CAIE Further Paper 1 (Further Paper 1) 2022 November

1 The cubic equation \(x ^ { 3 } + b x ^ { 2 } + d = 0\) has roots \(\alpha , \beta , \gamma\), where \(\alpha = \beta\) and \(d \neq 0\).

1. Show that \(4 b ^ { 3 } + 27 d = 0\).
2. Given that \(2 \alpha ^ { 2 } + \gamma ^ { 2 } = 3 b\), find the values of \(b\) and \(d\).

2 Prove by mathematical induction that, for all positive integers \(n , 7 ^ { 2 n } + 97 ^ { n } - 50\) is divisible by 48. [6]

3

1. By considering \(( 2 r + 1 ) ^ { 3 } - ( 2 r - 1 ) ^ { 3 }\), use the method of differences to prove that $$\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )$$ Let \(S _ { n } = 1 ^ { 2 } + 3 \times 2 ^ { 2 } + 3 ^ { 2 } + 3 \times 4 ^ { 2 } + 5 ^ { 2 } + 3 \times 6 ^ { 2 } + \ldots + \left( 2 + ( - 1 ) ^ { n } \right) n ^ { 2 }\).
2. Show that \(\mathrm { S } _ { 2 \mathrm { n } } = \frac { 1 } { 3 } \mathrm { n } ( 2 \mathrm { n } + 1 ) ( \mathrm { an } + \mathrm { b } )\), where \(a\) and \(b\) are integers to be determined.
3. State the value of \(\lim _ { n \rightarrow \infty } \frac { S _ { 2 n } } { n ^ { 3 } }\).

4 The plane \(\Pi\) contains the lines \(\mathbf { r } = 3 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } + \lambda ( - \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )\) and \(\mathbf { r } = 4 \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k } + \mu ( 3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k } )\).

1. Find a Cartesian equation of \(\Pi\), giving your answer in the form \(a x + b y + c z = d\).
   The line \(l\) passes through the point \(P\) with position vector \(2 \mathbf { i } + 3 \mathbf { j } + \mathbf { k }\) and is parallel to the vector \(\mathbf { j } + \mathbf { k }\).
2. Find the acute angle between \(I\) and \(\Pi\).
3. Find the position vector of the foot of the perpendicular from \(P\) to \(\Pi\).

5 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { r r } \frac { 1 } { 2 } \sqrt { 2 } & - \frac { 1 } { 2 } \sqrt { 2 }
\frac { 1 } { 2 } \sqrt { 2 } & \frac { 1 } { 2 } \sqrt { 2 } \end{array} \right) \left( \begin{array} { c c } 1 & k
0 & 1 \end{array} \right)\), where \(k\) is a constant.

1. The matrix \(\mathbf { M }\) represents a sequence of two geometrical transformations. State the type of each transformation, and make clear the order in which they are applied.
2. The triangle \(A B C\) in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto triangle \(D E F\). Find, in terms of \(k\), the single matrix which transforms triangle \(D E F\) onto triangle \(A B C\).
3. Find the set of values of \(k\) for which the transformation represented by \(\mathbf { M }\) has no invariant lines through the origin.

6

1. Show that the curve with Cartesian equation $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 36 \left( x ^ { 2 } - y ^ { 2 } \right)$$ has polar equation \(r ^ { 2 } = 36 \cos 2 \theta\).
   The curve \(C\) has polar equation \(r ^ { 2 } = 36 \cos 2 \theta\), for \(- \frac { 1 } { 4 } \pi \leqslant \theta \leqslant \frac { 1 } { 4 } \pi\).
2. Sketch \(C\) and state the maximum distance of a point on \(C\) from the pole.
3. Find the area of the region enclosed by \(C\).
4. Find the maximum distance of a point on \(C\) from the initial line, giving the answer in exact form.

7 The curve \(C\) has equation \(y = \frac { 5 x ^ { 2 } } { 5 x - 2 }\).

1. Find the equations of the asymptotes of \(C\).
2. Find the coordinates of the stationary points on \(C\).
3. Sketch \(C\).
4. Sketch the curve with equation \(y = \left| \frac { 5 x ^ { 2 } } { 5 x - 2 } \right|\) and find in exact form the set of values of \(x\) for which \(\left| \frac { 5 x ^ { 2 } } { 5 x - 2 } \right| < 2\).
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