CAIE FP1 (Further Pure Mathematics 1) 2008 June

1 The finite region enclosed by the line \(y = k x\), where \(k\) is a positive constant, the \(x\)-axis for \(0 \leqslant x \leqslant h\), and the line \(x = h\) is rotated through 1 complete revolution about the \(x\)-axis. Prove by integration that the centroid of the resulting cone is at a distance \(\frac { 3 } { 4 } h\) from the origin \(O\).
[0pt] [The volume of a cone of height \(h\) and base radius \(r\) is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).]

2 Given that $$u _ { n } = \ln \left( \frac { 1 + x ^ { n + 1 } } { 1 + x ^ { n } } \right)$$ where \(x > - 1\), find \(\sum _ { n = 1 } ^ { N } u _ { n }\) in terms of \(N\) and \(x\). Find the sum to infinity of the series $$u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots$$ when

1. \(- 1 < x < 1\),
2. \(x = 1\).

3 Show that if \(\lambda\) is an eigenvalue of the square matrix \(\mathbf { A }\) with \(\mathbf { e }\) as a corresponding eigenvector, and \(\mu\) is an eigenvalue of the square matrix \(\mathbf { B }\) for which \(\mathbf { e }\) is also a corresponding eigenvector, then \(\lambda + \mu\) is an eigenvalue of the matrix \(\mathbf { A } + \mathbf { B }\) with \(\mathbf { e }\) as a corresponding eigenvector. The matrix $$\mathbf { A } = \left( \begin{array} { r r r } 3 & - 1 & 0
- 4 & - 6 & - 6
5 & 11 & 10 \end{array} \right)$$ has \(\left( \begin{array} { r } 1
- 1
1 \end{array} \right)\) as an eigenvector. Find the corresponding eigenvalue. The other two eigenvalues of \(\mathbf { A }\) are 1 and 2, with corresponding eigenvectors \(\left( \begin{array} { r } 1
2
- 3 \end{array} \right)\) and \(\left( \begin{array} { r } 1
1
- 2 \end{array} \right)\) respectively. The matrix \(\mathbf { B }\) has eigenvalues \(2,3,1\) with corresponding eigenvectors \(\left( \begin{array} { r } 1
- 1
1 \end{array} \right) , \left( \begin{array} { r } 1
2
- 3 \end{array} \right)\), \(\left( \begin{array} { r } 1
1
- 2 \end{array} \right)\) respectively. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(( \mathbf { A } + \mathbf { B } ) ^ { 4 } = \mathbf { P D P } \mathbf { P } ^ { - 1 }\).
[0pt] [You are not required to evaluate \(\mathbf { P } ^ { - 1 }\).]

4 The curves \(C _ { 1 }\) and \(C _ { 2 }\) have polar equations $$r = \theta + 2 \quad \text { and } \quad r = \theta ^ { 2 }$$ respectively, where \(0 \leqslant \theta \leqslant \pi\).

1. Find the polar coordinates of the point of intersection of \(C _ { 1 }\) and \(C _ { 2 }\).
2. Sketch \(C _ { 1 }\) and \(C _ { 2 }\) on the same diagram.
3. Find the area bounded by \(C _ { 1 } , C _ { 2 }\) and the line \(\theta = 0\).

5 The equation $$x ^ { 3 } + x - 1 = 0$$ has roots \(\alpha , \beta , \gamma\). Show that the equation with roots \(\alpha ^ { 3 } , \beta ^ { 3 } , \gamma ^ { 3 }\) is $$y ^ { 3 } - 3 y ^ { 2 } + 4 y - 1 = 0$$ Hence find the value of \(\alpha ^ { 6 } + \beta ^ { 6 } + \gamma ^ { 6 }\).

6 The curve \(C\) is defined parametrically by $$x = 4 t - t ^ { 2 } \quad \text { and } \quad y = 1 - \mathrm { e } ^ { - t }$$ where \(0 \leqslant t < 2\). Show that at all points of \(C\), $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { ( t - 1 ) \mathrm { e } ^ { - t } } { 4 ( 2 - t ) ^ { 3 } }$$ Show that the mean value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) with respect to \(x\) over the interval \(0 \leqslant x \leqslant \frac { 7 } { 4 }\) is $$\frac { 4 e ^ { - \frac { 1 } { 2 } } - 3 } { 21 }$$

7 Prove by induction that $$\sum _ { r = 1 } ^ { n } \left( 3 r ^ { 5 } + r ^ { 3 } \right) = \frac { 1 } { 2 } n ^ { 3 } ( n + 1 ) ^ { 3 }$$ for all \(n \geqslant 1\). Use this result together with the List of Formulae (MF10) to prove that $$\sum _ { r = 1 } ^ { n } r ^ { 5 } = \frac { 1 } { 12 } n ^ { 2 } ( n + 1 ) ^ { 2 } \mathrm { Q } ( n )$$ where \(\mathrm { Q } ( n )\) is a quadratic function of \(n\) which is to be determined.

