OCR Further Additional Pure (Further Additional Pure) 2018 September

1

1. Write the number \(100011 _ { n }\), where \(n \geqslant 2\), as a polynomial in \(n\).
2. Show that \(n ^ { 2 } + n + 1\) is a factor of this expression.
3. Hence show that \(100011 _ { n }\) is composite in any number base \(n \geqslant 2\).

2 In this question, you must show detailed reasoning.
A curve is defined parametrically by \(x = t ^ { 3 } - 3 t + 1 , y = 3 t ^ { 2 } - 1\), for \(0 \leqslant t \leqslant 5\). Find, in exact form,

1. the length of the curve,
2. the area of the surface generated when the curve is rotated completely about the \(x\)-axis.

3 The function \(w = \mathrm { f } ( x , y , z )\) is given by \(\mathrm { f } ( x , y , z ) = x ^ { 2 } y z + 2 x y ^ { 2 } z + 3 x y z ^ { 2 } - 24 x y z\), for \(x , y , z \neq 0\).

1. (a) Find
   • \(\mathrm { f } _ { x }\),
   • \(\mathrm { f } _ { y }\),
   • \(\mathrm { f } _ { z }\).
     (b) Hence find the values of \(a , b , c\) and \(d\) for which \(w\) has a stationary value when \(d = \mathrm { f } ( a , b , c )\).
   • You are given that this stationary value is a local minimum of \(w\). Find values of \(x , y\) and \(z\) which show that it is not a global minimum of \(w\).

4 The points \(A , B , C\) and \(P\) have coordinates ( \(a , 0,0\) ), ( \(0 , b , 0\) ), ( \(0,0 , c\) ) and ( \(a , b , c\) ) respectively, where \(a , b\) and \(c\) are positive constants.
The plane \(\Pi\) contains \(A , B\) and \(C\).

1. (a) Use the scalar triple product to determine
   • the volume of tetrahedron \(O A B C\),
   • the volume of tetrahedron PABC.
     (b) Hence show that the distance from \(P\) to \(\Pi\) is twice the distance from \(O\) to \(\Pi\).
   • (a) Determine a vector which is normal to \(\Pi\).
     (b) Hence determine, in terms of \(a , b\) and \(c\) only, the distance from \(P\) to \(\Pi\). consisting of the powers of a particular, non-singular \(2 \times 2\) real matrix \(\mathbf { M } = \left( \begin{array} { l l } a & b
     c & d \end{array} \right)\), under the operation of
     matrix multiplication. matrix multiplication.
   • Explain why such a group is only possible if \(\operatorname { det } ( \mathbf { M } ) = 1\) or - 1 .
   • Write down values of \(a , b , c\) and \(d\) that would give a suitable matrix \(\mathbf { M }\) for which \(\mathbf { M } ^ { 6 } = \mathbf { I }\) and
   Student Q observes that their class has already found a group of order 6 in a previous task; a group
   Student Q observes that their class has already found a group of order 6 in a previous task; a group consisting of the powers of a particular, non-singular \(2 \times 2\) real matrix \(\mathbf { M } = \left( \begin{array} { l l } a & b
   c & d \end{array} \right)\), under the operation of
   matrix multiplication. \(\operatorname { det } ( \mathbf { M } ) = 1\).
   Explain why such a group is only possible if \(\operatorname { det } ( \mathbf { M } ) = 1\) or - 1 . Student Q believes that it is possible to construct a rational function f in the form \(\mathrm { f } ( x ) = \frac { a x + b } { c x + d }\) so that the group of functions is isomorphic to the matrix group which is generated by the matrix \(\mathbf { M }\) of part (iii).
2. (a) Write down and simplify the function f that, according to Student Q , corresponds to \(\mathbf { M }\).
   (b) By calculating \(\mathbf { M } ^ { 3 }\), show that Student Q's suggestion does not work.
   (c) Find a different function \(f\) that will satisfy the requirements of the task.

7 The members of the family of the sequences \(\left\{ u _ { n } \right\}\) satisfy the recurrence relation $$u _ { n + 1 } = 10 u _ { n } - u _ { n - 1 } \text { for } n \geqslant 1$$

1. Determine the general solution of (*).
2. The sequences \(\left\{ a _ { n } \right\}\) and \(\left\{ b _ { n } \right\}\) are members of this family of sequences, corresponding to the initial terms \(a _ { 0 } = 1 , a _ { 1 } = 5\) and \(b _ { 0 } = 0 , b _ { 1 } = 2\) respectively.
   (a) Find the next two terms of each sequence.
   (b) Prove that, for all non-negative integers \(n , \left( a _ { n } \right) ^ { 2 } - 6 \left( b _ { n } \right) ^ { 2 } = 1\).
   (c) Determine \(\lim _ { n \rightarrow \infty } \left( \frac { a _ { n } } { b _ { n } } \right)\). \section*{OCR} Oxford Cambridge and RSA