CAIE Further Paper 1 (Further Paper 1) 2021 November

1

1. Give full details of the geometrical transformation in the \(x - y\) plane represented by the matrix \(\left( \begin{array} { l l } 6 & 0
   0 & 6 \end{array} \right)\). Let \(\mathbf { A } = \left( \begin{array} { l l } 3 & 4
   2 & 2 \end{array} \right)\).
2. The triangle \(D E F\) in the \(x - y\) plane is transformed by \(\mathbf { A }\) onto triangle \(P Q R\). Given that the area of triangle \(D E F\) is \(13 \mathrm {~cm} ^ { 2 }\), find the area of triangle \(P Q R\).
3. Find the matrix \(\mathbf { B }\) such that \(\mathbf { A B } = \left( \begin{array} { l l } 6 & 0
   0 & 6 \end{array} \right)\).
4. Show that the origin is the only invariant point of the transformation in the \(x - y\) plane represented by \(\mathbf { A }\).

2 It is given that \(\mathrm { y } = \mathrm { xe } ^ { \mathrm { ax } }\), where \(a\) is a constant.
Prove by mathematical induction that, for all positive integers \(n\), $$\frac { d ^ { n } y } { d x ^ { n } } = \left( a ^ { n } x + n a ^ { n - 1 } \right) e ^ { a x }$$

3 Let \(S _ { n } = \sum _ { r = 1 } ^ { n } \ln \frac { r ( r + 2 ) } { ( r + 1 ) ^ { 2 } }\).

1. Using the method of differences, or otherwise, show that \(S _ { n } = \ln \frac { n + 2 } { 2 ( n + 1 ) }\).
   Let \(S = \sum _ { r = 1 } ^ { \infty } \ln \frac { r ( r + 2 ) } { ( r + 1 ) ^ { 2 } }\).
2. Find the least value of \(n\) such that \(\mathrm { S } _ { \mathrm { n } } - \mathrm { S } < 0.01\).

4 The cubic equation \(x ^ { 3 } + 2 x ^ { 2 } + 3 x + 3 = 0\) has roots \(\alpha , \beta , \gamma\).

1. Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\).
2. Show that \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = 1\).
3. Use standard results from the list of formulae (MF19) to show that $$\sum _ { r = 1 } ^ { n } \left( ( \alpha + r ) ^ { 3 } + ( \beta + r ) ^ { 3 } + ( \gamma + r ) ^ { 3 } \right) = n + \frac { 1 } { 4 } n ( n + 1 ) \left( a n ^ { 2 } + b n + c \right)$$ where \(a\), \(b\) and \(c\) are constants to be determined.

5 The curve \(C\) has polar equation \(r = 3 + 2 \sin \theta\), for \(- \pi < \theta \leqslant \pi\).

1. The diagram shows part of \(C\). Sketch the rest of \(C\) on the diagram.
   \includegraphics[max width=\textwidth, alt={}, center]{3dbf7021-79c0-4ebf-b96a-5ddeeed45011-08_865_702_408_1023} The straight line \(l\) has polar equation \(r \sin \theta = 2\).
2. Add \(l\) to the diagram in part (a) and find the polar coordinates of the points of intersection of \(C\) and \(l\).
3. The region \(R\) is enclosed by \(C\) and \(l\), and contains the pole. Find the area of \(R\), giving your answer in exact form.

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

1. Find the equations of the asymptotes of \(C\).
2. Show that there is no point on \(C\) for which \(0 < y < 12\).
3. Sketch C.
4. 1. Sketch the graphs of \(y = \left| \frac { x ^ { 2 } } { x - 3 } \right|\) and \(y = | x | - 3\) on a single diagram, stating the coordinates of the intersections with the axes.
   2. Use your sketch to find the set of values of \(c\) for which \(\left| \frac { x ^ { 2 } } { x - 3 } \right| \leqslant | x | + c\) has no solution. [1]

7 The points \(A , B , C\) have position vectors $$2 \mathbf { i } + 2 \mathbf { j } , \quad - \mathbf { j } + \mathbf { k } \quad \text { and } \quad 2 \mathbf { i } + \mathbf { j } - 7 \mathbf { k }$$ respectively, relative to the origin \(O\).

1. Find an equation of the plane \(O A B\), giving your answer in the form \(\mathbf { r } . \mathbf { n } = p\).
   The plane \(\Pi\) has equation \(\mathrm { x } - 3 \mathrm { y } - 2 \mathrm { z } = 1\).
2. Find the perpendicular distance of \(\Pi\) from the origin.
3. Find the acute angle between the planes \(O A B\) and \(\Pi\).
4. Find an equation for the common perpendicular to the lines \(O C\) and \(A B\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.