Hence show that the lines \(l _ { 1 }\) and \(l _ { 2 }\) do not intersect, and find the maximum length of \(P Q\) as \(t\) varies.
3. The plane \(\Pi _ { 1 }\) contains \(l _ { 1 }\) and \(P Q\); the plane \(\Pi _ { 2 }\) contains \(l _ { 2 }\) and \(P Q\). Find the angle between the planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), correct to the nearest tenth of a degree.

10 Find the eigenvalues and corresponding eigenvectors of the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r } 6 & 4 & 1
- 6 & - 1 & 3
8 & 8 & 4 \end{array} \right)$$ Hence find a non-singular matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(\mathbf { A } + \mathbf { A } ^ { 2 } + \mathbf { A } ^ { 3 } = \mathbf { P D P } \mathbf { P } ^ { - 1 }\).

The curve \(C\) has equation \(y = \frac { 5 ( x - 1 ) ( x + 2 ) } { ( x - 2 ) ( x + 3 ) }\).

1. Express \(y\) in the form \(P + \frac { Q } { x - 2 } + \frac { R } { x + 3 }\).
2. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) for exactly one value of \(x\) and find the corresponding value of \(y\).
3. Write down the equations of all the asymptotes of \(C\).
4. Find the set of values of \(k\) for which the line \(y = k\) does not intersect \(C\).

A curve has equation \(y = \frac { 2 } { 3 } x ^ { \frac { 3 } { 2 } }\), for \(x \geqslant 0\). The arc of the curve joining the origin to the point where \(x = 3\) is denoted by \(R\).

1. Find the length of \(R\).
2. Find the \(y\)-coordinate of the centroid of the region bounded by the \(x\)-axis, the line \(x = 3\) and \(R\).
3. Show that the area of the surface generated when \(R\) is rotated through one revolution about the \(y\)-axis is \(\frac { 232 } { 15 } \pi\).