CAIE FP1 (Further Pure Mathematics 1) 2010 November

1 The curve \(C\) has equation \(y = \frac { 1 } { 4 } \left( \mathrm { e } ^ { 2 x } + \mathrm { e } ^ { - 2 x } \right)\). Show that the length of the \(\operatorname { arc }\) of \(C\) from the point where \(x = 0\) to the point where \(x = \frac { 1 } { 2 }\) is \(\frac { \mathrm { e } ^ { 2 } - 1 } { 4 \mathrm { e } }\).

2 Use the method of differences to find \(S _ { N }\), where $$S _ { N } = \sum _ { n = 1 } ^ { N } \frac { 1 } { n ( n + 2 ) }$$ Deduce the value of \(\lim _ { N \rightarrow \infty } S _ { N }\).

3 A finite region \(R\) in the \(x - y\) plane is bounded by the curve with equation \(y = \sqrt { } x - \frac { 1 } { \sqrt { } x }\), the \(x\)-axis between \(x = 1\) and \(x = 4\), and the line \(x = 4\). Find the exact value of the \(y\)-coordinate of the centroid of \(R\).

4 Prove by mathematical induction that, for all non-negative integers \(n , 7 ^ { 2 n + 1 } + 5 ^ { n + 3 }\) is divisible by 44 .

5 Let \(I _ { n } = \int _ { 0 } ^ { 1 } ( 1 - x ) ^ { n } \sin x \mathrm {~d} x\) for \(n \geqslant 0\). Show that $$I _ { n + 2 } = 1 - ( n + 1 ) ( n + 2 ) I _ { n }$$ Hence find the value of \(I _ { 6 }\), correct to 4 decimal places.

6 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r r } 1 & 2 & - 1 & \alpha \\ 2 & 3 & - 1 & 0 \\ 2 & 1 & 2 & - 2 \\ 0 & 1 & - 3 & - 2 \end{array} \right)$$ Given that the dimension of the range space of T is 4 , show that \(\alpha \neq 1\). It is now given that \(\alpha = 1\). Show that the vectors $$\left( \begin{array} { l } 1 \\ 2 \\ 2 \\ 0 \end{array} \right) , \quad \left( \begin{array} { l } 2 \\ 3 \\ 1 \\ 1 \end{array} \right) \quad \text { and } \quad \left( \begin{array} { r } - 1 \\ - 1 \\ 2 \\ - 3 \end{array} \right)$$ form a basis for the range space of T . Given also that the vector \(\left( \begin{array} { c } p \\ 1 \\ 1 \\ q \end{array} \right)\) is in the range space of T , find a condition satisfied by \(p\) and \(q\).

7 The roots of the equation \(x ^ { 3 } + 4 x - 1 = 0\) are \(\alpha , \beta\) and \(\gamma\). Use the substitution \(y = \frac { 1 } { 1 + x }\) to show that the equation \(6 y ^ { 3 } - 7 y ^ { 2 } + 3 y - 1 = 0\) has roots \(\frac { 1 } { \alpha + 1 } , \frac { 1 } { \beta + 1 }\) and \(\frac { 1 } { \gamma + 1 }\). For the cases \(n = 1\) and \(n = 2\), find the value of $$\frac { 1 } { ( \alpha + 1 ) ^ { n } } + \frac { 1 } { ( \beta + 1 ) ^ { n } } + \frac { 1 } { ( \gamma + 1 ) ^ { n } }$$ Deduce the value of \(\frac { 1 } { ( \alpha + 1 ) ^ { 3 } } + \frac { 1 } { ( \beta + 1 ) ^ { 3 } } + \frac { 1 } { ( \gamma + 1 ) ^ { 3 } }\). Hence show that \(\frac { ( \beta + 1 ) ( \gamma + 1 ) } { ( \alpha + 1 ) ^ { 2 } } + \frac { ( \gamma + 1 ) ( \alpha + 1 ) } { ( \beta + 1 ) ^ { 2 } } + \frac { ( \alpha + 1 ) ( \beta + 1 ) } { ( \gamma + 1 ) ^ { 2 } } = \frac { 73 } { 36 }\).

