CAIE FP1 (Further Pure Mathematics 1) 2002 November

1 Given that $$u _ { n } = \mathrm { e } ^ { n x } - \mathrm { e } ^ { ( n + 1 ) x }$$ find \(\sum _ { n = 1 } ^ { N } \| _ { n }\) in terms of \(N\) and \(x\). Hence determine the set of values of \(x\) for which the infinite series $$u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots$$ is convergent and give the sum to infinity for cases where this exists.

2 The equation $$x ^ { 4 } + x ^ { 3 } + A x ^ { 2 } + 4 x - 2 = 0$$ where \(A\) is a constant, has roots \(\alpha , \beta , \gamma , \delta\). Find a polynomial equation whose roots are $$\frac { 1 } { \alpha } , \frac { 1 } { \beta } , \frac { 1 } { \gamma } , \frac { 1 } { \delta }$$ Given that $$\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } = \frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } } + \frac { 1 } { \delta ^ { 2 } }$$ find the value of \(A\).

3 It is given that, for \(n = 0,1,2,3 , \ldots\), $$a _ { n } = 17 ^ { 2 n } + 3 ( 9 ) ^ { n } + 20$$ Simplify \(a _ { n + 1 } - a _ { n }\), and hence prove by induction that \(a _ { n }\) is divisible by 24 for all \(n \geqslant 0\).

4 It is given that, for \(n \geqslant 0\), $$I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } e ^ { - x ^ { 2 } } d x$$

1. Find \(I _ { 1 }\) in terms of c .
2. Show that $$I _ { n + 2 } = \frac { n + 1 } { 2 } I _ { n } - \frac { 1 } { 2 \mathrm { e } }$$
3. Find \(I _ { 5 }\) in terms of \(e\).

5 The curve \(C\) has polar equation \(r \theta = 1\), for \(0 < \theta \leqslant 2 \pi\).

1. Use the fact that \(\frac { \sin \theta } { \theta }\) tends to 1 as \(\theta\) tends to 0 to show that the line with carresian equation \(y = 1\) is an asymptote to \(C\).
2. Sketch \(C\). The points \(P\) and \(Q\) on \(C\) correspond to \(\theta = \frac { 1 } { 6 } \pi\) and \(\theta = \frac { 1 } { 3 } \pi\) respectively.
3. Find the area of the sector \(O P Q\), where \(O\) is the origin.
4. Show that the length of the are \(P Q\) is $$\int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } \frac { \sqrt { } \left( 1 + \theta ^ { 2 } \right) } { \theta ^ { 2 } } \mathrm {~d} \theta$$

6 A curve has equation \(x ^ { 3 } + x y ^ { 2 } - y ^ { 3 } = 3\).

1. Show that there is no point of the curve at which \(\frac { d y } { d x } = 0\).
2. Find the values of \(\frac { d y } { d x }\) and \(\frac { d ^ { 2 } y } { d x ^ { 2 } }\) at the point \(( 1 , - 1 )\).

7 Given that \(z = \cos \theta + \mathrm { i } \sin \theta\), show that

1. \(z - \frac { 1 } { z } = 2 \mathrm { i } \sin \theta\).
2. \(z ^ { n } + z ^ { - n } = 2 \cos n \theta\). Hence show that $$\sin ^ { 6 } \theta = \frac { 1 } { 32 } ( 10 - 15 \cos 2 \theta + 6 \cos 4 \theta - \cos 6 \theta )$$ Find a similar expression for \(\cos ^ { 6 } \theta\), and hence express \(\cos ^ { 6 } \theta - \sin ^ { 6 } \theta\) in the fom \(a \cos 2 \theta + b \cos 6 \theta\).

8 The value of the assets of a large commercial organisation at time \(t\), measured in years, is \(
) \left( 10 ^ { 8 } y + 10 ^ { 9 } \right)\(. The variables \)y\( and \)t$ are related by the differential equation $$\frac { d ^ { 2 } y } { d t ^ { 2 } } + 5 \frac { d y } { d t } + 6 y = 15 \cos 3 t - 3 \sin 3 t$$ Find \(y\) in terms of \(t\), given that \(y = 3\) and \(\frac { \mathrm { d } y } { \mathrm {~d} t } = - 2\) when \(t = 0\). Show that, for large values of \(t\), the value of the assets is less than \(
) 9.5 \times 10 ^ { 8 }$ for about a third of the time.

