CAIE FP1 (Further Pure Mathematics 1) 2018 November

1 The roots of the cubic equation $$x ^ { 3 } - 5 x ^ { 2 } + 13 x - 4 = 0$$ are \(\alpha , \beta , \gamma\).

1. Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\).
2. Find the value of \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }\).

2 It is given that $$\mathbf { A } = \left( \begin{array} { r r r } 2 & 3 & 1
0 & - 2 & 1
0 & 0 & 1 \end{array} \right)$$

1. Find the eigenvalue of \(\mathbf { A }\) corresponding to the eigenvector \(\left( \begin{array} { l } 1
   0
   0 \end{array} \right)\).
2. Write down the negative eigenvalue of \(\mathbf { A }\) and find a corresponding eigenvector.
3. Find an eigenvalue and a corresponding eigenvector of the matrix \(\mathbf { A } + \mathbf { A } ^ { 6 }\).

3 The curve \(C\) has polar equation \(r = a \cos 3 \theta\), for \(- \frac { 1 } { 6 } \pi \leqslant \theta \leqslant \frac { 1 } { 6 } \pi\), where \(a\) is a positive constant.

1. Sketch \(C\).
2. Find the area of the region enclosed by \(C\), showing full working.
3. Using the identity \(\cos 3 \theta \equiv 4 \cos ^ { 3 } \theta - 3 \cos \theta\), find a cartesian equation of \(C\).

4

1. Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + x = 4 \sin t$$
2. State an approximate solution for large positive values of \(t\).

5 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r r } 3 & 2 & 0 & 1
6 & 5 & - 1 & 3
9 & 8 & - 2 & 5
- 3 & - 2 & 0 & - 1 \end{array} \right)$$

1. Find the rank of \(\mathbf { M }\).
   Let \(K\) be the null space of T .
2. Find a basis for \(K\).
3. Find the general solution of $$\mathbf { M } \mathbf { x } = \left( \begin{array} { r } 2
   5
   8
   - 2 \end{array} \right) .$$

6 It is given that \(y = \mathrm { e } ^ { x } u\), where \(u\) is a function of \(x\). The \(r\) th derivatives \(\frac { \mathrm { d } ^ { r } y } { \mathrm {~d} x ^ { r } }\) and \(\frac { \mathrm { d } ^ { r } u } { \mathrm {~d} x ^ { r } }\) are denoted by \(y ^ { ( r ) }\) and \(u ^ { ( r ) }\) respectively. Prove by mathematical induction that, for all positive integers \(n\), $$y ^ { ( n ) } = \mathrm { e } ^ { x } \left( \binom { n } { 0 } u + \binom { n } { 1 } u ^ { ( 1 ) } + \binom { n } { 2 } u ^ { ( 2 ) } + \ldots + \binom { n } { r } u ^ { ( r ) } + \ldots + \binom { n } { n } u ^ { ( n ) } \right)$$ [You may use without proof the result \(\binom { k } { r } + \binom { k } { r - 1 } = \binom { k + 1 } { r }\).]

8
- 2 \end{array} \right) .$$ 6 It is given that \(y = \mathrm { e } ^ { x } u\), where \(u\) is a function of \(x\). The \(r\) th derivatives \(\frac { \mathrm { d } ^ { r } y } { \mathrm {~d} x ^ { r } }\) and \(\frac { \mathrm { d } ^ { r } u } { \mathrm {~d} x ^ { r } }\) are denoted by \(y ^ { ( r ) }\) and \(u ^ { ( r ) }\) respectively. Prove by mathematical induction that, for all positive integers \(n\), $$y ^ { ( n ) } = \mathrm { e } ^ { x } \left( \binom { n } { 0 } u + \binom { n } { 1 } u ^ { ( 1 ) } + \binom { n } { 2 } u ^ { ( 2 ) } + \ldots + \binom { n } { r } u ^ { ( r ) } + \ldots + \binom { n } { n } u ^ { ( n ) } \right)$$ [You may use without proof the result \(\binom { k } { r } + \binom { k } { r - 1 } = \binom { k + 1 } { r }\).]
7 Let $$S _ { N } = \sum _ { r = 1 } ^ { N } ( 3 r + 1 ) ( 3 r + 4 ) \quad \text { and } \quad T _ { N } = \sum _ { r = 1 } ^ { N } \frac { 1 } { ( 3 r + 1 ) ( 3 r + 4 ) } .$$

