OCR Further Additional Pure AS (Further Additional Pure AS) 2017 December

1 Solve \(12 x \equiv 3 ( \bmod 99 )\).

2

1. For non-zero vectors \(\mathbf { x }\) and \(\mathbf { y }\), explain the geometrical significance of the statement \(\mathbf { x } \times \mathbf { y } = \mathbf { 0 }\).
2. The points \(P\) and \(Q\) have position vectors \(\mathbf { p } = 2 \mathbf { i } + 7 \mathbf { j } - \mathbf { k }\) and \(\mathbf { q } = \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k }\) respectively. Find, in the form \(( \mathbf { r } - \mathbf { a } ) \times \mathbf { d } = \mathbf { 0 }\), the equation of line \(P Q\).

3 The surface with equation \(z = x ^ { 3 } + y ^ { 3 } - 6 x y\) has two stationary points; one at the origin and the second at the point \(A\). Determine the coordinates of \(A\).

4

1. The binary operation is defined on \(\mathbb { Z }\) by \(a\) b \(b = a + b - a b\) for all \(a , b \in \mathbb { Z }\). Prove that is associative on \(\mathbb { Z }\). The operation ∘ is defined on the set \(A = \{ 0,2,3,4,5,6 \}\) by \(a \circ b = a + b - a b ( \bmod 7 )\) for all \(a , b \in A\).
2. Complete the Cayley table for \(\left( A , { } ^ { \circ } \right)\) given in the Printed Answer Booklet.
3. Prove that \(( A , \circ )\) is a group. You may assume that the operation is associative.
4. List all the subgroups of \(( A , \circ )\).

5 Given that \(n\) is a positive integer greater than 2 , prove that

1. \(\quad 10201 _ { n }\) is a square number.
2. \(\quad 1221 _ { n }\) is a composite number.

6 For real constants \(a\) and \(b\), the sequence \(U _ { 1 } , U _ { 2 } , U _ { 3 } , \ldots\) is given by $$U _ { 1 } = a \text { and } U _ { n } = \left( U _ { n - 1 } \right) ^ { 2 } - b \text { for } n \geqslant 2 .$$

1. Determine the behaviour of the sequence in the case where \(a = 1\) and \(b = 3\).
2. In the case where \(b = 6\), find the values of \(a\) for which the sequence is constant.
3. In the case where \(a = - 1\) and \(b = 8\), prove that \(U _ { n }\) is divisible by 7 for all even values of \(n\).

7 The points \(A ( 0,35,120 ) , B ( 28,21,120 )\) and \(C ( 96,35 , - 72 )\) lie on the sphere \(S\), with centre \(O\) and radius 125 . Triangle \(A B C\) is denoted by \(\triangle\).

1. Find, in simplest surd form, the area of \(\Delta\). The points \(A , B\) and \(C\) also form a spherical triangle, \(T\), on the surface of \(S\). Each 'side' of \(T\) is the shorter arc of the circle, centre \(O\) and radius 125, which passes through two of the given vertices of \(T\). In order to find an approximation to the area of the spherical triangle, \(T\) is being modelled by \(\Delta\).
2. (a) State the assumption being made in using this model.
   (b) Say, giving a reason, whether the model gives an under-estimate or an over-estimate of the area of \(T\).

8 A surface \(S\) has equation \(z = 8 y ^ { 3 } - 6 x ^ { 2 } y + 60 x y - 15 x ^ { 2 } + 186 x - 150 y - 100\).

1. (a) Find any stationary points of the section of \(S\) given by \(y = - 3\).
   (b) Find any stationary points of the section of \(S\) given by \(x = - 1\).
2. Show that the surface \(S\) has a least one saddle point. \section*{OCR} Oxford Cambridge and RSA