CAIE Further Paper 1 (Further Paper 1) 2022 June

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

1. Use the method of differences to find \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \frac { 1 } { ( \mathrm { ar } + 1 ) ( \mathrm { ar } + \mathrm { a } + 1 ) }\) in terms of \(n\) and \(a\).
2. Find the value of \(a\) for which \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { ( a r + 1 ) ( a r + a + 1 ) } = \frac { 1 } { 6 }\).

2 The points \(A , B , C\) have position vectors $$4 \mathbf { i } - 4 \mathbf { j } + \mathbf { k } , \quad - 4 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k } , \quad 4 \mathbf { i } - \mathbf { j } - 2 \mathbf { k } ,$$ respectively, relative to the origin \(O\).

1. Find the equation of the plane \(A B C\), giving your answer in the form \(a x + b y + c z = d\).
2. Find the perpendicular distance from \(O\) to the plane \(A B C\).
3. The point \(D\) has position vector \(2 \mathbf { i } + 3 \mathbf { j } - 3 \mathbf { k }\). Find the coordinates of the point of intersection of the line \(O D\) with the plane \(A B C\).

3 The sequence of positive numbers \(u _ { 1 } , u _ { 2 } , u _ { 3 } , \ldots\) is such that \(u _ { 1 } > 4\) and, for \(n \geqslant 1\), $$u _ { n + 1 } = \frac { u _ { n } ^ { 2 } + u _ { n } + 12 } { 2 u _ { n } }$$

1. By considering \(\mathrm { u } _ { \mathrm { n } + 1 } - 4\), or otherwise, prove by mathematical induction that \(\mathrm { u } _ { \mathrm { n } } > 4\) for all positive integers \(n\).
2. Show that \(u _ { n + 1 } < u _ { n }\) for \(n \geqslant 1\).

4 The cubic equation \(2 x ^ { 3 } + 5 x ^ { 2 } - 6 = 0\) has roots \(\alpha , \beta , \gamma\).

1. Find a cubic equation whose roots are \(\frac { 1 } { \alpha ^ { 3 } } , \frac { 1 } { \beta ^ { 3 } } , \frac { 1 } { \gamma ^ { 3 } }\).
2. Find the value of \(\frac { 1 } { \alpha ^ { 6 } } + \frac { 1 } { \beta ^ { 6 } } + \frac { 1 } { \gamma ^ { 6 } }\).
3. Find also the value of \(\frac { 1 } { \alpha ^ { 9 } } + \frac { 1 } { \beta ^ { 9 } } + \frac { 1 } { \gamma ^ { 9 } }\).

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

1. Show that \(C\) has no vertical asymptotes and state the equation of the horizontal asymptote of \(C\).
2. Find the coordinates of the stationary points on \(C\).
3. Sketch \(C\), stating the coordinates of the intersections with the axes.
4. Sketch the curve with equation \(y = \left| \frac { 2 x ^ { 2 } - x - 1 } { x ^ { 2 } + x + 1 } \right|\) and state the set of values of \(k\) for which \(\left| \frac { 2 x ^ { 2 } - x - 1 } { x ^ { 2 } + x + 1 } \right| = k\) has 4 distinct real solutions.

6 The curve \(C\) has polar equation \(r ^ { 2 } = \tan ^ { - 1 } \left( \frac { 1 } { 2 } \theta \right)\), where \(0 \leqslant \theta \leqslant 2\).

1. Sketch \(C\) and state, in exact form, the greatest distance of a point on \(C\) from the pole.
2. Find the exact value of the area of the region bounded by \(C\) and the half-line \(\theta = 2\).
   Now consider the part of \(C\) where \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
3. Show that, at the point furthest from the half-line \(\theta = \frac { 1 } { 2 } \pi\), $$\left( \theta ^ { 2 } + 4 \right) \tan ^ { - 1 } \left( \frac { 1 } { 2 } \theta \right) \sin \theta - \cos \theta = 0$$ and verify that this equation has a root between 0.6 and 0.7 .
   \(7 \quad\) The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l l } 1 & 2 & 3
   4 & k & 6
   7 & 8 & 9 \end{array} \right)\).
4. Find the set of values of \(k\) for which \(\mathbf { A }\) is non-singular.
5. Given that \(\mathbf { A }\) is non-singular, find, in terms of \(k\), the entries in the top row of \(\mathbf { A } ^ { - 1 }\).
6. Given that \(\mathbf { B } = \left( \begin{array} { l l l } 1 & 0 & 0
   0 & 1 & 0 \end{array} \right)\), give an example of a matrix \(\mathbf { C }\) such that \(\mathbf { B A C } = \left( \begin{array} { l l } 2 & 1
   k & 4 \end{array} \right)\).
7. Find the set of values of \(k\) for which the transformation in the \(x - y\) plane represented by \(\left( \begin{array} { l l } 2 & 1
   k & 4 \end{array} \right)\) has two distinct invariant lines through the origin.
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