OCR Further Additional Pure (Further Additional Pure) 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 \(k\).

2 Find the volume of tetrahedron OABC , where O is the origin, \(\mathrm { A } = ( 2,3,1 ) , \mathrm { B } = ( - 4,2,5 )\) and \(\mathrm { C } = ( 1,4,4 )\).

3 Given \(z = x \sin y + y \cos x\), show that \(\frac { \partial ^ { 2 } z } { \partial x ^ { 2 } } + \frac { \partial ^ { 2 } z } { \partial y ^ { 2 } } + z = 0\).

4

1. Solve the recurrence relation \(u _ { n + 2 } = 4 u _ { n + 1 } - 4 u _ { n }\) for \(n \geq 0\), given that \(u _ { 0 } = 1\) and \(u _ { 1 } = 1\).
2. Show that each term of the sequence \(\left\{ u _ { n } \right\}\) is an integer.

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. (a) Evaluate \(I _ { 1 }\).
   (b) 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. (a) Determine the coordinates of \(P\).
   (b) Determine 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 an integer 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 find 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. (a) Given that \(N _ { 0 } = 8\), find the value of \(N _ { 4 }\) for each of the two models.
   (b) Which of the two models gives values for \(N _ { t }\) with the more appropriate level of precision?

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 \(G\). [You may use the result that matrix multiplication is associative.]
3. Determine a subgroup of \(G\) of order 17.

9

1. (a) Prove that \(p \equiv \pm 1 ( \bmod 6 )\) for all primes \(p > 3\).
   (b) Hence or otherwise prove that \(p ^ { 2 } - 1 \equiv 0 ( \bmod 24 )\) for all primes \(p > 3\).
2. Given that \(p\) is an odd prime, determine the residue of \(2 ^ { p ^ { 2 } - 1 }\) modulo \(p\).
3. Let \(p\) and \(q\) be distinct primes greater than 3 . Prove that \(p ^ { q - 1 } + q ^ { p - 1 } \equiv 1 ( \bmod p q )\). \section*{END OF QUESTION PAPER} {www.ocr.org.uk}) after the live examination series.
   If OCR has unwittingly failed to correctly acknowledge or clear any third-party content in this assessment material, OCR will be happy to correct its mistake at the earliest possible opportunity. For queries or further information please contact the Copyright Team, First Floor, 9 Hills Road, Cambridge CB2 1GE.
   OCR is part of the Cambridge Assessment Group; Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }