CAIE FP1 (Further Pure Mathematics 1) 2018 November

1 The vectors \(\mathbf { a } , \mathbf { b } , \mathbf { c }\) and \(\mathbf { d }\) in \(\mathbb { R } ^ { 3 }\) are given by $$\mathbf { a } = \left( \begin{array} { l } 1

2
1 \end{array} \right) , \quad \mathbf { b } = \left( \begin{array} { l } 2
9
0 \end{array} \right) , \quad \mathbf { c } = \left( \begin{array} { l }

4 \end{array} \right) \quad \text { and } \quad \mathbf { d } = \left( \begin{array} { r } 0
- 8
3 \end{array} \right) .$$

1. Show that \(\{ \mathbf { a } , \mathbf { b } , \mathbf { c } \}\) is a basis for \(\mathbb { R } ^ { 3 }\).
2. Express \(\mathbf { d }\) in terms of \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\).
   2 The roots of the equation $$x ^ { 3 } + p x ^ { 2 } + q x + r = 0$$ are \(\alpha , 2 \alpha , 4 \alpha\), where \(p , q , r\) and \(\alpha\) are non-zero real constants.
3. Show that $$2 p \alpha + q = 0$$
4. Show that $$p ^ { 3 } r - q ^ { 3 } = 0$$ 3 The sequence of positive numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is such that \(u _ { 1 } < 3\) and, for \(n \geqslant 1\), $$u _ { n + 1 } = \frac { 4 u _ { n } + 9 } { u _ { n } + 4 }$$
5. By considering \(3 - u _ { n + 1 }\), or otherwise, prove by mathematical induction that \(u _ { n } < 3\) for all positive integers \(n\).
6. Show that \(u _ { n + 1 } > u _ { n }\) for \(n \geqslant 1\).
   4 A curve is defined parametrically by $$x = t - \frac { 1 } { 2 } \sin 2 t \quad \text { and } \quad y = \sin ^ { 2 } t$$ The arc of the curve joining the point where \(t = 0\) to the point where \(t = \pi\) is rotated through one complete revolution about the \(x\)-axis. The area of the surface generated is denoted by \(S\).
7. Show that $$S = a \pi \int _ { 0 } ^ { \pi } \sin ^ { 3 } t \mathrm {~d} t$$ where the constant \(a\) is to be found.
8. Using the result \(\sin 3 t = 3 \sin t - 4 \sin ^ { 3 } t\), find the exact value of \(S\).

5 It is given that \(\lambda\) is an eigenvalue of the matrix \(\mathbf { A }\) with \(\mathbf { e }\) as a corresponding eigenvector, and \(\mu\) is an eigenvalue of the matrix \(\mathbf { B }\) for which \(\mathbf { e }\) is also a corresponding eigenvector.

1. Show that \(\lambda + \mu\) is an eigenvalue of the matrix \(\mathbf { A } + \mathbf { B }\) with \(\mathbf { e }\) as a corresponding eigenvector.
   The matrix \(\mathbf { A }\), given by $$\mathbf { A } = \left( \begin{array} { r r r } 2 & 0 & 1
   - 1 & 2 & 3
   1 & 0 & 2 \end{array} \right)$$ has \(\left( \begin{array} { l } 1
   2
   1 \end{array} \right) , \left( \begin{array} { r } 1
   4
   - 1 \end{array} \right)\) and \(\left( \begin{array} { l } 0
   1
   0 \end{array} \right)\) as eigenvectors.
2. Find the corresponding eigenvalues.
   The matrix \(\mathbf { B }\) has eigenvalues 4, 5 and 1 with corresponding eigenvectors \(\left( \begin{array} { l } 1
   2
   1 \end{array} \right) , \left( \begin{array} { r } 1
   4
   - 1 \end{array} \right)\) and \(\left( \begin{array} { l } 0
   1
   0 \end{array} \right)\) respectively.
3. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(( \mathbf { A } + \mathbf { B } ) ^ { 3 } = \mathbf { P D P } ^ { - 1 }\).

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

1. Find the equations of the asymptotes of \(C\).
2. Show that \(C\) intersects the \(x\)-axis twice.
3. Justifying your answer, find the number of stationary points on \(C\).
4. Sketch \(C\), stating the coordinates of its point of intersection with the \(y\)-axis.

