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

1 The points \(A , B\) and \(C\) have position vectors \(6 \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k } , 13 \mathbf { i } + 2 \mathbf { j } + 5 \mathbf { k }\) and \(16 \mathbf { i } + 6 \mathbf { j } + 3 \mathbf { k }\) respectively.

1. Using the vector product, calculate the area of triangle \(A B C\).
2. Hence find, in simplest surd form, the perpendicular distance from \(C\) to the line through \(A\) and \(B\).

2 The surface with equation \(z = 6 x ^ { 3 } + \frac { 1 } { 9 } y ^ { 2 } + x ^ { 2 } y\) has two stationary points.

1. Verify that one of these stationary points is at the origin.
2. Find the coordinates of the second stationary point.

3 Given that \(n\) is a positive integer, show that the numbers ( \(4 n + 1\) ) and ( \(6 n + 1\) ) are co-prime.

4 The group \(G\) consists of a set of six matrices under matrix multiplication. Two of the elements of \(G\) are \(\mathbf { A } = \left( \begin{array} { l l } 0 & 1
1 & 0 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } 1 & - 1
0 & - 1 \end{array} \right)\).

1. Determine each of the following:
   • \(\mathbf { A } ^ { 2 }\)
   • \(\mathbf { B } ^ { 2 }\)
   • Determine all the elements of \(G\).
   • State the order of each non-identity element of \(G\).
   • State, with justification, whether \(G\) is
   • abelian
   • cyclic.

5 For integers \(a\) and \(b\), with \(a \geqslant 0\) and \(0 \leqslant b \leqslant 99\), the numbers \(M\) and \(N\) are such that $$M = 100 a + b \text { and } N = a - 9 b .$$

1. By considering the number \(M + 2 N\), show that \(17 \mid M\) if and only if \(17 \mid N\).
2. Demonstrate step-by-step how an algorithm based on the result of part (i) can be used to show that 2058376813901 is a multiple of 17 .

6 The Fibonacci sequence \(\left\{ F _ { n } \right\}\) is defined by \(F _ { 0 } = 0 , F _ { 1 } = 1\) and \(F _ { n } = F _ { n - 1 } + F _ { n - 2 }\) for all \(n \geqslant 2\).

1. Show that \(F _ { n + 5 } = 5 F _ { n + 1 } + 3 F _ { n }\)
2. Prove that \(F _ { n }\) is a multiple of 5 when \(n\) is a multiple of 5 .

7 The 'parabolic' TV satellite dish in the diagram can be modelled by the surface generated by the rotation of part of a parabola around a vertical \(z\)-axis. The model is represented by part of the surface with equation \(z = \mathrm { f } ( x , y )\) and \(O\) is on the surface. The point \(P\) is on the rim of the dish and directly above the \(x\)-axis.
The object, \(B\), modelled as a point on the \(z\)-axis is the receiving box which collects the TV signals reflected by the dish.
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1. The horizontal plane \(\Pi _ { 1 }\), containing the point \(P\), intersects the surface of the model in a contour of the surface.
   (a) Sketch this contour in the Printed Answer Booklet.
   (b) State a suitable equation for this contour.
2. A second plane, \(\Pi _ { 2 }\), containing both \(P\) and the \(z\)-axis, intersects the surface of the model in a section of the surface.
   (a) Sketch this section in the Printed Answer Booklet.
   (b) State a suitable equation for this section.
3. A proposed equation for the surface is \(z = a x ^ { 2 } + b y ^ { 2 }\). What can you say about the constants \(a\) and \(b\) within this equation? Justify your answers.
4. The real TV satellite dish has the following measurements (in metres): the height of \(P\) above \(O\) is 0.065 and the perimeter of the rim is 2.652 . Using this information, calculate correct to three decimal places the values of
   • \(a\) and \(b\),
   • any other constants stated within the answers to parts (i)(b) and (ii)(b).
   • Incoming satellite signals arrive at the dish in linear "beams" travelling parallel to the \(z\)-axis. They are then 'bounced' off the dish to the receiving box at \(B\).
   • On the diagram for part (ii)(a) in the Printed Answer Booklet draw some of these beams and mark \(B\).
   • If the values of \(a\) and \(b\) were changed, what would happen?