CAIE FP1 (Further Pure Mathematics 1) 2008 November

1 The curve \(C\) is defined parametrically by $$x = t ^ { 4 } - 4 \ln t , \quad y = 4 t ^ { 2 }$$ Show that the length of the arc of \(C\) from the point where \(t = 2\) to the point where \(t = 4\) is $$240 + 4 \ln 2 .$$

2 Let \(y = \mathrm { e } ^ { x }\). Find the mean value of \(y\) with respect to \(x\) over the interval \(0 \leqslant x \leqslant 2\). Show that the mean value of \(x\) with respect to \(y\) over the interval \(1 \leqslant y \leqslant \mathrm { e } ^ { 2 }\) is \(\frac { \mathrm { e } ^ { 2 } + 1 } { \mathrm { e } ^ { 2 } - 1 }\).

3 The curve \(C\) has polar equation $$r = \left( \frac { 1 } { 2 } \pi - \theta \right) ^ { 2 } ,$$ where \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). Draw a sketch of \(C\). Find the area of the region bounded by \(C\) and the initial line, leaving your answer in terms of \(\pi\).

4 The matrix \(\mathbf { A }\) has \(\lambda\) as an eigenvalue with \(\mathbf { e }\) as a corresponding eigenvector. Show that \(\mathbf { e }\) is an eigenvector of \(\mathbf { A } ^ { 2 }\) and state the corresponding eigenvalue. Given that one eigenvalue of \(\mathbf { A }\) is 3 , find an eigenvalue of the matrix \(\mathbf { A } ^ { 4 } + 3 \mathbf { A } ^ { 2 } + 2 \mathbf { I }\), justifying your answer.

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

6 The matrix \(\mathbf { A }\) is defined by $$\mathbf { A } = \left( \begin{array} { r r r r } 1 & - 1 & - 2 & - 3
- 2 & 1 & 7 & 2
- 3 & 3 & 6 & \alpha
7 & - 6 & - 17 & - 17 \end{array} \right) .$$

1. Show that if \(\alpha = 9\) then the rank of \(\mathbf { A }\) is 2, and find a basis for the null space of \(\mathbf { A }\) in this case.
2. Find the rank of \(\mathbf { A }\) when \(\alpha \neq 9\).

7 Let \(I _ { n } = \int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + x ^ { 4 } \right) ^ { n } } \mathrm {~d} x\). By considering \(\frac { \mathrm { d } } { \mathrm { d } x } \left( \frac { x } { \left( 1 + x ^ { 4 } \right) ^ { n } } \right)\), show that $$4 n I _ { n + 1 } = \frac { 1 } { 2 ^ { n } } + ( 4 n - 1 ) I _ { n }$$ Given that \(I _ { 1 } = 0.86697\), correct to 5 decimal places, find \(I _ { 3 }\).

8 Find \(y\) in terms of \(t\), given that $$5 \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + 6 \frac { \mathrm {~d} y } { \mathrm {~d} t } + 5 y = 15 + 12 t + 5 t ^ { 2 }$$ and that \(y = \frac { \mathrm { d } y } { \mathrm {~d} t } = 0\) when \(t = 0\).

9 Use induction to prove that $$\sum _ { n = 1 } ^ { N } \frac { 4 n + 1 } { n ( n + 1 ) ( 2 n - 1 ) ( 2 n + 1 ) } = 1 - \frac { 1 } { ( N + 1 ) ( 2 N + 1 ) }$$ Show that $$\sum _ { n = N + 1 } ^ { 2 N } \frac { 4 n + 1 } { n ( n + 1 ) ( 2 n - 1 ) ( 2 n + 1 ) } < \frac { 3 } { 8 N ^ { 2 } }$$

10 Use de Moivre's theorem to express \(\cos 8 \theta\) as a polynomial in \(\cos \theta\). Hence

1. express \(\cos 8 \theta\) as a polynomial in \(\sin \theta\),
2. find the exact value of $$4 x ^ { 4 } - 8 x ^ { 3 } + 5 x ^ { 2 } - x$$ where \(x = \cos ^ { 2 } \left( \frac { 1 } { 8 } \pi \right)\).

11 The plane \(\Pi _ { 1 }\) has equation $$\mathbf { r } = \mathbf { i } + 2 \mathbf { j } + \mathbf { k } + \theta ( 2 \mathbf { j } - \mathbf { k } ) + \phi ( 3 \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k } )$$ Find a vector normal to \(\Pi _ { 1 }\) and hence show that the equation of \(\Pi _ { 1 }\) can be written as \(2 x + 3 y + 6 z = 14\). The line \(l\) has equation $$\mathbf { r } = 3 \mathbf { i } + 8 \mathbf { j } + 2 \mathbf { k } + t ( 4 \mathbf { i } + 6 \mathbf { j } + 5 \mathbf { k } )$$ The point on \(l\) where \(t = \lambda\) is denoted by \(P\). Find the set of values of \(\lambda\) for which the perpendicular distance of \(P\) from \(\Pi _ { 1 }\) is not greater than 4 . The plane \(\Pi _ { 2 }\) contains \(l\) and the point with position vector \(\mathbf { i } + 2 \mathbf { j } + \mathbf { k }\). Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).

The curve \(C\) has equation $$y = \frac { ( x - 2 ) ( x - a ) } { ( x - 1 ) ( x - 3 ) } ,$$ where \(a\) is a constant not equal to 1,2 or 3 .

1. Write down the equations of the asymptotes of \(C\).
2. Show that \(C\) meets the asymptote parallel to the \(x\)-axis at the point where \(x = \frac { 2 a - 3 } { a - 2 }\).
3. Show that the \(x\)-coordinates of any stationary points on \(C\) satisfy $$( a - 2 ) x ^ { 2 } + ( 6 - 4 a ) x + ( 5 a - 6 ) = 0$$ and hence find the set of values of \(a\) for which \(C\) has stationary points.
4. Sketch the graph of \(C\) for
   (a) \(a > 3\),
   (b) \(2 < a < 3\).

The roots of the equation $$x ^ { 4 } - 5 x ^ { 2 } + 2 x - 1 = 0$$ are \(\alpha , \beta , \gamma , \delta\). Let \(S _ { n } = \alpha ^ { n } + \beta ^ { n } + \gamma ^ { n } + \delta ^ { n }\).

1. Show that $$S _ { n + 4 } - 5 S _ { n + 2 } + 2 S _ { n + 1 } - S _ { n } = 0 .$$
2. Find the values of \(S _ { 2 }\) and \(S _ { 4 }\).
3. Find the value of \(S _ { 3 }\) and hence find the value of \(S _ { 6 }\).
4. Hence find the value of $$\alpha ^ { 2 } \left( \beta ^ { 4 } + \gamma ^ { 4 } + \delta ^ { 4 } \right) + \beta ^ { 2 } \left( \gamma ^ { 4 } + \delta ^ { 4 } + \alpha ^ { 4 } \right) + \gamma ^ { 2 } \left( \delta ^ { 4 } + \alpha ^ { 4 } + \beta ^ { 4 } \right) + \delta ^ { 2 } \left( \alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 } \right) .$$ \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }