CAIE FP1 (Further Pure Mathematics 1) 2019 June

1 Prove by mathematical induction that \(3 ^ { 3 n } - 1\) is divisible by 13 for every positive integer \(n\).

2 The curve \(C\) has polar equation \(r ^ { 2 } = \ln ( 1 + \theta )\), for \(0 \leqslant \theta \leqslant 2 \pi\).

1. Sketch \(C\).
2. Using the substitution \(u = 1 + \theta\), or otherwise, find the area of the region bounded by \(C\) and the initial line, leaving your answer in an exact form.

3

1. Write down the fifth roots of unity.
2. Find all the roots of the equation $$z ^ { 10 } + z ^ { 5 } + 1 = 0$$ giving each root in the form \(\mathrm { e } ^ { \mathrm { i } \theta }\).

4

1. Use the method of differences to show that \(\sum _ { r = 1 } ^ { N } \frac { 1 } { ( 3 r + 1 ) ( 3 r - 2 ) } = \frac { 1 } { 3 } - \frac { 1 } { 3 ( 3 N + 1 ) }\).
2. Find the limit, as \(N \rightarrow \infty\), of \(\sum _ { r = N + 1 } ^ { N ^ { 2 } } \frac { N } { ( 3 r + 1 ) ( 3 r - 2 ) }\).

5 The linear transformation \(\mathrm { T } : \mathbb { R } ^ { 4 } \rightarrow \mathbb { R } ^ { 4 }\) is represented by the matrix \(\mathbf { M }\), where $$\mathbf { M } = \left( \begin{array} { r r r r } 1 & 2 & 0 & 4
5 & 2 & 1 & - 3
4 & 0 & 1 & - 7
- 2 & 4 & - 1 & \alpha \end{array} \right)$$ It is given that the rank of \(\mathbf { M }\) is 2 .

1. Find the value of \(\alpha\) and state a basis for the range space of T .
2. Obtain a basis for the null space of T .

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

1. Obtain the equations of the asymptotes of \(C\).
2. Find the coordinates of the stationary points of \(C\).
3. Sketch \(C\).

7 The line \(l _ { 1 }\) passes through the points \(A ( - 3,1,4 )\) and \(B ( - 1,5,9 )\). The line \(l _ { 2 }\) passes through the points \(C ( - 2,6,5 )\) and \(D ( - 1,7,5 )\).

1. Find the shortest distance between the lines \(l _ { 1 }\) and \(l _ { 2 }\).
2. Find the acute angle between the line \(l _ { 2 }\) and the plane containing \(A , B\) and \(D\).

8 Find the particular solution of the differential equation $$9 \frac { \mathrm {~d} ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 6 \frac { \mathrm {~d} x } { \mathrm {~d} t } + x = 50 \sin t$$ given that when \(t = 0 , x = 0\) and \(\frac { \mathrm { d } x } { \mathrm {~d} t } = 0\).

9 A cubic equation \(x ^ { 3 } + b x ^ { 2 } + c x + d = 0\) has real roots \(\alpha , \beta\) and \(\gamma\) such that $$\begin{aligned} \frac { 1 } { \alpha } + \frac { 1 } { \beta } + \frac { 1 } { \gamma } & = - \frac { 5 } { 12 }
\alpha \beta \gamma & = - 12
\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } & = 90 \end{aligned}$$

1. Find the values of \(c\) and \(d\).
2. Express \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\) in terms of \(b\).
3. Show that \(b ^ { 3 } - 15 b + 126 = 0\).
4. Given that \(3 + \mathrm { i } \sqrt { } ( 12 )\) is a root of \(y ^ { 3 } - 15 y + 126 = 0\), deduce the value of \(b\).

10 Let \(I _ { n } = \int _ { \frac { 1 } { 4 } \pi } ^ { \frac { 1 } { 2 } \pi } \cot ^ { n } x \mathrm {~d} x\), where \(n \geqslant 0\).

1. By considering \(\frac { \mathrm { d } } { \mathrm { d } x } \left( \cot ^ { n + 1 } x \right)\), or otherwise, show that $$I _ { n + 2 } = \frac { 1 } { n + 1 } - I _ { n }$$ The curve \(C\) has equation \(y = \cot x\), for \(\frac { 1 } { 4 } \pi \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
2. Find, in an exact form, the \(y\)-coordinate of the centroid of the region enclosed by \(C\), the line \(x = \frac { 1 } { 4 } \pi\) and the \(x\)-axis.

A \(3 \times 3\) matrix \(\mathbf { A }\) has distinct eigenvalues 2, 1, 3, with corresponding eigenvectors $$\left( \begin{array} { l } 1
1
0 \end{array} \right) , \quad \left( \begin{array} { r } - 1
0
b \end{array} \right) , \quad \left( \begin{array} { r } 0
1
- 1 \end{array} \right)$$ respectively, where \(b\) is a positive constant.

1. Find \(\mathbf { A }\) in terms of \(b\).
2. Find \(\mathbf { A } ^ { - 1 } \left( \begin{array} { r } 0
   2
   - 2 \end{array} \right)\).
3. It is given that $$\mathbf { A } ^ { n } \left( \begin{array} { l } 1
   1
   0 \end{array} \right) = \left( \begin{array} { l } 4
   4
   0 \end{array} \right) \quad \text { and } \quad \mathbf { A } ^ { n } \left( \begin{array} { r } - 1
   0
   b \end{array} \right) = \left( \begin{array} { c } - 1
   0
   b ^ { - 1 } \end{array} \right) .$$ Find the values of \(n\) and \(b\).

The positive variables \(y\) and \(t\) are related by $$y = a ^ { t }$$ where \(a\) is a positive constant.

1. (a) By differentiating \(\ln y\) with respect to \(t\), show that \(\frac { \mathrm { d } y } { \mathrm {~d} t } = a ^ { t } \ln a\).
   (b) Write down \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } }\).
2. Determine the set of values of \(a\) for which the infinite series $$y + \frac { \mathrm { d } y } { \mathrm {~d} t } + \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} t ^ { 2 } } + \frac { \mathrm { d } ^ { 3 } y } { \mathrm {~d} t ^ { 3 } } + \ldots$$ is convergent.
   A curve has parametric equations $$x = t ^ { a } , \quad y = a ^ { t }$$
3. Find \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }\) in terms of \(a\) and \(t\), and show that, when \(t = 2\), $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 ^ { 1 - 2 a } ( 1 - a + 2 \ln a ) \ln a$$ If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.