8

1. Given that $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 2 } \pi } t ^ { n } \sin t \mathrm {~d} t$$ show that, for \(n \geqslant 2\), $$I _ { n } = n \left( \frac { \pi } { 2 } \right) ^ { n - 1 } - n ( n - 1 ) I _ { n - 2 } .$$
2. A curve \(C\) in the \(x - y\) plane is defined parametrically in terms of \(t\). It is given that $$\frac { \mathrm { d } x } { \mathrm {~d} t } = t ^ { 4 } ( 1 - \cos 2 t ) \quad \text { and } \quad \frac { \mathrm { d } y } { \mathrm {~d} t } = t ^ { 4 } \sin 2 t .$$ Find the length of the arc of \(C\) from the point where \(t = 0\) to the point where \(t = \frac { 1 } { 2 } \pi\).

9 The curve \(C\) has equation $$y = \frac { x ^ { 2 } - 2 x + \lambda } { x + 1 }$$ where \(\lambda\) is a constant. Show that the equations of the asymptotes of \(C\) are independent of \(\lambda\). Find the value of \(\lambda\) for which the \(x\)-axis is a tangent to \(C\), and sketch \(C\) in this case. Sketch \(C\) in the case \(\lambda = - 4\), giving the exact coordinates of the points of intersection of \(C\) with the \(x\)-axis.

10 By considering \(\sum _ { n = 1 } ^ { N } z ^ { 2 n - 1 }\), where \(z = \mathrm { e } ^ { \mathrm { i } \theta }\), show that $$\sum _ { n = 1 } ^ { N } \cos ( 2 n - 1 ) \theta = \frac { \sin ( 2 N \theta ) } { 2 \sin \theta }$$ where \(\sin \theta \neq 0\). Deduce that $$\sum _ { n = 1 } ^ { N } ( 2 n - 1 ) \sin \left[ \frac { ( 2 n - 1 ) \pi } { N } \right] = - N \operatorname { cosec } \frac { \pi } { N }$$

11 Show that, with a suitable value of the constant \(\alpha\), the substitution \(y = x ^ { \alpha } w\) reduces the differential equation $$2 x ^ { 2 } \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + \left( 3 x ^ { 2 } + 8 x \right) \frac { \mathrm { d } y } { \mathrm {~d} x } + \left( x ^ { 2 } + 6 x + 4 \right) y = \mathrm { f } ( x )$$ to $$2 \frac { \mathrm {~d} ^ { 2 } w } { \mathrm {~d} x ^ { 2 } } + 3 \frac { \mathrm {~d} w } { \mathrm {~d} x } + w = \mathrm { f } ( x )$$ Find the general solution for \(y\) in the case where \(\mathrm { f } ( x ) = 6 \sin 2 x + 7 \cos 2 x\).

The position vectors of the points \(A , B , C , D\) are
\(7 \mathbf { i } + 4 \mathbf { j } - \mathbf { k }\),
\(3 \mathbf { i } + 5 \mathbf { j } - 2 \mathbf { k }\),
\(2 \mathbf { i } + 6 \mathbf { j } + 3 \mathbf { k }\),
\(2 \mathbf { i } + 7 \mathbf { j } + \lambda \mathbf { k }\)
respectively. It is given that the shortest distance between the line \(A B\) and the line \(C D\) is 3 .

1. Show that \(\lambda ^ { 2 } - 5 \lambda + 4 = 0\).
2. Find the acute angle between the planes through \(A , B , D\) corresponding to the values of \(\lambda\) satisfying the equation in part (i).

The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix $$\left( \begin{array} { r r r r } 1 & 2 & - 1 & - 1
1 & 3 & - 1 & 0
1 & 0 & 3 & 1
0 & 3 & - 4 & - 1 \end{array} \right) .$$ The range space of T is denoted by \(V\).

1. Determine the dimension of \(V\).
2. Show that the vectors \(\left( \begin{array} { l } 1
   1
   1
   0 \end{array} \right) , \left( \begin{array} { l } 2
   3
   0
   3 \end{array} \right) , \left( \begin{array} { r } - 1
   - 1
   3
   - 4 \end{array} \right)\) are linearly independent.
3. Write down a basis of \(V\). The set of elements of \(\mathbb { R } ^ { 4 }\) which do not belong to \(V\) is denoted by \(W\).
4. State, with a reason, whether \(W\) is a vector space.
5. Show that if the vector \(\left( \begin{array} { l } x
   y
   z
   t \end{array} \right)\) belongs to \(W\) then \(y - z - t \neq 0\).