OCR Further Additional Pure (Further Additional Pure) 2017 Specimen

1 A curve is given by \(x = t ^ { 2 } - 2 \ln t , y = 4 t\) for \(t > 0\). When the arc of the curve between the points where \(t = 1\) and \(t = 4\) is rotated through \(2 \pi\) radians about the \(x\)-axis, a surface of revolution is formed with surface area \(A\).
Given that \(A = k \pi\), where \(k\) is an integer, write down an integral which gives \(A\) and find the value of \(k\).

5 In this question you must show detailed reasoning.
It is given that \(I _ { n } = \int _ { 0 } ^ { \pi } \sin ^ { n } \theta \mathrm {~d} \theta\) for \(n \geq 0\).

1. Prove that \(I _ { n } = \frac { n - 1 } { n } I _ { n - 2 }\) for \(n \geq 2\).
2. Evaluate \(I _ { 1 }\) and use the reduction formula to determine the exact value of \(\int _ { 0 } ^ { \pi } \cos ^ { 2 } \theta \sin ^ { 5 } \theta \mathrm {~d} \theta\).

6 A surface \(S\) has equation \(z = \mathrm { f } ( x , y )\), where \(\mathrm { f } ( x , y ) = 2 x ^ { 2 } - y ^ { 2 } + 3 x y + 17 y\). It is given that \(S\) has a single stationary point, \(P\).

1. Determine the coordinates, and the nature, of \(P\).
2. Find the equation of the tangent plane to \(S\) at the point \(Q ( 1,2,38 )\).

7 In order to rescue them from extinction, a particular species of ground-nesting birds is introduced into a nature reserve. The number of breeding pairs of these birds in the nature reserve, \(t\) years after their introduction, is denoted by \(N _ { t }\). The initial number of breeding pairs is given by \(N _ { 0 }\). An initial discrete population model is proposed for \(N _ { t }\). $$\text { Model I: } N _ { t + 1 } = \frac { 6 } { 5 } N _ { t } \left( 1 - \frac { 1 } { 900 } N _ { t } \right)$$

1. (a) For Model I, show that the steady state values of the number of breeding pairs are 0 and 150 .
   (b) Show that \(N _ { t + 1 } - N _ { t } < 150 - N _ { t }\) when \(N _ { t }\) lies between 0 and 150 .
   (c) Hence determine the long-term behaviour of the number of breeding pairs of this species of birds in the nature reserve predicted by Model I when \(N _ { 0 } \in ( 0,150 )\). An alternative discrete population model is proposed for \(N _ { t }\). $$\text { Model II: } N _ { t + 1 } = \operatorname { INT } \left( \frac { 6 } { 5 } N _ { t } \left( 1 - \frac { 1 } { 900 } N _ { t } \right) \right)$$
2. Given that \(N _ { 0 } = 8\), find the value of \(N _ { 4 }\) for each of the two models and give a reason why Model II may be considered more suitable.

8 The set \(X\) consists of all \(2 \times 2\) matrices of the form \(\left( \begin{array} { r r } x & - y
y & x \end{array} \right)\), where \(x\) and \(y\) are real numbers which are not both zero.

1. (a) The matrices \(\left( \begin{array} { c c } a & - b
   b & a \end{array} \right)\) and \(\left( \begin{array} { c c } c & - d
   d & c \end{array} \right)\) are both elements of \(X\). Show that \(\left( \begin{array} { c c } a & - b
   b & a \end{array} \right) \left( \begin{array} { c c } c & - d
   d & c \end{array} \right) = \left( \begin{array} { c c } p & - q
   q & p \end{array} \right)\) for some real numbers \(p\) and \(q\) to be found in terms of \(a , b , c\) and \(d\).
   (b) Prove by contradiction that \(p\) and \(q\) are not both zero.
2. Prove that \(X\), under matrix multiplication, forms a group \(G\).
   [0pt] [You may use the result that matrix multiplication is associative.]
3. Determine a subgroup of \(G\) of order 17.