OCR Further Additional Pure (Further Additional Pure) 2017 Specimen

A curve is given by \(x = t^2 - 2\ln t\), \(y = 4t\) 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 \(k\). [4]

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

Given \(z = x\sin y + y\cos x\), show that \(\frac{\partial^2 z}{\partial x^2} + \frac{\partial^2 z}{\partial y^2} + z = 0\). [5]

1. Solve the recurrence relation \(u_{n+2} = 4u_{n+1} - 4u_n\) for \(n \geq 0\), given that \(u_0 = 1\) and \(u_1 = 1\). [4]
2. Show that each term of the sequence \(\{u_n\}\) is an integer. [2]

In this question you must show detailed reasoning. It is given that \(I_n = \int_0^\pi \sin^n \theta \, d\theta\) for \(n \geq 0\).

1. Prove that \(I_n = \frac{n-1}{n} I_{n-2}\) for \(n \geq 2\). [5]
2. Evaluate \(I_1\) and use the reduction formula to determine the exact value of \(\int_0^\pi \cos^2 \theta \sin^5 \theta \, d\theta\). [4]

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

1. Determine the coordinates, and the nature, of \(P\). [8]
2. Find the equation of the tangent plane to \(S\) at the point \(Q(1, 2, 38)\). [2]

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 \(N_t\). Model I: \(N_{t+1} = \frac{6}{5}N_t\left(1 - \frac{1}{900}N_t\right)\)

1. 1. For Model I, show that the steady state values of the number of breeding pairs are 0 and 150. [3]
   2. Show that \(N_{t+1} - N_t < 150 - N_t\) when \(N_t\) lies between 0 and 150. [3]
   3. 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 \(N_0 \in (0, 150)\). [2]
   An alternative discrete population model is proposed for \(N_t\). Model II: \(N_{t+1} = \text{INT}\left(\frac{6}{5}N_t\left(1 - \frac{1}{900}N_t\right)\right)\)
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. [3]

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

1. 1. The matrices \(\begin{pmatrix} a & -b \\ b & a \end{pmatrix}\) and \(\begin{pmatrix} c & -d \\ d & c \end{pmatrix}\) are both elements of \(X\). Show that \(\begin{pmatrix} a & -b \\ b & a \end{pmatrix}\begin{pmatrix} c & -d \\ d & c \end{pmatrix} = \begin{pmatrix} p & -q \\ q & p \end{pmatrix}\) for some real numbers \(p\) and \(q\) to be found in terms of \(a\), \(b\), \(c\) and \(d\). [2]
   2. Prove by contradiction that \(p\) and \(q\) are not both zero. [5]
2. Prove that \(X\), under matrix multiplication, forms a group \(G\). [You may use the result that matrix multiplication is associative.] [4]
3. Determine a subgroup of \(G\) of order 17. [2]

1. 1. Prove that \(p \equiv \pm 1 \pmod{6}\) for all primes \(p > 3\). [2]
   2. Hence or otherwise prove that \(p^2 - 1 \equiv 0 \pmod{24}\) for all primes \(p > 3\). [3]
2. Given that \(p\) is an odd prime, determine the residue of \(2^{p^2-1}\) modulo \(p\). [4]
3. Let \(p\) and \(q\) be distinct primes greater than 3. Prove that \(p^{q-1} + q^{p-1} \equiv 1 \pmod{pq}\). [5]