CAIE Further Paper 1 (Further Paper 1) 2023 June

1 Let \(\mathbf { A } = \left( \begin{array} { l l } 3 & 0
1 & 1 \end{array} \right)\).

1. Prove by mathematical induction that, for all positive integers \(n\), $$2 \mathbf { A } ^ { n } = \left( \begin{array} { l l }

2 \times 3 ^ { n } & 0
3 ^ { n } - 1 & 2 \end{array} \right)$$ (b) Find, in terms of \(n\), the inverse of \(\mathbf { A } ^ { n }\).
2 The cubic equation \(x ^ { 3 } + 4 x ^ { 2 } + 6 x + 1 = 0\) has roots \(\alpha , \beta , \gamma\).
(a) Find the value of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\).
(b) Use standard results from the list of formulae (MF19) to show that $$\sum _ { r = 1 } ^ { n } \left( ( \alpha + r ) ^ { 2 } + ( \beta + r ) ^ { 2 } + ( \gamma + r ) ^ { 2 } \right) = n \left( n ^ { 2 } + a n + b \right)$$ where \(a\) and \(b\) are constants to be determined.

3

1. Use the method of differences to find \(\sum _ { \mathrm { r } = 1 } ^ { \mathrm { n } } \frac { 1 } { ( \mathrm { kr } + 1 ) ( \mathrm { kr } - \mathrm { k } + 1 ) }\) in terms of \(n\) and \(k\), where \(k\) is a positive constant.
2. Deduce the value of \(\sum _ { \mathrm { r } = 1 } ^ { \infty } \frac { 1 } { ( \mathrm { kr } + 1 ) ( \mathrm { kr } - \mathrm { k } + 1 ) }\).
3. Find also \(\sum _ { \mathrm { r } = \mathrm { n } } ^ { \mathrm { n } ^ { 2 } } \frac { 1 } { ( \mathrm { kr } + 1 ) ( \mathrm { kr } - \mathrm { k } + 1 ) }\) in terms of \(n\) and \(k\).

4 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l } \mathrm { a } & \mathrm { b } ^ { 2 }
\mathrm { c } ^ { 2 } & \mathrm { a } \end{array} \right)\), where \(a , b , c\) are real constants and \(b \neq 0\).

1. Show that \(\mathbf { M }\) does not represent a rotation about the origin.
2. Find the equations of the invariant lines, through the origin, of the transformation in the \(x - y\) plane represented by \(\mathbf { M }\).
   It is given that \(\mathbf { M }\) represents the sequence of two transformations in the \(x - y\) plane given by an enlargement, centre the origin, scale factor 5 followed by a shear, \(x\)-axis fixed, with \(( 0,1 )\) mapped to \(( 5,1 )\).
3. Find \(\mathbf { M }\).
4. The triangle \(D E F\) in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto triangle \(P Q R\). Given that the area of triangle \(D E F\) is \(12 \mathrm {~cm} ^ { 2 }\), find the area of triangle \(P Q R\).

5 The curve \(C\) has polar equation \(r ^ { 2 } = \frac { 1 } { \theta ^ { 2 } + 1 }\), for \(0 \leqslant \theta \leqslant \pi\).

1. Sketch \(C\) and state the polar coordinates of the point of \(C\) furthest from the pole.
2. Find the area of the region enclosed by \(C\), the initial line, and the half-line \(\theta = \pi\).
3. Show that, at the point of \(C\) furthest from the initial line, $$\left( \theta + \frac { 1 } { \theta } \right) \cot \theta - 1 = 0$$ and verify that this equation has a root between 1.1 and 1.2.

6 The curve \(C\) has equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 2 } + 2 \mathrm { x } - 15 } { \mathrm { x } - 2 }\).

1. Find the equations of the asymptotes of \(C\).
2. Show that \(C\) has no stationary points.
3. Sketch \(C\), stating the coordinates of the intersections with the axes.
4. Sketch the curve with equation \(\mathrm { y } = \left| \frac { \mathrm { x } ^ { 2 } - 2 \mathrm { x } - 15 } { \mathrm { x } - 2 } \right|\).
5. Find the set of values of \(x\) for which \(\left| \frac { 2 x ^ { 2 } + 4 x - 30 } { x - 2 } \right| < 15\).

7 The plane \(\Pi _ { 1 }\) has equation \(r = - 4 \mathbf { j } - 3 \mathbf { k } + \lambda ( \mathbf { i } - \mathbf { j } + \mathbf { k } ) + \mu ( \mathbf { i } + \mathbf { j } - \mathbf { k } )\).

1. Obtain an equation of \(\Pi _ { 1 }\) in the form \(\mathrm { px } + \mathrm { qy } + \mathrm { rz } = \mathrm { d }\).
2. The plane \(\Pi _ { 2 }\) has equation \(\mathbf { r } . ( - 5 \mathbf { i } + 3 \mathbf { j } + 5 \mathbf { k } ) = 4\). Find a vector equation of the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
   The line \(l\) passes through the point \(A\) with position vector \(a \mathbf { i } + a \mathbf { j } + ( a - 7 ) \mathbf { k }\) and is parallel to \(( 1 - b ) \mathbf { i } + b \mathbf { j } + b \mathbf { k }\), where \(a\) and \(b\) are positive constants.
3. Given that the perpendicular distance from \(A\) to \(\Pi _ { 1 }\) is \(\sqrt { 2 }\), find the value of \(a\).
4. Given that the obtuse angle between \(l\) and \(\Pi _ { 1 }\) is \(\frac { 3 } { 4 } \pi\), find the exact value of \(b\).
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