CAIE FP1 (Further Pure Mathematics 1) 2017 June

1 The roots of the cubic equation \(x ^ { 3 } + 2 x ^ { 2 } - 3 = 0\) are \(\alpha , \beta\) and \(\gamma\).

1. By using the substitution \(y = \frac { 1 } { x ^ { 2 } }\), find the cubic equation with roots \(\frac { 1 } { \alpha ^ { 2 } } , \frac { 1 } { \beta ^ { 2 } }\) and \(\frac { 1 } { \gamma ^ { 2 } }\).
2. Hence find the value of \(\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } }\).
3. Find also the value of \(\frac { 1 } { \alpha ^ { 2 } \beta ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } \gamma ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } \alpha ^ { 2 } }\).

2

1. Verify that \(\frac { 2 r + 1 } { r ( r + 1 ) ( r + 2 ) } = \frac { 1 } { 2 } \left\{ \frac { ( 2 r + 1 ) ( 2 r + 3 ) } { ( r + 1 ) ( r + 2 ) } - \frac { ( 2 r - 1 ) ( 2 r + 1 ) } { r ( r + 1 ) } \right\}\).
2. Hence show that \(\sum _ { r = 1 } ^ { n } \frac { 2 r + 1 } { r ( r + 1 ) ( r + 2 ) } = \frac { 1 } { 2 } \left\{ \frac { ( 2 n + 1 ) ( 2 n + 3 ) } { ( n + 1 ) ( n + 2 ) } - \frac { 3 } { 2 } \right\}\).
3. Deduce the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 2 r + 1 } { r ( r + 1 ) ( r + 2 ) }\).

3 Prove, by mathematical induction, that \(\sum _ { r = 1 } ^ { n } r \ln \left( \frac { r + 1 } { r } \right) = \ln \left( \frac { ( n + 1 ) ^ { n } } { n ! } \right)\) for all positive integers \(n\). [6]

4 A curve \(C\) has equation \(x ^ { 3 } - 3 x y + y ^ { 2 } = 4\). Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at the point \(( 0,2 )\) of \(C\).

5 A curve \(C\) has parametric equations $$x = \frac { 2 } { 5 } t ^ { \frac { 5 } { 2 } } - 2 t ^ { \frac { 1 } { 2 } } , \quad y = \frac { 4 } { 3 } t ^ { \frac { 3 } { 2 } } , \quad \text { for } 1 \leqslant t \leqslant 4$$

1. Find the exact value of the arc length of \(C\).
2. Find also the exact value of the surface area generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.

6 Let \(I _ { n }\) denote \(\int _ { 0 } ^ { 2 } \left( 4 + x ^ { 2 } \right) ^ { - n } \mathrm {~d} x\).

1. Find \(\frac { \mathrm { d } } { \mathrm { d } x } \left( x \left( 4 + x ^ { 2 } \right) ^ { - n } \right)\) and hence show that $$8 n I _ { n + 1 } = ( 2 n - 1 ) I _ { n } + 2 \times 8 ^ { - n } .$$
2. Use the result for integrating \(\frac { 1 } { x ^ { 2 } + a ^ { 2 } }\) with respect to \(x\), in the List of Formulae (MF10), to find the value of \(I _ { 1 }\) and deduce that $$I _ { 3 } = \frac { 3 } { 1024 } \pi + \frac { 1 } { 128 }$$

7

1. Use de Moivre's theorem to prove that $$\tan 4 \theta = \frac { 4 \tan \theta - 4 \tan ^ { 3 } \theta } { 1 - 6 \tan ^ { 2 } \theta + \tan ^ { 4 } \theta } .$$
2. Hence find the solutions of the equation $$t ^ { 4 } - 4 t ^ { 3 } - 6 t ^ { 2 } + 4 t + 1 = 0$$ giving your answers in the form \(\tan k \pi\), where \(k\) is a rational number.

8 Find the solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 6 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 9 x = 18 t ^ { 2 } + 6 t + 1$$ given that, when \(t = 0 , x = 3\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 0\).

9 The plane \(\Pi _ { 1 }\) passes through the points \(( 1,2,1 )\) and \(( 5 , - 2,9 )\) and is parallel to the vector \(\mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }\).