9 The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), which meet in the line \(/\), have vector equations $$\begin{aligned} & \mathbf { r } = 2 \mathbf { i } + 4 \mathbf { j } + 6 \mathbf { k } + \theta _ { 1 } ( 2 \mathbf { i } + 3 \mathbf { k } ) + \phi _ { 1 } ( - 4 \mathbf { j } + 5 \mathbf { k } ) ,
& \mathbf { r } = 2 \mathbf { i } + 4 \mathbf { j } + 6 \mathbf { k } + \theta _ { 2 } ( 3 \mathbf { j } + \mathbf { k } ) + \phi _ { 2 } ( - \mathbf { i } + \mathbf { j } + 2 \mathbf { k } ) , \end{aligned}$$ respectively. Find a vector equation of the line \(l\) in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\). Find a vector equation of the plane \(\Pi _ { 3 }\) which contains \(l\) and which passes through the point with position vector \(4 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }\). Find also the equation of \(\Pi _ { 3 }\) in the form \(a x + b y + c z = d\). Deduce, or prove otherwise, that the system of equations $$\begin{aligned} & 6 x - 5 y - 4 z = - 32
& 5 x - y + 3 z = 24
& 9 x - 2 y + 5 z = 40 \end{aligned}$$ has an infinite number of solutions.

10 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { H }\), where $$\mathbf { H } = \left( \begin{array} { r r r r } 1 & 2 & - 3 & - 5
- 1 & 4 & 5 & 1
2 & 3 & 0 & - 3
- 3 & 5 & 7 & 2 \end{array} \right)$$

1. Find the dimension of the range space of T .
2. Find a basis for the null space of \(T\).
3. It is given that \(\mathbf { x }\) satisfies the equation $$\mathbf { H } \mathbf { x } = \left( \begin{array} { r } 2
   - 10
   - 1
   - 15 \end{array} \right)$$ Using the fact that $$\mathbf { H } \left( \begin{array} { r } 1
   - 3
   1
   - 2 \end{array} \right) = \left( \begin{array} { r } 2
   - 10
   - 1
   - 15 \end{array} \right) ,$$ find the least possible value of \(| \mathbf { x } |\).
   [0pt] [For the vector \(\mathbf { x } = \left( \begin{array} { c } x _ { 1 }
   x _ { 2 }
   x _ { 3 }
   x _ { 4 } \end{array} \right) , | \mathbf { x } | = \sqrt { } \left( x _ { 1 } ^ { 2 } + x _ { 2 } ^ { 2 } + x _ { 3 } ^ { 2 } + x _ { 4 } ^ { 2 } \right)\).]

The vector \(\mathbf { e }\) is an eigenvector of the square matrix \(\mathbf { G }\). Show that

1. \(\mathbf { e }\) is an eigenvector of \(\mathbf { G } + k \mathbf { I }\), where \(k\) is a scalar and \(\mathbf { I }\) is an identity matrix,
2. \(\mathbf { e }\) is an cigenvector of \(\mathbf { G } ^ { 2 }\). Find the eigenvalues, and corresponding eigenvectors, of the matrices \(\mathbf { A }\) and \(\mathbf { B } ^ { 2 }\), where $$\mathbf { A } = \left( \begin{array} { r r r } 3 & - 3 & 0
   1 & 0 & 1
   - 1 & 3 & 2 \end{array} \right) \quad \text { and } \quad \mathbf { B } = \left( \begin{array} { r r r } - 5 & - 3 & 0
   1 & - 8 & 1
   - 1 & 3 & - 6 \end{array} \right)$$

The curve \(C\) has equation $$y = \frac { ( x - a ) ( x - b ) } { x - c }$$ where \(a , b , c\) are constants, and it is given that \(0 < a < b < c\).

1. Express \(y\) in the form $$x + P + \frac { Q } { x - c }$$ giving the constants \(P\) and \(Q\) in terms of \(a , b\) and \(c\).
2. Find the equations of the asymptotes of \(C\).
3. Show that \(C\) has two stationary points.
4. Given also that \(a + b > c\), sketch \(C\), showing the asymptotes and the coordinates of the points of intersection of \(C\) with the axes.