CAIE Further Paper 1 (Further Paper 1) 2023 June

1 Prove by mathematical induction that, for all positive integers \(n , 5 ^ { 3 n } + 32 ^ { n } - 33\) is divisible by 31 .

2

1. Use standard results from the list of formulae (MF19) to show that $$\sum _ { r = 1 } ^ { n } \left( 6 r ^ { 2 } + 6 r - 5 \right) = a n ^ { 3 } + b n ^ { 2 } + c n$$ where \(a\), \(b\) and \(c\) are integers to be determined.
2. Use the method of differences to find \(\sum _ { r = 1 } ^ { n } \frac { 6 r ^ { 2 } + 6 r - 5 } { r ^ { 2 } + r }\) in terms of \(n\).
3. Find also \(\sum _ { r = n + 1 } ^ { 2 n } \frac { 6 r ^ { 2 } + 6 r - 5 } { r ^ { 2 } + r }\) in terms of \(n\).

3 The equation \(x ^ { 4 } - x ^ { 2 } + 2 x + 5 = 0\) has roots \(\alpha , \beta , \gamma , \delta\).

1. Find a quartic equation whose roots are \(\alpha ^ { 2 } , \beta ^ { 2 } , \gamma ^ { 2 } , \delta ^ { 2 }\) and state the value of \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } + \delta ^ { 2 }\).
2. Find the value of \(\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } } + \frac { 1 } { \gamma ^ { 2 } } + \frac { 1 } { \delta ^ { 2 } }\).
3. Find the value of \(\alpha ^ { 4 } + \beta ^ { 4 } + \gamma ^ { 4 } + \delta ^ { 4 }\).

4 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { r r } \cos 2 \theta & - \sin 2 \theta
\sin 2 \theta & \cos 2 \theta \end{array} \right) \left( \begin{array} { l l } 1 & k
0 & 1 \end{array} \right)\), where \(0 < \theta < \pi\) and \(k\) is a non-zero constant. The matrix \(\mathbf { M }\) represents a sequence of two geometrical transformations, one of which is a shear.

1. Describe fully the other transformation and state the order in which the transformations are applied.
2. Write \(\mathbf { M } ^ { - 1 }\) as the product of two matrices, neither of which is \(\mathbf { I }\).
3. Find, in terms of \(k\), the value of \(\tan \theta\) for which \(\mathbf { M - I }\) is singular.
4. Given that \(k = 2 \sqrt { 3 }\) and \(\theta = \frac { 1 } { 3 } \pi\), show that the invariant points of the transformation represented by \(\mathbf { M }\) lie on the line \(3 y + \sqrt { 3 } x = 0\).

5

1. Show that the curve with Cartesian equation $$x ^ { 2 } - y ^ { 2 } = a$$ where \(a\) is a positive constant, has polar equation \(r ^ { 2 } = a \sec 2 \theta\).
   The curve \(C\) has polar equation \(r ^ { 2 } = \operatorname { asec } 2 \theta\), where \(a\) is a positive constant, for \(0 \leqslant \theta < \frac { 1 } { 4 } \pi\).
2. Sketch \(C\) and state the minimum distance of \(C\) from the pole.
3. Find, in terms of \(a\), the exact value of the area of the region enclosed by \(C\), the initial line, and the half-line \(\theta = \frac { 1 } { 12 } \pi\). [You may use any result from the list of formulae (MF19) without proof.] [4]

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

1. Find the equation of the plane \(A B C\), giving your answer in the form \(a x + b y + c z = d\).
2. Find the perpendicular distance from \(O\) to the plane \(A B C\).
3. Find a vector equation of the common perpendicular to the lines \(O C\) and \(A B\).

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

1. Find the equations of the asymptotes of \(C\).
2. Find the coordinates of the turning points on \(C\).
3. Sketch \(C\).
4. Sketch the curves with equations \(y = \left| \frac { x ^ { 2 } + 2 x + 1 } { x - 3 } \right|\) and \(y ^ { 2 } = \frac { x ^ { 2 } + 2 x + 1 } { x - 3 }\) on a single diagram, clearly identifying each curve. If you use the following page to complete the answer to any question, the question number must be clearly shown.