CAIE Further Paper 1 (Further Paper 1) 2020 November

1 The cubic equation \(\mathrm { x } ^ { 3 } + \mathrm { bx } ^ { 2 } + \mathrm { cx } + \mathrm { d } = 0\), where \(b , c\) and \(d\) are constants, has roots \(\alpha , \beta , \gamma\). It is given that \(\alpha \beta \gamma = - 1\).

1. State the value of \(d\).
2. Find a cubic equation, with coefficients in terms of \(b\) and \(c\), whose roots are \(\alpha + 1 , \beta + 1 , \gamma + 1\).
3. Given also that \(\gamma + 1 = - \alpha - 1\), deduce that \(( \mathrm { c } - 2 \mathrm {~b} + 3 ) ( \mathrm { b } - 3 ) = \mathrm { b } - \mathrm { c }\).

2 Prove by mathematical induction that \(7 ^ { 2 n } - 1\) is divisible by 12 for every positive integer \(n\).

3

1. By simplifying \(\left( x ^ { n } - \sqrt { x ^ { 2 n } + 1 } \right) \left( x ^ { n } + \sqrt { x ^ { 2 n } + 1 } \right)\), show that \(\frac { 1 } { x ^ { n } - \sqrt { x ^ { 2 n } + 1 } } = - x ^ { n } - \sqrt { x ^ { 2 n } + 1 }\). [1]
   Let \(u _ { n } = x ^ { n + 1 } + \sqrt { x ^ { 2 n + 2 } + 1 } + \frac { 1 } { x ^ { n } - \sqrt { x ^ { 2 n } + 1 } }\).
2. Use the method of differences to find \(\sum _ { \mathrm { n } = 1 } ^ { \mathrm { N } } \mathrm { u } _ { \mathrm { n } }\) in terms of \(N\) and \(x\).
3. Deduce the set of 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.

4 The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are given by $$\mathbf { A } = \left( \begin{array} { l l } 0 & 1
1 & 0 \end{array} \right) \text { and } \mathbf { B } = \left( \begin{array} { c c } \frac { 1 } { 2 } & - \frac { 1 } { 2 } \sqrt { 3 }
\frac { 1 } { 2 } \sqrt { 3 } & \frac { 1 } { 2 } \end{array} \right)$$

1. Give full details of the geometrical transformation in the \(x - y\) plane represented by \(\mathbf { A }\).
2. Give full details of the geometrical transformation in the \(x - y\) plane represented by \(\mathbf { B }\).
   The triangle \(D E F\) in the \(x - y\) plane is transformed by \(\mathbf { A B }\) onto triangle \(P Q R\).
3. Show that the triangles \(D E F\) and \(P Q R\) have the same area.
4. Find the matrix which transforms triangle \(P Q R\) onto triangle \(D E F\).
5. Find the equations of the invariant lines, through the origin, of the transformation in the \(x - y\) plane represented by \(\mathbf { A B }\).

5 The curve \(C\) has polar equation \(r = \ln ( 1 + \pi - \theta )\), for \(0 \leqslant \theta \leqslant \pi\).

1. Sketch \(C\) and state the polar coordinates of the point of \(C\) furthest from the pole.
2. Using the substitution \(u = 1 + \pi - \theta\), or otherwise, show that the area of the region enclosed by \(C\) and the initial line is $$\frac { 1 } { 2 } ( 1 + \pi ) \ln ( 1 + \pi ) ( \ln ( 1 + \pi ) - 2 ) + \pi$$
3. Show that, at the point of \(C\) furthest from the initial line, $$( 1 + \pi - \theta ) \ln ( 1 + \pi - \theta ) - \tan \theta = 0$$ and verify that this equation has a root between 1.2 and 1.3.

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

1. The curve \(C _ { 1 }\) has equation \(\mathrm { y } = \frac { \mathrm { x } - \mathrm { a } } { \mathrm { x } - 2 \mathrm { a } }\). Sketch \(C _ { 1 }\). The curve \(C _ { 2 }\) has equation \(\mathrm { y } = \left( \frac { \mathrm { x } - \mathrm { a } } { \mathrm { x } - 2 \mathrm { a } } \right) ^ { 2 }\). The curve \(C _ { 3 }\) has equation \(\mathrm { y } = \left| \frac { \mathrm { x } - \mathrm { a } } { \mathrm { x } - 2 \mathrm { a } } \right|\).
2. 1. Find the coordinates of any stationary points of \(C _ { 2 }\).
   2. Find also the coordinates of any points of intersection of \(C _ { 2 }\) and \(C _ { 3 }\).
3. Sketch \(C _ { 2 }\) and \(C _ { 3 }\) on a single diagram, clearly identifying each curve. Hence find the set of values of \(x\) for which \(\left( \frac { x - a } { x - 2 a } \right) ^ { 2 } \leqslant \left| \frac { x - a } { x - 2 a } \right|\).

7 The points \(A , B , C\) have position vectors $$- 2 \mathbf { i } + 2 \mathbf { j } - \mathbf { k } , \quad - 2 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } , \quad - 2 \mathbf { j } + \mathbf { k } ,$$ 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 acute angle between the planes \(O B C\) and \(A B C\).
   The point \(D\) has position vector \(t \mathbf { i } - \mathbf { j }\).
3. Given that the shortest distance between the lines \(A B\) and \(C D\) is \(\sqrt { \mathbf { 1 0 } }\), find the value of \(t\).
   If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.