OCR Further Additional Pure AS (Further Additional Pure AS) 2018 March

1 Use standard divisibility tests to show that the number $$N = 91039173588$$

• is divisible by 9
• is divisible by 11
• is not divisible by 8 .

2 The surface \(S\) has equation \(z = x ^ { 2 } y - 8 x y ^ { 2 } + \frac { x } { y }\) for \(y \neq 0\).

1. (a) Find the following.
   • \(\frac { \partial z } { \partial x }\)
   • \(\frac { \partial z } { \partial y }\)
     (b) Find the coordinates of all stationary points of \(S\).
   • Find all four second partial derivatives of \(z\) with respect to \(x\) and/or \(y\).

3 In this question you must show detailed reasoning.

1. The sequence \(\left\{ A _ { n } \right\}\) is given by \(A _ { 1 } = \sqrt { 3 }\) and \(A _ { n + 1 } = ( \sqrt { 3 } + 1 ) A _ { n }\) for \(n \geqslant 1\). Find an expression for \(A _ { n }\) in terms of \(n\).
2. The sequence \(\left\{ B _ { n } \right\}\) is given by the formula $$B _ { n } = \frac { 1 } { \sqrt { 3 } } \left( ( \sqrt { 3 } + 1 ) ^ { n } - ( \sqrt { 3 } - 1 ) ^ { n } \right) \text { for } n \geqslant 1 .$$ Explain why \(B _ { n } \rightarrow \frac { 1 } { \sqrt { 3 } } ( \sqrt { 3 } + 1 ) ^ { n }\) as \(n \rightarrow \infty\).
3. The sequence \(\left\{ C _ { n } \right\}\) converges and is defined by \(C _ { n } = \frac { A _ { n } } { B _ { n } }\) for \(n \geqslant 1\). Identify the limit of \(C n\) as \(n \rightarrow \infty\).

4 The group \(G\) consists of the symmetries of the equilateral triangle \(A B C\) under the operation of composition of transformations (which may be assumed to be associative). Three elements of \(G\) are

• \(\boldsymbol { i }\), the identity
• \(\boldsymbol { j }\), the reflection in the vertical line of symmetry of the triangle
• \(\boldsymbol { k }\), the anticlockwise rotation of \(120 ^ { \circ }\) about the centre of the triangle.

These are shown in the diagram below.
\includegraphics[max width=\textwidth, alt={}, center]{0b4458dc-4f82-40e4-adcf-cbffca088389-3_204_531_735_772}
\includegraphics[max width=\textwidth, alt={}, center]{0b4458dc-4f82-40e4-adcf-cbffca088389-3_211_543_975_762}
\includegraphics[max width=\textwidth, alt={}, center]{0b4458dc-4f82-40e4-adcf-cbffca088389-3_216_543_1215_762}

1. Explain why the order of \(G\) is 6 .
2. Determine
   • the order of \(\boldsymbol { j }\),
   • the order of \(\boldsymbol { k }\).
   • - Express, in terms of \(\boldsymbol { j }\) and/or \(\boldsymbol { k }\), each of the remaining three elements of \(G\).
   • Draw a diagram for each of these elements.
   • Is the operation of composition of transformations on \(G\) commutative? Justify your answer.
   • List all the proper subgroups of \(G\).

5 The points \(A\) and \(B\) have position vectors \(\mathbf { a }\) and \(\mathbf { b }\) respectively, relative to a fixed origin \(O\).

1. (a) Prove that \(\mathbf { a } \times ( \mathbf { b } - \mathbf { a } ) = \mathbf { a } \times \mathbf { b }\).
   (b) Determine the relationship between \(\mathbf { a } \times ( \mathbf { b } - \mathbf { a } )\) and \(\mathbf { b } \times ( \mathbf { b } - \mathbf { a } )\).
2. The point \(D\) is on the line \(A B\). \(O D\) is perpendicular to \(A B\). By considering the area of triangle \(O A B\), show
   that \(| O D | = \frac { | \mathbf { a } \times \mathbf { b } | } { | \mathbf { b } - \mathbf { a } | }\).

6 You are given that \(n\) is an integer.

1. (a) Show that \(\operatorname { hcf } ( 2 n + 1,3 n + 2 ) = 1\).
   (b) Hence prove that, if \(( 2 n + 1 )\) divides \(\left( 36 n ^ { 2 } + 3 n - 14 \right)\), then \(( 2 n + 1 )\) divides \(( 12 n - 7 )\).
2. Use the results of part (i) to find all integers \(n\) for which \(\frac { 36 n ^ { 2 } + 3 n - 14 } { 2 n + 1 }\) is also an integer.

7 Irrational numbers can be modelled by sequences \(\left\{ u _ { n } \right\}\) of rational numbers of the form $$u _ { 0 } = 1 \text { and } u _ { n + 1 } = a + \frac { 1 } { b + u _ { n } } \text { for } n \geqslant 0 \text {, }$$ where \(a\) and \(b\) are non-negative integer constants.

1. (a) The constants \(a = 1\) and \(b = 0\) produce the irrational number \(\omega\). State the value of \(\omega\) correct to six decimal places.
   (b) By setting \(u _ { n + 1 }\) and \(u _ { n }\) equal to \(\omega\), determine the exact value of \(\omega\).
2. Use the method of part (i) (b) to find the exact value of the irrational number produced by taking \(a = 0\) and \(b = 2\).
3. Find positive integers \(a\) and \(b\) which would produce the irrational number \(2 + \sqrt { 10 }\). \section*{END OF QUESTION PAPER}