8 The curves \(C _ { 1 }\) and \(C _ { 2 }\) have polar equations given by $$\begin{array} { l l r } C _ { 1 } : & r = 3 \sin \theta , & 0 \leqslant \theta < \pi , \\ C _ { 2 } : & r = 1 + \sin \theta , & - \pi < \theta \leqslant \pi . \end{array}$$

1. Find the polar coordinates of the points, other than the pole, where \(C _ { 1 }\) and \(C _ { 2 }\) meet.
2. In a single diagram, draw sketch graphs of \(C _ { 1 }\) and \(C _ { 2 }\).
3. Show that the area of the region which is inside \(C _ { 1 }\) but outside \(C _ { 2 }\) is \(\pi\).

9 Find the eigenvalues and corresponding eigenvectors of the matrix $$\mathbf { A } = \left( \begin{array} { r r r } 3 & - 1 & 0 \\ - 1 & 2 & - 1 \\ 0 & - 1 & 3 \end{array} \right)$$ Find a non-singular matrix \(\mathbf { M }\) and a diagonal matrix \(\mathbf { D }\) such that \(( \mathbf { A } - 2 \mathbf { I } ) ^ { 3 } = \mathbf { M D M } ^ { - 1 }\), where \(\mathbf { I }\) is the \(3 \times 3\) identity matrix.

10 By using de Moivre's theorem to express \(\sin 5 \theta\) and \(\cos 5 \theta\) in terms of \(\sin \theta\) and \(\cos \theta\), show that $$\tan 5 \theta = \frac { 5 t - 10 t ^ { 3 } + t ^ { 5 } } { 1 - 10 t ^ { 2 } + 5 t ^ { 4 } }$$ where \(t = \tan \theta\). Show that the roots of the equation \(x ^ { 4 } - 10 x ^ { 2 } + 5 = 0\) are \(\tan \left( \frac { 1 } { 5 } n \pi \right)\) for \(n = 1,2,3,4\). By considering the product of the roots of this equation, find the exact value of \(\tan \left( \frac { 1 } { 5 } \pi \right) \tan \left( \frac { 2 } { 5 } \pi \right)\).

11 It is given that \(x \neq 0\) and $$x \frac { \mathrm {~d} ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 2 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 4 x y = 8 x ^ { 2 } + 16$$ Show that if \(z = x y\) then $$\frac { \mathrm { d } ^ { 2 } z } { \mathrm {~d} x ^ { 2 } } + 4 z = 8 x ^ { 2 } + 16$$ Find \(y\) in terms of \(x\), given that \(y = 0\) and \(\frac { \mathrm { d } y } { \mathrm {~d} x } = - 2\) when \(x = \frac { 1 } { 2 } \pi\).

The curve \(C\) has equation $$y = \frac { x ^ { 2 } + 2 \lambda x } { x ^ { 2 } - 2 x + \lambda }$$ where \(\lambda\) is a constant and \(\lambda \neq - 1\).

1. Show that \(C\) has at most two stationary points.
2. Show that if \(C\) has exactly two stationary points then \(\lambda > - \frac { 5 } { 4 }\).
3. Find the set of values of \(\lambda\) such that \(C\) has two vertical asymptotes.
4. Find the \(x\)-coordinates of the points of intersection of \(C\) with
   1. the \(x\)-axis,
   2. the horizontal asymptote.
   3. Sketch \(C\) in each of the cases
      (a) \(\lambda < - 2\),
      (b) \(\lambda > 2\).

The plane \(\Pi _ { 1 }\) has equation \(\mathbf { r } = 2 \mathbf { i } + \mathbf { j } + 4 \mathbf { k } + \lambda ( 2 \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k } ) + \mu ( - \mathbf { i } + \mathbf { k } )\). Obtain a cartesian equation of \(\Pi _ { 1 }\) in the form \(p x + q y + r z = d\). The plane \(\Pi _ { 2 }\) has equation \(\mathbf { r } . ( \mathbf { i } - 4 \mathbf { j } + 5 \mathbf { k } ) = 12\). Find a vector equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\). The line \(l\) passes through the point \(A\) with position vector \(a \mathbf { i } + ( 2 a + 1 ) \mathbf { j } - 3 \mathbf { k }\) and is parallel to \(3 c \mathbf { i } - 3 \mathbf { j } + c \mathbf { k }\), where \(a\) and \(c\) are positive constants. Given that the perpendicular distance from \(A\) to \(\Pi _ { 1 }\) is \(\frac { 15 } { \sqrt { } 6 }\) and that the acute angle between \(l\) and \(\Pi _ { 1 }\) is \(\sin ^ { - 1 } \left( \frac { 2 } { \sqrt { } 6 } \right)\), find the values of \(a\) and \(c\).