1. Find the cartesian equation of \(\Pi _ { 1 }\).
   The plane \(\Pi _ { 2 }\) contains the lines $$\mathbf { r } = 2 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } - \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 2 \mathbf { i } - 3 \mathbf { j } + \mathbf { k } + \mu ( 2 \mathbf { i } + 3 \mathbf { j } - \mathbf { k } ) .$$
2. Find the cartesian equation of \(\Pi _ { 2 }\).
3. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).

10 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { l l l } 6 & - 8 & 7 \\ 7 & - 9 & 7 \\ 6 & - 6 & 5 \end{array} \right)$$

1. Given that \(\left( \begin{array} { l } 1 \\ 1 \\ 0 \end{array} \right)\) is an eigenvector of \(\mathbf { A }\), find the corresponding eigenvalue.
2. Given also that - 1 is an eigenvalue of \(\mathbf { A }\), find a corresponding eigenvector.
3. It is given that the determinant of \(\mathbf { A }\) is equal to the product of the eigenvalues of \(\mathbf { A }\). Use this result to find the third eigenvalue of \(\mathbf { A }\), and find also a corresponding eigenvector.
4. Write down matrices \(\mathbf { P }\) and \(\mathbf { D }\) such that \(\mathbf { P } ^ { - 1 } \mathbf { A P } = \mathbf { D }\), where \(\mathbf { D }\) is a diagonal matrix, and hence find the matrix \(\mathbf { A } ^ { n }\) in terms of \(n\), where \(n\) is a positive integer.

A curve \(C\) has polar equation \(r = 2 a \cos \left( 2 \theta + \frac { 1 } { 2 } \pi \right)\) for \(0 \leqslant \theta < 2 \pi\), where \(a\) is a positive constant.

1. Show that \(r = - 2 a \sin 2 \theta\) and sketch \(C\).
2. Deduce that the cartesian equation of \(C\) is $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { \frac { 3 } { 2 } } = - 4 a x y .$$
3. Find the area of one loop of \(C\).
4. Show that, at the points (other than the pole) at which a tangent to \(C\) is parallel to the initial line, $$2 \tan \theta = - \tan 2 \theta .$$

The matrix \(\mathbf { A }\), given by $$\mathbf { A } = \left( \begin{array} { r r r r } 1 & - 1 & 0 & 2 \\ 3 & - 1 & 4 & 0 \\ 5 & - 8 & - 6 & 19 \\ - 2 & 3 & 2 & - 7 \end{array} \right) ,$$ represents a transformation from \(\mathbb { R } ^ { 4 }\) to \(\mathbb { R } ^ { 4 }\).

1. Find the rank of \(\mathbf { A }\) and show that \(\left\{ \left( \begin{array} { r } 2 \\ 2 \\ - 1 \\ 0 \end{array} \right) , \left( \begin{array} { l } 1 \\ 3 \\ 0 \\ 1 \end{array} \right) \right\}\) is a basis for the null space of the transformation. \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
2. Show that if $$\mathbf { A x } = p \left( \begin{array} { r } 1 \\ 3 \\ 5 \\ - 2 \end{array} \right) + q \left( \begin{array} { r } - 1 \\ - 1 \\ - 8 \\ 3 \end{array} \right) ,$$ where \(p\) and \(q\) are given real numbers, then $$\mathbf { x } = \left( \begin{array} { c } p + 2 \lambda + \mu \\ q + 2 \lambda + 3 \mu \\ - \lambda \\ \mu \end{array} \right) ,$$ where \(\lambda\) and \(\mu\) are real numbers. \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
3. Find the values of \(p\) and \(q\) such that $$p \left( \begin{array} { r } 1 \\ 3 \\ 5 \\ - 2 \end{array} \right) + q \left( \begin{array} { r } - 1 \\ - 1 \\ - 8 \\ 3 \end{array} \right) = \left( \begin{array} { r } 3 \\ 7 \\ 18 \\ - 7 \end{array} \right)$$ \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)
4. Find the solution of the equation \(\mathbf { A x } = \left( \begin{array} { r } 3 \\ 7 \\ 18 \\ - 7 \end{array} \right)\) of the form \(\mathbf { x } = \left( \begin{array} { l } 4 \\ 9 \\ \alpha \\ \beta \end{array} \right)\), where \(\alpha\) and \(\beta\) are positive integers to be found. \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\) \(\_\_\_\_\)