CAIE Further Paper 1 (Further Paper 1) 2020 June

1 Let \(a\) be a positive constant.

1. Sketch the curve with equation \(\mathrm { y } = \frac { \mathrm { ax } } { \mathrm { x } + 7 }\).
2. Sketch the curve with equation \(y = \left| \frac { a x } { x + 7 } \right|\) and find the set of values of \(x\) for which \(\left| \frac { a x } { x + 7 } \right| > \frac { a } { 2 }\).

2 The cubic equation \(6 \mathrm { x } ^ { 3 } + \mathrm { px } ^ { 2 } - 3 \mathrm { x } - 5 = 0\), where \(p\) is a constant, has roots \(\alpha , \beta , \gamma\).

1. Find a cubic equation whose roots are \(\alpha ^ { 2 } , \beta ^ { 2 } , \gamma ^ { 2 }\).
2. It is given that \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = 2 ( \alpha + \beta + \gamma )\).
   1. Find the value of \(p\).
   2. Find the value of \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }\).

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

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

4

1. By first expressing \(\frac { 1 } { r ^ { 2 } - 1 }\) in partial fractions, show that $$\sum _ { r = 2 } ^ { n } \frac { 1 } { r ^ { 2 } - 1 } = \frac { 3 } { 4 } - \frac { a n + b } { 2 n ( n + 1 ) }$$ where \(a\) and \(b\) are integers to be found.
2. Deduce the value of \(\sum _ { r = 2 } ^ { \infty } \frac { 1 } { r ^ { 2 } - 1 }\).
3. Find the limit, as \(n \rightarrow \infty\), of \(\sum _ { r = n + 1 } ^ { 2 n } \frac { n } { r ^ { 2 } - 1 }\).

5 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\mathbf { r } = 3 \mathbf { i } + 3 \mathbf { k } + \lambda ( \mathbf { i } + 4 \mathbf { j } + 4 \mathbf { k } )\) and \(\mathbf { r } = 3 \mathbf { i } - 5 \mathbf { j } - 6 \mathbf { k } + \mu ( 5 \mathbf { j } + 6 \mathbf { k } )\) respectively.

1. Find the shortest distance between \(l _ { 1 }\) and \(l _ { 2 }\).
   The plane \(\Pi\) contains \(l _ { 1 }\) and is parallel to the vector \(\mathbf { i } + \mathbf { k }\).
2. Find the equation of \(\Pi\), giving your answer in the form \(a x + b y + c z = d\).
3. Find the acute angle between \(l _ { 2 }\) and \(\Pi\).

6 Let \(\mathbf { A } = \left( \begin{array} { l l } 2 & 0
1 & 1 \end{array} \right)\).

1. The transformation in the \(x - y\) plane represented by \(\mathbf { A } ^ { - 1 }\) transforms a triangle of area \(30 \mathrm {~cm} ^ { 2 }\) into a triangle of area \(d \mathrm {~cm} ^ { 2 }\). Find the value of \(d\).
2. Prove by mathematical induction that, for all positive integers \(n\), $$\mathbf { A } ^ { n } = \left( \begin{array} { c c } 2 ^ { n } & 0
   2 ^ { n } - 1 & 1 \end{array} \right)$$
3. The line \(y = 2 x\) is invariant under the transformation in the \(x - y\) plane represented by \(\mathbf { A } ^ { n } \mathbf { B }\), where \(\mathbf { B } = \left( \begin{array} { r l } 1 & 0
   33 & 0 \end{array} \right)\). Find the value of \(n\).

7 The curve \(C _ { 1 }\) has polar equation \(r = \theta \cos \theta\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).

1. The point on \(C _ { 1 }\) furthest from the line \(\theta = \frac { 1 } { 2 } \pi\) is denoted by \(P\). Show that, at \(P\), $$2 \theta \tan \theta - 1 = 0$$ and verify that this equation has a root between 0.6 and 0.7 .
   The curve \(C _ { 2 }\) has polar equation \(r = \theta \sin \theta\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the pole, denoted by \(O\), and at another point \(Q\).
2. Find the polar coordinates of \(Q\), giving your answers in exact form.
3. Sketch \(C _ { 1 }\) and \(C _ { 2 }\) on the same diagram.
4. Find, in terms of \(\pi\), the area of the region bounded by the arc \(O Q\) of \(C _ { 1 }\) and the arc \(O Q\) of \(C _ { 2 }\). [7]
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.