CAIE Further Paper 1 (Further Paper 1) 2020 Specimen

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1. Given that \(\mathrm { f } ( r ) = \frac { 1 } { ( r + 1 ) ( r + 2 ) }\), show that $$\mathrm { f } ( r - 1 ) - \mathrm { f } ( r ) = \frac { 2 } { r ( r + 1 ) ( r + 2 ) } .$$
2. Hence find \(\sum _ { r = 1 } ^ { n } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }\).
3. Deduce the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 1 } { r ( r + 1 ) ( r + 2 ) }\).

2 It is given that \(\phi ( n ) = 5 ^ { n } ( 4 n + 1 ) - 1\), for \(n = 1,2,3 , \ldots\).
Prove, by mathematical induction, that \(\phi ( n )\) is divisible by 8 for every positive integer \(n\).

3 The curve \(C\) has polar equation \(r = 2 + 2 \cos \theta\), for \(0 \leqslant \theta \leqslant \pi\).

1. Sketch \(C\).
2. Find the area of the region enclosed by \(C\) and the initial line.
3. Show that the Cartesian equation of \(C\) can be expressed as \(4 \left( x ^ { 2 } + y ^ { 2 } \right) = \left( x ^ { 2 } + y ^ { 2 } - 2 x \right) ^ { 2 }\).

4 The cubic equation $$z ^ { 3 } - z ^ { 2 } - z - 5 = 0$$ has roots \(\alpha , \beta\) and \(\gamma\).

1. Show that the value of \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }\) is 19 .
2. Find the value of \(\alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 }\).
3. Find a cubic equation with roots \(\alpha + 1 , \beta + 1\) and \(\gamma + 1\), giving your answer in the form $$p x ^ { 3 } + q x ^ { 2 } + r x + s = 0 ,$$ where \(p , q , r\) and \(s\) are constants to be determined.

5 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { r r }

6 The position vectors of the points \(A , B , C , D\) are $$2 \mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k } , \quad - 2 \mathbf { i } + 5 \mathbf { j } - 4 \mathbf { k } , \quad \mathbf { i } + 4 \mathbf { j } + \mathbf { k } , \quad \mathbf { i } + 5 \mathbf { j } + m \mathbf { k } ,$$ respectively, where \(m\) is an integer. It is given that the shortest distance between the line through \(A\) and \(B\) and the line through \(C\) and \(D\) is 3 .

1. Show that the only possible value of \(m\) is 2 .
2. Find the shortest distance of \(D\) from the line through \(A\) and \(C\).
3. Show that the acute angle between the planes \(A C D\) and \(B C D\) is \(\cos ^ { - 1 } \left( \frac { 1 } { \sqrt { 3 } } \right)\).

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

1. State the equations of the asymptotes of \(C\).
2. Show that \(y \leqslant \frac { 25 } { 12 }\) at all points on \(C\).
3. Find the coordinates of any stationary points of \(C\).
4. Sketch \(C\), stating the coordinates of any intersections of \(C\) with the coordinate axes and the asymptotes.
5. Sketch the curve with equation \(y = \left| \frac { 2 x ^ { 2 } - 3 x - 2 } { x ^ { 2 } - 2 x + 1 } \right|\) and find the set of values of \(x\) for which \(\left| \frac { 2 x ^ { 2 } - 3 x - 2 } { x ^ { 2 } - 2 x + 1 } \right| < 2\).