7

1. Use de Moivre's theorem to show that $$\sin 8 \theta = 8 \sin \theta \cos \theta \left( 1 - 10 \sin ^ { 2 } \theta + 24 \sin ^ { 4 } \theta - 16 \sin ^ { 6 } \theta \right) .$$
2. Use the equation \(\frac { \sin 8 \theta } { \sin 2 \theta } = 0\) to find the roots of $$16 x ^ { 6 } - 24 x ^ { 4 } + 10 x ^ { 2 } - 1 = 0$$ in the form \(\sin k \pi\), where \(k\) is rational.

9
0 \end{array} \right) , \quad \mathbf { c } = \left( \begin{array} { l } 3
3
4 \end{array} \right) \quad \text { and } \quad \mathbf { d } = \left( \begin{array} { r } 0
- 8
3 \end{array} \right) .$$

1. Show that \(\{ \mathbf { a } , \mathbf { b } , \mathbf { c } \}\) is a basis for \(\mathbb { R } ^ { 3 }\).
2. Express \(\mathbf { d }\) in terms of \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\).
   2 The roots of the equation $$x ^ { 3 } + p x ^ { 2 } + q x + r = 0$$ are \(\alpha , 2 \alpha , 4 \alpha\), where \(p , q , r\) and \(\alpha\) are non-zero real constants.
3. Show that $$2 p \alpha + q = 0$$
4. Show that $$p ^ { 3 } r - q ^ { 3 } = 0$$ 3 The sequence of positive numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is such that \(u _ { 1 } < 3\) and, for \(n \geqslant 1\), $$u _ { n + 1 } = \frac { 4 u _ { n } + 9 } { u _ { n } + 4 }$$
5. By considering \(3 - u _ { n + 1 }\), or otherwise, prove by mathematical induction that \(u _ { n } < 3\) for all positive integers \(n\).
6. Show that \(u _ { n + 1 } > u _ { n }\) for \(n \geqslant 1\).
   4 A curve is defined parametrically by $$x = t - \frac { 1 } { 2 } \sin 2 t \quad \text { and } \quad y = \sin ^ { 2 } t$$ The arc of the curve joining the point where \(t = 0\) to the point where \(t = \pi\) is rotated through one complete revolution about the \(x\)-axis. The area of the surface generated is denoted by \(S\).
7. Show that $$S = a \pi \int _ { 0 } ^ { \pi } \sin ^ { 3 } t \mathrm {~d} t$$ where the constant \(a\) is to be found.
8. Using the result \(\sin 3 t = 3 \sin t - 4 \sin ^ { 3 } t\), find the exact value of \(S\).
   5 It is given that \(\lambda\) is an eigenvalue of the matrix \(\mathbf { A }\) with \(\mathbf { e }\) as a corresponding eigenvector, and \(\mu\) is an eigenvalue of the matrix \(\mathbf { B }\) for which \(\mathbf { e }\) is also a corresponding eigenvector.
9. Show that \(\lambda + \mu\) is an eigenvalue of the matrix \(\mathbf { A } + \mathbf { B }\) with \(\mathbf { e }\) as a corresponding eigenvector.
   The matrix \(\mathbf { A }\), given by $$\mathbf { A } = \left( \begin{array} { r r r } 2 & 0 & 1
   - 1 & 2 & 3
   1 & 0 & 2 \end{array} \right)$$ has \(\left( \begin{array} { l } 1
   2
   1 \end{array} \right) , \left( \begin{array} { r } 1
   4
   - 1 \end{array} \right)\) and \(\left( \begin{array} { l } 0
   1
   0 \end{array} \right)\) as eigenvectors.
10. Find the corresponding eigenvalues.
    The matrix \(\mathbf { B }\) has eigenvalues 4, 5 and 1 with corresponding eigenvectors \(\left( \begin{array} { l } 1
    2
    1 \end{array} \right) , \left( \begin{array} { r } 1
    4
    - 1 \end{array} \right)\) and \(\left( \begin{array} { l } 0
    1
    0 \end{array} \right)\) respectively.
11. Find a matrix \(\mathbf { P }\) and a diagonal matrix \(\mathbf { D }\) such that \(( \mathbf { A } + \mathbf { B } ) ^ { 3 } = \mathbf { P D P } ^ { - 1 }\).
    6 The curve \(C\) has equation $$y = \frac { x ^ { 2 } + a x - 1 } { x + 1 }$$ where \(a\) is constant and \(a > 1\).
12. Find the equations of the asymptotes of \(C\).
13. Show that \(C\) intersects the \(x\)-axis twice.
14. Justifying your answer, find the number of stationary points on \(C\).
15. Sketch \(C\), stating the coordinates of its point of intersection with the \(y\)-axis. 7
16. Use de Moivre's theorem to show that $$\sin 8 \theta = 8 \sin \theta \cos \theta \left( 1 - 10 \sin ^ { 2 } \theta + 24 \sin ^ { 4 } \theta - 16 \sin ^ { 6 } \theta \right) .$$
17. Use the equation \(\frac { \sin 8 \theta } { \sin 2 \theta } = 0\) to find the roots of $$16 x ^ { 6 } - 24 x ^ { 4 } + 10 x ^ { 2 } - 1 = 0$$ in the form \(\sin k \pi\), where \(k\) is rational.
    8 The plane \(\Pi _ { 1 }\) has equation $$\mathbf { r } = \left( \begin{array} { l } 5
    1
    0 \end{array} \right) + s \left( \begin{array} { r } - 4
    1
    3 \end{array} \right) + t \left( \begin{array} { l } 0
    1
    2 \end{array} \right)$$
18. Find a cartesian equation of \(\Pi _ { 1 }\).
    The plane \(\Pi _ { 2 }\) has equation \(3 x + y - z = 3\).
19. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), giving your answer in degrees.
20. Find an equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\). [5]
    9 The curve \(C\) has polar equation $$r = 5 \sqrt { } ( \cot \theta ) ,$$ where \(0.01 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
21. Find the area of the finite region bounded by \(C\) and the line \(\theta = 0.01\), showing full working. Give your answer correct to 1 decimal place.
    Let \(P\) be the point on \(C\) where \(\theta = 0.01\).
22. Find the distance of \(P\) from the initial line, giving your answer correct to 1 decimal place.
23. Find the maximum distance of \(C\) from the initial line.
24. Sketch \(C\).

