CAIE Further Paper 1 (Further Paper 1) 2022 June

1

1. Sketch the curve with equation \(\mathrm { y } = \frac { \mathrm { x } + 1 } { \mathrm { x } - 1 }\).
2. Sketch the curve with equation \(\mathrm { y } = \frac { | \mathrm { x } | + 1 } { | \mathrm { x } | - 1 }\) and find the set of values of x for which \(\frac { | \mathrm { x } | + 1 } { | \mathrm { x } | - 1 } < - 2\).

2 The cubic equation \(x ^ { 3 } + 5 x ^ { 2 } + 10 x - 2 = 0\) has roots \(\alpha , \beta , \gamma\).

1. Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\).
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2. Show that the matrix \(\left( \begin{array} { c c c } 1 & \alpha & \beta
   \alpha & 1 & \gamma
   \beta & \gamma & 1 \end{array} \right)\) is singular.

3 A curve \(C\) has equation \(\mathrm { y } = \frac { \mathrm { a } \mathrm { x } ^ { 2 } + \mathrm { x } - 1 } { \mathrm { x } - 1 }\), where \(a\) is a positive constant.

1. Find the equations of the asymptotes of \(C\).
2. Show that there is no point on \(C\) for which \(1 < \mathrm { y } < 1 + 4 \mathrm { a }\).
3. Sketch C. You do not need to find the coordinates of the intersections with the axes.

4 Let \(\mathrm { u } _ { \mathrm { r } } = \mathrm { e } ^ { \mathrm { rx } } \left( \mathrm { e } ^ { 2 \mathrm { x } } - 2 \mathrm { e } ^ { \mathrm { x } } + 1 \right)\).

1. Using the method of differences, or otherwise, find \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \mathrm { u } _ { \mathrm { r } }\) in terms of \(n\) and \(x\).
2. Deduce the set of non-zero values of \(x\) for which the infinite series $$u _ { 1 } + u _ { 2 } + u _ { 3 } + \ldots$$ is convergent and give the sum to infinity when this exists.
3. Using a standard result from the list of formulae (MF19), find \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \ln \mathrm { u } _ { \mathrm { r } }\) in terms of \(n\) and \(x\).

5 Let \(\mathbf { A } = \left( \begin{array} { l l } 1 & a
0 & 1 \end{array} \right)\), where \(a\) is a positive constant.

1. State the type of the geometrical transformation in the \(x - y\) plane represented by \(\mathbf { A }\).
2. Prove by mathematical induction that, for all positive integers \(n\), $$\mathbf { A } ^ { \mathrm { n } } = \left( \begin{array} { c c } 1 & \mathrm { na }
   0 & 1 \end{array} \right)$$ Let \(\mathbf { B } = \left( \begin{array} { c c } b & b
   a ^ { - 1 } & a ^ { - 1 } \end{array} \right)\), where \(b\) is a positive constant.
3. Find the equations of the invariant lines, through the origin, of the transformation in the \(x - y\) plane represented by \(\mathbf { A } ^ { n } \mathbf { B }\).

6 The curve \(C\) has Cartesian equation \(x ^ { 2 } + x y + y ^ { 2 } = a\), where \(a\) is a positive constant.

1. Show that the polar equation of \(C\) is \(r ^ { 2 } = \frac { 2 a } { 2 + \sin 2 \theta }\).
2. Sketch the part of \(C\) for \(0 \leqslant \theta \leqslant \frac { 1 } { 4 } \pi\). The region \(R\) is enclosed by this part of \(C\), the initial line and the half-line \(\theta = \frac { 1 } { 4 } \pi\).
3. It is given that \(\sin 2 \theta\) may be expressed as \(\frac { 2 \tan \theta } { 1 + \tan ^ { 2 } \theta }\). Use this result to show that the area of \(R\) is $$\frac { 1 } { 2 } a \int _ { 0 } ^ { \frac { 1 } { 4 } \pi } \frac { 1 + \tan ^ { 2 } \theta } { 1 + \tan \theta + \tan ^ { 2 } \theta } \mathrm {~d} \theta$$ and use the substitution \(t = \tan \theta\) to find the exact value of this area.

7 The position vectors of the points \(A , B , C , D\) are $$7 \mathbf { i } + 4 \mathbf { j } - \mathbf { k } , \quad 11 \mathbf { i } + 3 \mathbf { j } , \quad 2 \mathbf { i } + 6 \mathbf { j } + 3 \mathbf { k } , \quad 2 \mathbf { i } + 7 \mathbf { j } + \lambda \mathbf { k }$$ respectively.

1. Given that the shortest distance between the line \(A B\) and the line \(C D\) is 3 , show that \(\lambda ^ { 2 } - 5 \lambda + 4 = 0\).
   Let \(\Pi _ { 1 }\) be the plane \(A B D\) when \(\lambda = 1\).
   Let \(\Pi _ { 2 }\) be the plane \(A B D\) when \(\lambda = 4\).
2. 1. Write down an equation of \(\Pi _ { 1 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \mathbf { s b } + \mathbf { t c }\).
   2. Find an equation of \(\Pi _ { 2 }\), giving your answer in the form \(a x + b y + c z = d\).
3. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
   If you use the following page to complete the answer to any question, the question number must be clearly shown.