CAIE Further Paper 1 (Further Paper 1) 2024 November

1 The matrix \(\mathbf { M }\) represents the sequence of two transformations in the \(x - y\) plane given by a stretch parallel to the \(x\)-axis, scale factor \(k ( k \neq 0 )\), followed by a shear, \(x\)-axis fixed, with \(( 0,1 )\) mapped to \(( k , 1 )\).

1. Show that \(\mathbf { M } = \left( \begin{array} { c c } k & k
   0 & 1 \end{array} \right)\).
2. The transformation represented by \(\mathbf { M }\) has a line of invariant points. Find, in terms of \(k\), the equation of this line.
   The unit square \(S\) in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto the parallelogram \(P\).
3. Find, in terms of \(k\), a matrix which transforms \(P\) onto \(S\).
4. Given that the area of \(P\) is \(3 k ^ { 2 }\) units \({ } ^ { 2 }\), find the possible values of \(k\).

2 Prove by mathematical induction that, for all positive integers \(n\), $$\frac { \mathrm { d } ^ { n } } { \mathrm {~d} x ^ { n } } \left( \tan ^ { - 1 } x \right) = P _ { n } ( x ) \left( 1 + x ^ { 2 } \right) ^ { - n } ,$$ where \(P _ { n } ( x )\) is a polynomial of degree \(n - 1\).
\includegraphics[max width=\textwidth, alt={}, center]{beb69d0f-3f83-49bf-b9a2-329ddc7243fa-04_2718_42_107_2007}
\includegraphics[max width=\textwidth, alt={}, center]{beb69d0f-3f83-49bf-b9a2-329ddc7243fa-05_2726_35_97_20}

4

1. Use the method of differences to find \(\sum _ { r = 1 } ^ { n } \frac { 5 k } { ( 5 r + k ) ( 5 r + 5 + k ) }\) in terms of \(n\) and \(k\), where \(k\) is a positive constant.
   \includegraphics[max width=\textwidth, alt={}, center]{beb69d0f-3f83-49bf-b9a2-329ddc7243fa-08_2715_35_110_2012}
   \includegraphics[max width=\textwidth, alt={}, center]{beb69d0f-3f83-49bf-b9a2-329ddc7243fa-09_2723_35_101_20} It is given that \(\sum _ { r = 1 } ^ { \infty } \frac { 5 k } { ( 5 r + k ) ( 5 r + 5 + k ) } = \frac { 1 } { 3 }\).
2. Find the value of \(k\).
3. Hence find \(\sum _ { r = n } ^ { n ^ { 2 } } \frac { 5 k } { ( 5 r + k ) ( 5 r + 5 + k ) }\) in terms of \(n\).
   $$\left( x ^ { 2 } + y ^ { 2 } \right) ^ { 2 } = 6 x y$$ has polar equation \(r ^ { 2 } = 3 \sin 2 \theta\).
   The curve \(C\) has polar equation \(r ^ { 2 } = 3 \sin 2 \theta\), for \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
4. Sketch \(C\) and state the maximum distance of a point on \(C\) from the pole.
   \includegraphics[max width=\textwidth, alt={}, center]{beb69d0f-3f83-49bf-b9a2-329ddc7243fa-10_2716_35_108_2012}
5. Find the area of the region enclosed by \(C\).
6. Find the maximum distance of a point on \(C\) from the initial line.

6 The curve \(C\) has equation \(y = \frac { 4 x ^ { 2 } + x + 1 } { 2 x ^ { 2 } - 7 x + 3 }\).

1. Find the equations of the asymptotes of \(C\).
2. Find the coordinates of any stationary points on \(C\).
   \includegraphics[max width=\textwidth, alt={}, center]{beb69d0f-3f83-49bf-b9a2-329ddc7243fa-13_2720_40_106_18}
3. Sketch \(C\), stating the coordinates of any intersections with the axes.
4. Sketch the curve with equation \(y = \left| \frac { 4 x ^ { 2 } + x + 1 } { 2 x ^ { 2 } - 7 x + 3 } \right|\) and state the set of values of \(k\) for which \(\left| \frac { 4 x ^ { 2 } + x + 1 } { 2 x ^ { 2 } - 7 x + 3 } \right| = k\) has 4 distinct real solutions.

7 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\mathbf { r } = \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k } + \lambda ( 2 \mathbf { i } + \mathbf { j } + \mathbf { k } )\) and \(\mathbf { r } = \mathbf { i } - 2 \mathbf { j } + 9 \mathbf { k } + \mu ( \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k } )\) respectively. The plane \(\Pi _ { 1 }\) contains \(l _ { 1 }\) and is parallel to \(l _ { 2 }\).

1. Find the equation of \(\Pi _ { 1 }\), giving your answer in the form \(a x + b y + c z = d\).
   The plane \(\Pi _ { 2 }\) contains \(l _ { 2 }\) and the point with coordinates \(( 2 , - 1,7 )\).
2. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\).
   \includegraphics[max width=\textwidth, alt={}, center]{beb69d0f-3f83-49bf-b9a2-329ddc7243fa-15_2723_35_101_20} The point \(P\) on \(l _ { 1 }\) and the point \(Q\) on \(l _ { 2 }\) are such that \(P Q\) is perpendicular to both \(l _ { 1 }\) and \(l _ { 2 }\).
3. Find a vector equation for \(P Q\).
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