10

1. Find the particular solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 10 x = 37 \sin 3 t$$ given that \(x = 3\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 0\) when \(t = 0\).
2. Show that, for large positive values of \(t\) and for any initial conditions, $$x \approx \sqrt { } ( 37 ) \sin ( 3 t - \phi ) ,$$ where the constant \(\phi\) is such that \(\tan \phi = 6\).

1. By considering \(( 2 r + 1 ) ^ { 2 } - ( 2 r - 1 ) ^ { 2 }\), use the method of differences to prove that $$\sum _ { r = 1 } ^ { n } r = \frac { 1 } { 2 } n ( n + 1 )$$
2. By considering \(( 2 r + 1 ) ^ { 4 } - ( 2 r - 1 ) ^ { 4 }\), use the method of differences and the result given in part (i) to prove that $$\sum _ { r = 1 } ^ { n } r ^ { 3 } = \frac { 1 } { 4 } n ^ { 2 } ( n + 1 ) ^ { 2 }$$ The sums \(S\) and \(T\) are defined as follows: $$\begin{aligned} & S = 1 ^ { 3 } + 2 ^ { 3 } + 3 ^ { 3 } + 4 ^ { 3 } + \ldots + ( 2 N ) ^ { 3 } + ( 2 N + 1 ) ^ { 3 } ,
   & T = 1 ^ { 3 } + 3 ^ { 3 } + 5 ^ { 3 } + 7 ^ { 3 } + \ldots + ( 2 N - 1 ) ^ { 3 } + ( 2 N + 1 ) ^ { 3 } . \end{aligned}$$
3. Use the result given in part (ii) to show that \(S = ( 2 N + 1 ) ^ { 2 } ( N + 1 ) ^ { 2 }\).
4. Hence, or otherwise, find an expression in terms of \(N\) for \(T\), factorising your answer as far as possible.
5. Deduce the value of \(\frac { S } { T }\) as \(N \rightarrow \infty\).

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

1. Show that \(A ( - 1,1 )\) is the only point on \(C\) with \(x\)-coordinate equal to - 1 .
   For \(n \geqslant 1\), let \(A _ { n }\) denote the value of \(\frac { \mathrm { d } ^ { n } y } { \mathrm {~d} x ^ { n } }\) at the point \(A ( - 1,1 )\).
2. Show that \(A _ { 1 } = 0\).
3. Show that \(A _ { 2 } = \frac { 2 } { 5 }\).
   Let \(I _ { n } = \int _ { - 1 } ^ { 0 } x ^ { n } \frac { \mathrm {~d} ^ { n } y } { \mathrm {~d} x ^ { n } } \mathrm {~d} x\).
4. Show that for \(n \geqslant 2\), $$I _ { n } = ( - 1 ) ^ { n + 1 } A _ { n - 1 } - n I _ { n - 1 } .$$
5. Deduce the value of \(I _ { 3 }\) in terms of \(I _ { 1 }\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.