CAIE Further Paper 1 (Further Paper 1) 2021 June

1

1. Show that $$\tan ( r + 1 ) - \tan r = \frac { \sin 1 } { \cos ( r + 1 ) \cos r }$$ Let \(\mathrm { u } _ { \mathrm { r } } = \frac { 1 } { \cos ( \mathrm { r } + 1 ) \cos \mathrm { r } }\).
2. Use the method of differences to find \(\sum _ { r = 1 } ^ { n } u _ { r }\).
3. Explain why the infinite series \(u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots\) does not converge.

2 The cubic equation \(2 x ^ { 3 } - 4 x ^ { 2 } + 3 = 0\) has roots \(\alpha , \beta , \gamma\). Let \(\mathrm { S } _ { \mathrm { n } } = \alpha ^ { \mathrm { n } } + \beta ^ { \mathrm { n } } + \gamma ^ { \mathrm { n } }\).

1. State the value of \(S _ { 1 }\) and find the value of \(S _ { 2 }\).
2. 1. Express \(\mathrm { S } _ { \mathrm { n } + 3 }\) in terms of \(\mathrm { S } _ { \mathrm { n } + 2 }\) and \(\mathrm { S } _ { \mathrm { n } }\).
   2. Hence, or otherwise, find the value of \(S _ { 4 }\).
3. Use the substitution \(\mathrm { y } = \mathrm { S } _ { 1 } - \mathrm { x }\), where \(S _ { 1 }\) is the numerical value found in part (a), to find and simplify an equation whose roots are \(\alpha + \beta , \beta + \gamma , \gamma + \alpha\).
4. Find the value of \(\frac { 1 } { \alpha + \beta } + \frac { 1 } { \beta + \gamma } + \frac { 1 } { \gamma + \alpha }\).

3

1. Prove by mathematical induction that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } \left( 5 r ^ { 4 } + r ^ { 2 } \right) = \frac { 1 } { 2 } n ^ { 2 } ( n + 1 ) ^ { 2 } ( 2 n + 1 )$$
2. Use the result given in part (a) together with the List of formulae (MF19) to find \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \mathrm { r } ^ { 4 }\) in terms of \(n\), fully factorising your answer.

4 The matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) are given by $$\mathbf { A } = \left( \begin{array} { c c c } 2 & k & k
5 & - 1 & 3
1 & 0 & 1 \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { c c } 1 & 0
0 & 1
1 & 0 \end{array} \right) \text { and } \quad \mathbf { C } = \left( \begin{array} { r c c } 0 & 1 & 1
- 1 & 2 & 0 \end{array} \right)$$ where \(k\) is a real constant.

1. Find \(\mathbf { C A B }\).
2. Given that \(\mathbf { A }\) is singular, find the value of \(k\).
3. Using the value of \(k\) from part (b), find the equations of the invariant lines, through the origin, of the transformation in the \(x - y\) plane represented by \(\mathbf { C A B }\).

5 The curve \(C\) has polar equation \(r = \frac { 1 } { \pi - \theta } - \frac { 1 } { \pi }\), where \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).

1. Sketch \(C\).
2. Show that the area of the region bounded by the half-line \(\theta = \frac { 1 } { 2 } \pi\) and \(C\) is \(\frac { 3 - 4 \ln 2 } { 4 \pi }\).

6 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\mathbf { r } = - \mathbf { i } - 2 \mathbf { j } + \mathbf { k } + s ( 2 \mathbf { i } - 3 \mathbf { j } )\) and \(\mathbf { r } = 3 \mathbf { i } - 2 \mathbf { k } + t ( 3 \mathbf { i } - \mathbf { j } + 3 \mathbf { k } )\) respectively. The plane \(\Pi _ { 1 }\) contains \(l _ { 1 }\) and the point \(P\) with position vector \(- 2 \mathbf { i } - 2 \mathbf { j } + 4 \mathbf { k }\).

1. Find an equation of \(\Pi _ { 1 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b } + \mu \mathbf { c }\).
   The plane \(\Pi _ { 2 }\) contains \(l _ { 2 }\) and is parallel to \(l _ { 1 }\).
2. Find an equation of \(\Pi _ { 2 }\), giving your answer in the form \(\mathrm { ax } + \mathrm { by } + \mathrm { cz } = \mathrm { d }\).
3. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
4. The point \(Q\) is such that \(\overrightarrow { \mathrm { OQ } } = - 5 \overrightarrow { \mathrm { OP } }\). Find the position vector of the foot of the perpendicular from the point \(Q\) to \(\Pi _ { 2 }\).

7 The curve \(C\) has equation \(y = \frac { x ^ { 2 } - x - 3 } { 1 + x - x ^ { 2 } }\).

1. Find the equations of the asymptotes of \(C\).
2. Find the coordinates of any stationary points on \(C\).
3. Sketch \(C\), stating the coordinates of the intersections with the axes.
4. Sketch the curve with equation \(y = \left| \frac { x ^ { 2 } - x - 3 } { 1 + x - x ^ { 2 } } \right|\) and find in exact form the set of values of \(x\) for which \(\left| \frac { x ^ { 2 } - x - 3 } { 1 + x - x ^ { 2 } } \right| < 3\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.