CAIE Further Paper 1 (Further Paper 1) 2024 June

1 The matrix \(\mathbf { A }\) is given by $$\mathbf { A } = \left( \begin{array} { c c c } k & 1 & 0
6 & 5 & 2
- 1 & 3 & - k \end{array} \right)$$ where \(k\) is a real constant.

1. Show that \(\mathbf { A }\) is non-singular.
2. Given that \(\mathbf { A } ^ { - 1 } = \left( \begin{array} { c c c } 3 & 0 & - 1
   1 & 0 & 0
   - \frac { 23 } { 2 } & \frac { 1 } { 2 } & 3 \end{array} \right)\), find the value of \(k\).

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

1. Find a cubic equation whose roots are \(\alpha ^ { 2 } + 1 , \beta ^ { 2 } + 1 , \gamma ^ { 2 } + 1\).
   \includegraphics[max width=\textwidth, alt={}, center]{7eb2abb1-68f4-4cc8-8314-f436906d6c4e-04_2714_34_143_2012}
2. Find the value of \(\left( \alpha ^ { 2 } + 1 \right) ^ { 2 } + \left( \beta ^ { 2 } + 1 \right) ^ { 2 } + \left( \gamma ^ { 2 } + 1 \right) ^ { 2 }\).
3. Find the value of \(\left( \alpha ^ { 2 } + 1 \right) ^ { 3 } + \left( \beta ^ { 2 } + 1 \right) ^ { 3 } + \left( \gamma ^ { 2 } + 1 \right) ^ { 3 }\).

3 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l } 1 & 2
0 & 1 \end{array} \right) \left( \begin{array} { l l } 7 & 0
0 & 1 \end{array} \right)\).

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. [4]
2. Find the equations of the invariant lines, through the origin, of the transformation represented by \(\mathbf { M }\).
   The triangle \(D E F\) in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto triangle \(P Q R\) .
3. Given that the area of triangle \(P Q R\) is \(35 \mathrm {~cm} ^ { 2 }\) ,find the area of triangle \(D E F\) .

4

1. Prove by mathematical induction that, for all positive integers \(n\), $$\sum _ { r = 1 } ^ { n } r ^ { 2 } = \frac { 1 } { 6 } n ( n + 1 ) ( 2 n + 1 )$$ \includegraphics[max width=\textwidth, alt={}, center]{7eb2abb1-68f4-4cc8-8314-f436906d6c4e-08_2716_35_143_2012} The sum \(S _ { n }\) is defined by \(S _ { n } = \sum _ { r = 1 } ^ { n } r ^ { 4 }\).
2. Using the identity $$( 2 r + 1 ) ^ { 5 } - ( 2 r - 1 ) ^ { 5 } \equiv 160 r ^ { 4 } + 80 r ^ { 2 } + 2$$ show that \(S _ { n } = \frac { 1 } { 30 } n ( n + 1 ) ( 2 n + 1 ) \left( 3 n ^ { 2 } + 3 n - 1 \right)\).
3. Find the value of \(\lim _ { n \rightarrow \infty } \left( n ^ { - 5 } S _ { n } \right)\).

5 The lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations \(\mathbf { r } = \mathbf { i } + 4 \mathbf { j } - \mathbf { k } + \lambda ( \mathbf { j } - 2 \mathbf { k } )\) and \(\mathbf { r } = - 3 \mathbf { i } + 4 \mathbf { j } + \mu ( \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )\) respectively.

1. Find the shortest distance between \(l _ { 1 }\) and \(l _ { 2 }\).
   The plane \(\Pi _ { 1 }\) contains \(l _ { 1 }\) and is parallel to \(l _ { 2 }\).
2. Obtain an equation of \(\Pi _ { 1 }\) in the form \(p x + q y + r z = s\).
   \includegraphics[max width=\textwidth, alt={}, center]{7eb2abb1-68f4-4cc8-8314-f436906d6c4e-10_2715_40_144_2007}
3. The point \(( 1,1,1 )\) lies on the plane \(\Pi _ { 2 }\). It is given that the line of intersection of the planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) passes through the point ( \(0,0,2\) ) and is parallel to the vector \(\mathbf { i } + 4 \mathbf { j } - 3 \mathbf { k }\). Obtain an equation of \(\Pi _ { 2 }\) in the form \(a x + b y + c z = d\).

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

1. Show that \(C\) has no vertical asymptotes and state the equation of the horizontal asymptote. [2]
2. Find the coordinates of any stationary points on \(C\).
   \includegraphics[max width=\textwidth, alt={}, center]{7eb2abb1-68f4-4cc8-8314-f436906d6c4e-12_2715_35_144_2012}
   \includegraphics[max width=\textwidth, alt={}, center]{7eb2abb1-68f4-4cc8-8314-f436906d6c4e-13_2718_33_141_23}
3. Sketch \(C\), stating the coordinates of the intersections with the axes.
4. Sketch \(y ^ { 2 } = \frac { x + 1 } { x ^ { 2 } + 3 }\), stating the coordinates of the stationary points and the intersections with the axes.

7 The curve \(C\) has polar equation \(r ^ { 2 } = \sin 2 \theta \cos \theta\), for \(0 \leqslant \theta \leqslant \pi\).

1. Sketch \(C\) and state the equation of the line of symmetry.
2. Find a Cartesian equation for \(C\).
   
3. Find the total area enclosed by \(C\).
4. Find the greatest distance of a point on \(C\) from the pole.
   \includegraphics[max width=\textwidth, alt={}, center]{7eb2abb1-68f4-4cc8-8314-f436906d6c4e-16_2718_36_141_2011} If you use the following page to complete the answer to any question, the question number must be clearly shown.
   

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