CAIE Further Paper 1 (Further Paper 1) 2023 November

1

1. By considering \(( r + 1 ) ^ { 2 } - r ^ { 2 }\), use the method of differences to prove that $$\sum _ { r = 1 } ^ { n } r = \frac { 1 } { 2 } n ( n + 1 )$$
2. Given that \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } ( \mathrm { r } + \mathrm { a } ) = \mathrm { n }\), find \(a\) in terms of \(n\).

2 Prove by mathematical induction that, for all positive integers \(n\), $$1 + 2 x + 3 x ^ { 2 } + \ldots + n x ^ { n - 1 } = \frac { 1 - ( n + 1 ) x ^ { n } + n x ^ { n + 1 } } { ( 1 - x ) ^ { 2 } }$$

3 The quartic equation \(\mathrm { x } ^ { 4 } + \mathrm { bx } ^ { 3 } + \mathrm { cx } ^ { 2 } + \mathrm { dx } - 2 = 0\) has roots \(\alpha , \beta , \gamma , \delta\). It is given that $$\alpha + \beta + \gamma + \delta = 3 , \quad \alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 } = 5 , \quad \alpha ^ { - 1 } + \beta ^ { - 1 } + \gamma ^ { - 1 } + \delta ^ { - 1 } = 6$$

1. Find the values of \(b , c\) and \(d\).
2. Given also that \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } + \delta ^ { 3 } = - 27\), find the value of \(\alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 } + \delta ^ { 4 }\).

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

1. Find the shortest distance between \(l _ { 1 }\) and \(l _ { 2 }\).
   The plane \(\Pi\) contains \(l _ { 1 }\) and the point with position vector \(- \mathbf { i } - 3 \mathbf { j } - 4 \mathbf { k }\).
2. Find an equation of \(\Pi\), giving your answer in the form \(a x + b y + c z = d\).

5 Let \(k\) be a constant. The matrices \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) are given by $$\mathbf { A } = \left( \begin{array} { l l l } 1 & k & 3
2 & 1 & 3
3 & 2 & 5 \end{array} \right) , \quad \mathbf { B } = \left( \begin{array} { r r } 0 & - 2
- 1 & 3
0 & 0 \end{array} \right) \quad \text { and } \quad \mathbf { C } = \left( \begin{array} { r r r } - 2 & - 1 & 1
1 & 1 & 3 \end{array} \right)$$ It is given that \(\mathbf { A }\) is singular.

1. Show that \(\mathbf { C A B } = \left( \begin{array} { r r } 3 & - 7
   - 9 & 3 \end{array} \right)\).
2. Find the equations of the invariant lines, through the origin, of the transformation in the \(x - y\) plane represented by \(\mathbf { C A B }\).
3. The matrices \(\mathbf { D } , \mathbf { E }\) and \(\mathbf { F }\) represent geometrical transformations in the \(x - y\) plane.
   • D represents an enlargement, centre the origin.
   • E represents a stretch parallel to the \(x\)-axis.
   • F represents a reflection in the line \(y = x\).
   Given that \(\mathbf { C A B } = \mathbf { D } - 9 \mathbf { E F }\), find \(\mathbf { D } , \mathbf { E }\) and \(\mathbf { F }\).

6

1. Show that the curve with Cartesian equation $$\left( x - \frac { 1 } { 2 } \right) ^ { 2 } + y ^ { 2 } = \frac { 1 } { 4 }$$ has polar equation \(r = \cos \theta\).
   The curves \(C _ { 1 }\) and \(C _ { 2 }\) have polar equations $$r = \cos \theta \quad \text { and } \quad r = \sin 2 \theta$$ respectively, where \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\). The curves \(C _ { 1 }\) and \(C _ { 2 }\) intersect at the pole and at another point \(P\).
2. Find the polar coordinates of \(P\).
3. In a single diagram sketch \(C _ { 1 }\) and \(C _ { 2 }\), clearly identifying each curve, and mark the point \(P\).
4. The region \(R\) is enclosed by \(C _ { 1 }\) and \(C _ { 2 }\) and includes the line \(O P\). Find, in exact form, the area of \(R\).

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

1. Find the equations of the asymptotes of \(C\).
2. Find the coordinates of any stationary points on \(C\), giving your answers correct to 1 decimal place.
3. Sketch \(C\), stating the coordinates of any intersections with the axes.
4. Sketch the curve with equation \(\mathrm { y } = \frac { 1 } { \mathrm { f } ( \mathrm { x } ) }\).
5. Find the set of values for which \(\frac { 1 } { \mathrm { f } ( x ) } < \mathrm { f } ( x )\).
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