1. Use standard results from the List of Formulae (MF10) to show that $$S _ { N } = N \left( 3 N ^ { 2 } + 12 N + 13 \right)$$
2. Use the method of differences to show that $$T _ { N } = \frac { 1 } { 12 } - \frac { 1 } { 3 ( 3 N + 4 ) } .$$
3. Deduce that \(\frac { S _ { N } } { T _ { N } }\) is an integer.
4. Find \(\lim _ { N \rightarrow \infty } \frac { S _ { N } } { N ^ { 3 } T _ { N } }\).
   8
5. By considering the binomial expansion of \(\left( z + \frac { 1 } { z } \right) ^ { 6 }\), where \(z = \cos \theta + i \sin \theta\), express \(\cos ^ { 6 } \theta\) in the form $$\frac { 1 } { 32 } ( p + q \cos 2 \theta + r \cos 4 \theta + s \cos 6 \theta ) ,$$ where \(p , q , r\) and \(s\) are integers to be determined.
6. Hence find the exact value of $$\int _ { - \frac { 1 } { 2 } \pi } ^ { \frac { 1 } { 2 } \pi } \cos ^ { 6 } \left( \frac { 1 } { 2 } x \right) \mathrm { d } x$$

9 The curve \(C\) has equation $$y = \frac { 5 x ^ { 2 } + 5 x + 1 } { x ^ { 2 } + x + 1 } .$$

1. Find the equation of the asymptote of \(C\).
2. Show that, for all real values of \(x , - \frac { 1 } { 3 } \leqslant y < 5\).
3. Find the coordinates of any stationary points of \(C\).
4. Sketch \(C\), stating the coordinates of any intersections with the \(y\)-axis.

10 The position vectors of the points \(A , B , C , D\) are $$\mathbf { i } + \mathbf { j } + 3 \mathbf { k } , \quad 3 \mathbf { i } + 4 \mathbf { j } + 5 \mathbf { k } , \quad - \mathbf { i } + 3 \mathbf { k } , \quad m \mathbf { j } + 4 \mathbf { k } ,$$ respectively, where \(m\) is a constant.

1. Show that the lines \(A B\) and \(C D\) are parallel when \(m = \frac { 3 } { 2 }\).
2. Given that \(m \neq \frac { 3 } { 2 }\), find the shortest distance between the lines \(A B\) and \(C D\).
3. When \(m = 2\), find the acute angle between the planes \(A B C\) and \(A B D\), giving your answer in degrees.

The curve \(C\) is defined parametrically by $$x = 18 t - t ^ { 2 } \quad \text { and } \quad y = 8 t ^ { \frac { 3 } { 2 } }$$ where \(0 < t \leqslant 4\).

1. Show that at all points of \(C\), $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = \frac { 3 ( 9 + t ) } { 2 t ^ { \frac { 1 } { 2 } } ( 9 - t ) ^ { 3 } }$$
2. Show that the mean value of \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) with respect to \(x\) over the interval \(0 < x \leqslant 56\) is \(\frac { 3 } { 70 }\).
3. Find the area of the surface generated when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis, showing full working.

Let \(I _ { n } = \int _ { 1 } ^ { \sqrt { } 2 } \left( x ^ { 2 } - 1 \right) ^ { n } \mathrm {~d} x\).

1. Show that, for \(n \geqslant 1\), $$( 2 n + 1 ) I _ { n } = \sqrt { } 2 - 2 n I _ { n - 1 } .$$
2. Using the substitution \(x = \sec \theta\), show that $$I _ { n } = \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \tan ^ { 2 n + 1 } \theta \sec \theta \mathrm {~d} \theta$$
3. Deduce the exact value of $$\int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { \sin ^ { 7 } \theta } { \cos ^ { 8 } \theta } \mathrm {~d} \theta$$ If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.