CAIE Further Paper 1 (Further Paper 1) 2024 June

1 The cubic equation \(2 x ^ { 3 } + x ^ { 2 } - p x - 5 = 0\), where \(p\) is a positive constant, has roots \(\alpha , \beta , \gamma\).

1. State, in terms of \(p\), the value of \(\alpha \beta + \beta \gamma + \gamma \alpha\).
2. Find the value of \(\alpha ^ { 2 } \beta \gamma + \alpha \beta ^ { 2 } \gamma + \alpha \beta \gamma ^ { 2 }\).
3. Deduce a cubic equation whose roots are \(\alpha \beta , \beta \gamma , \alpha \gamma\).
4. Given that \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = \frac { 1 } { 3 }\), find the value of \(p\).

2 Prove by mathematical induction that \(6 ^ { 4 n } + 38 ^ { n } - 2\) is divisible by 74 for all positive integers \(n\).

3

1. Use standard results from the list of formulae (MF19) to show that $$\sum _ { r = 1 } ^ { N } r ( r + 1 ) ( 3 r + 4 ) = \frac { 1 } { 12 } N ( N + 1 ) ( N + 2 ) ( 9 N + 19 )$$
2. Express \(\frac { 3 r + 4 } { r ( r + 1 ) }\) in partial fractions and hence use the method of differences to find $$\sum _ { r = 1 } ^ { N } \frac { 3 r + 4 } { r ( r + 1 ) } \left( \frac { 1 } { 4 } \right) ^ { r + 1 }$$ in terms of \(N\).
3. Deduce the value of \(\sum _ { r = 1 } ^ { \infty } \frac { 3 r + 4 } { r ( r + 1 ) } \left( \frac { 1 } { 4 } \right) ^ { r + 1 }\).

4 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { c c } \frac { 1 } { 2 } & - \frac { 1 } { 2 } \sqrt { 3 }
\frac { 1 } { 2 } \sqrt { 3 } & \frac { 1 } { 2 } \end{array} \right) \left( \begin{array} { c c } 14 & 0
0 & 1 \end{array} \right)\).

1. The matrix \(\mathbf { M }\) represents a sequence of two geometrical transformations in the \(x - y\) plane. Give full details of each transformation, and make clear the order in which they are applied.
2. Write \(\mathbf { M } ^ { - 1 }\) as the product of two matrices, neither of which is \(\mathbf { I }\).
3. Find the equations of the invariant lines, through the origin, of the transformation represented by \(\mathbf { M }\).
4. The triangle \(A B C\) in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto triangle \(D E F\). Given that the area of triangle \(D E F\) is \(28 \mathrm {~cm} ^ { 2 }\), find the area of triangle \(A B C\).

5 The points \(A , B , C\) have position vectors $$2 \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k } , \quad 2 \mathbf { i } + 4 \mathbf { j } - \mathbf { k } , \quad - 3 \mathbf { i } - 3 \mathbf { j } + 4 \mathbf { k }$$ respectively, relative to the origin \(O\).

1. Find the equation of the plane \(A B C\), giving your answer in the form \(a x + b y + c z = d\).
   The point \(D\) has position vector \(2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k }\).
2. Find the perpendicular distance from \(D\) to the plane \(A B C\).
3. Find the shortest distance between the lines \(A B\) and \(C D\).

6 The curve \(C\) has equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 2 } + \mathrm { ax } + 1 } { \mathrm { x } + 2 }\), where \(a > \frac { 5 } { 2 }\).

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 point of intersection with the \(y\)-axis and labelling the asymptotes.
4. 1. Sketch the curve with equation \(\mathrm { y } = \left| \frac { \mathrm { x } ^ { 2 } + \mathrm { ax } + 1 } { \mathrm { x } + 2 } \right|\).
   2. On your sketch in part (i), draw the line \(\mathrm { y } = \mathrm { a }\).
   3. It is given that \(\left| \frac { \mathrm { x } ^ { 2 } + \mathrm { ax } + 1 } { \mathrm { x } + 2 } \right| < \mathrm { a }\) for \(- 5 - \sqrt { 14 } < x < - 3\) and \(- 5 + \sqrt { 14 } < x < 3\). Find the value of \(a\).

7 The curve \(C\) has polar equation \(r ^ { 2 } = ( \pi - \theta ) \tan ^ { - 1 } ( \pi - \theta )\), 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 \(u = \pi - \theta\), or otherwise, find the area of the region enclosed by \(C\) and the initial line.
3. Show that, at the point of \(C\) furthest from the initial line, $$2 ( \pi - \theta ) \tan ^ { - 1 } ( \pi - \theta ) \cot \theta - \frac { \pi - \theta } { 1 + ( \pi - \theta ) ^ { 2 } } - \tan ^ { - 1 } ( \pi - \theta ) = 0$$ and verify that this equation has a root for \(\theta\) between 1.2 and 1.3.
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