CAIE FP1 (Further Pure Mathematics 1) 2017 November

1 Find \(\sum _ { r = 1 } ^ { n } ( 4 r - 3 ) ( 4 r + 1 )\), giving your answer in its simplest form.

2 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 5 x = 4 - 5 t ^ { 2 }$$

3

1. Show that \(\frac { \mathrm { d } ^ { n + 1 } } { \mathrm {~d} x ^ { n + 1 } } \left( x ^ { n + 1 } \ln x \right) = \frac { \mathrm { d } ^ { n } } { \mathrm {~d} x ^ { n } } \left( x ^ { n } + ( n + 1 ) x ^ { n } \ln x \right)\).
2. Prove by mathematical induction that, for all positive integers \(n\), $$\frac { \mathrm { d } ^ { n } } { \mathrm {~d} x ^ { n } } \left( x ^ { n } \ln x \right) = n ! \left( \ln x + 1 + \frac { 1 } { 2 } + \ldots + \frac { 1 } { n } \right)$$

4 The cubic equation \(2 x ^ { 3 } - 3 x ^ { 2 } + 4 x - 10 = 0\) has roots \(\alpha , \beta\) and \(\gamma\).

1. Find the value of \(( \alpha + 1 ) ( \beta + 1 ) ( \gamma + 1 )\).
2. Find the value of \(( \beta + \gamma ) ( \gamma + \alpha ) ( \alpha + \beta )\).

5 The curve \(C\) has equation \(2 x ^ { 3 } + 3 x ^ { 2 } y - 3 y ^ { 3 } - 16 = 0\).

1. Find the coordinates of the point \(A\) on \(C\) at which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) and \(x \neq 0\).
2. Find the value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) at \(A\).

6 The points \(A , B\) and \(C\) have position vectors \(2 \mathbf { i } - \mathbf { j } + \mathbf { k } , 3 \mathbf { i } + 4 \mathbf { j } - \mathbf { k }\) and \(- \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k }\) respectively.

1. Find the area of the triangle \(A B C\).
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2. Find the perpendicular distance of the point \(A\) from the line \(B C\).
3. Find the cartesian equation of the plane through \(A , B\) and \(C\).

7 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r r } 1 & - 1 & - 2 & 3
5 & - 3 & - 4 & 25
6 & - 4 & - 6 & 28
7 & - 5 & - 8 & 31 \end{array} \right)$$

1. Find the rank of \(\mathbf { A }\) and a basis for the null space of T .
2. Find the matrix product \(\mathbf { A } \left( \begin{array} { r } - 1
   1
   - 1
   1 \end{array} \right)\) and hence find the general solution of the equation \(\mathbf { A } \mathbf { x } = \left( \begin{array} { r } 3
   21
   24
   27 \end{array} \right)\).
   \(8 \quad\) Let \(I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \sec ^ { n } x \mathrm {~d} x\) for \(n > 0\).

1. Find the value of \(I _ { 2 }\).
2. Show that, for \(n > 2\), $$( n - 1 ) I _ { n } = 2 ^ { \frac { 1 } { 2 } n - 1 } + ( n - 2 ) I _ { n - 2 }$$
3. The curve \(C\) has equation \(y = \sec ^ { 3 } x\) for \(0 \leqslant x \leqslant \frac { 1 } { 4 } \pi\). The region \(R\) is bounded by \(C\), the \(x\)-axis, the \(y\)-axis and the line \(x = \frac { 1 } { 4 } \pi\). Find the volume of revolution generated when \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis.

9 The curve \(C\) has equation $$y = \frac { 3 x - 9 } { ( x - 2 ) ( x + 1 ) }$$

1. Find the equations of the asymptotes of \(C\).
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2. Show that there is no point on \(C\) for which \(\frac { 1 } { 3 } < y < 3\).
3. Find the coordinates of the turning points of \(C\).
4. Sketch \(C\).

10

1. Use de Moivre's theorem to show that $$\sin 5 \theta = 5 \sin \theta - 20 \sin ^ { 3 } \theta + 16 \sin ^ { 5 } \theta$$
2. Hence explain why the roots of the equation \(16 x ^ { 4 } - 20 x ^ { 2 } + 5 = 0\) are \(x = \pm \sin \frac { 1 } { 5 } \pi\) and \(x = \pm \sin \frac { 2 } { 5 } \pi\).
3. Without using a calculator, find the exact values of $$\sin \frac { 1 } { 5 } \pi \sin \frac { 2 } { 5 } \pi \sin \frac { 3 } { 5 } \pi \sin \frac { 4 } { 5 } \pi \quad \text { and } \quad \sin ^ { 2 } \left( \frac { 1 } { 5 } \pi \right) + \sin ^ { 2 } \left( \frac { 2 } { 5 } \pi \right) .$$

1. The vector \(\mathbf { e }\) is an eigenvector of the matrix \(\mathbf { A }\), with corresponding eigenvalue \(\lambda\), and is also an eigenvector of the matrix \(\mathbf { B }\), with corresponding eigenvalue \(\mu\). Show that \(\mathbf { e }\) is an eigenvector of the matrix \(\mathbf { A B }\) with corresponding eigenvalue \(\lambda \mu\).
2. Find the eigenvalues and corresponding eigenvectors of the matrix \(\mathbf { A }\), where $$\mathbf { A } = \left( \begin{array} { r r r } 0 & 1 & 3
   3 & 2 & - 3
   1 & 1 & 2 \end{array} \right) .$$
3. The matrix \(\mathbf { B }\), where $$\mathbf { B } = \left( \begin{array} { r r r } 3 & 6 & 1
   1 & - 2 & - 1
   6 & 6 & - 2 \end{array} \right) ,$$ has eigenvectors \(\left( \begin{array} { r } 1
   - 1
   0 \end{array} \right) , \left( \begin{array} { r } 1
   - 1
   1 \end{array} \right)\) and \(\left( \begin{array} { l } 1
   0
   1 \end{array} \right)\). Find the eigenvalues of the matrix \(\mathbf { A B }\), and state corresponding eigenvectors.

The polar equation of a curve \(C\) is \(r = a ( 1 + \cos \theta )\) for \(0 \leqslant \theta < 2 \pi\), where \(a\) is a positive constant.

1. Sketch \(C\).
2. Show that the cartesian equation of \(C\) is $$x ^ { 2 } + y ^ { 2 } = a \left( x + \sqrt { } \left( x ^ { 2 } + y ^ { 2 } \right) \right)$$
3. Find the area of the sector of \(C\) between \(\theta = 0\) and \(\theta = \frac { 1 } { 3 } \pi\).
4. Find the arc length of \(C\) between the point where \(\theta = 0\) and the point where \(\theta = \frac { 1 } { 3 } \pi\).