OCR Further Additional Pure (Further Additional Pure) 2018 September

1. Write the number \(100011_n\), where \(n \geq 2\), as a polynomial in \(n\). [1]
2. Show that \(n^2 + n + 1\) is a factor of this expression. [2]
3. Hence show that \(100011_n\) is composite in any number base \(n \geq 2\). [2]

In this question, you must show detailed reasoning. A curve is defined parametrically by \(x = t^3 - 3t + 1\), \(y = 3t^2 - 1\), for \(0 \leq t \leq 5\). Find, in exact form,

1. the length of the curve, [6]
2. the area of the surface generated when the curve is rotated completely about the \(x\)-axis. [4]

The function \(w = f(x, y, z)\) is given by \(f(x, y, z) = x^2yz + 2xy^2z + 3xyz^2 - 24xyz\), for \(x, y, z \neq 0\).

1. 1. Find
   2. Hence find the values of \(a\), \(b\), \(c\) and \(d\) for which \(w\) has a stationary value when \(d = f(a, b, c)\). [5]
2. You are given that this stationary value is a local minimum of \(w\). Find values of \(x\), \(y\) and \(z\) which show that it is not a global minimum of \(w\). [2]

The points \(A\), \(B\), \(C\) and \(P\) have coordinates \((a, 0, 0)\), \((0, b, 0)\), \((0, 0, c)\) and \((a, b, c)\) respectively, where \(a\), \(b\) and \(c\) are positive constants. The plane \(\Pi\) contains \(A\), \(B\) and \(C\).

1. 1. Use the scalar triple product to determine
   2. Hence show that the distance from \(P\) to \(\Pi\) is twice the distance from \(O\) to \(\Pi\). [2]
2. 1. Determine a vector which is normal to \(\Pi\). [2]
   2. Hence determine, in terms of \(a\), \(b\) and \(c\) only, the distance from \(P\) to \(\Pi\). [3]

1. You are given that \(N = \binom{p-1}{r}\), where \(p\) is a prime number and \(r\) is an integer such that \(1 \leq r \leq p - 1\). By considering the number \(N \times r!\), prove that \(N \equiv (-1)^r \pmod{p}\). [5]
2. You are given that \(M = \binom{2p}{p}\), where \(p\) is an odd prime number. Prove that \(M \equiv 2 \pmod{p}\). [6]

A class of students is set the task of finding a group of functions, under composition of functions, of order 6. Student P suggests that this can be achieved by finding a function \(f\) for which \(f^6(x) = x\) and using this as a generator for the group.

1. Explain why the suggestion by Student P might not work. [2]

Student Q observes that their class has already found a group of order 6 in a previous task; a group consisting of the powers of a particular, non-singular \(2 \times 2\) real matrix \(\mathbf{M} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\), under the operation of matrix multiplication.

1. Explain why such a group is only possible if \(\det(\mathbf{M}) = 1\) or \(-1\). [2]
2. Write down values of \(a\), \(b\), \(c\) and \(d\) that would give a suitable matrix \(\mathbf{M}\) for which \(\mathbf{M}^6 = \mathbf{I}\) and \(\det(\mathbf{M}) = 1\). [1]

Student Q believes that it is possible to construct a rational function \(f\) in the form \(f(x) = \frac{ax + b}{cx + d}\) so that the group of functions is isomorphic to the matrix group which is generated by the matrix \(\mathbf{M}\) of part (iii).

1. 1. Write down and simplify the function \(f\) that, according to Student Q, corresponds to \(\mathbf{M}\). [1]
   2. By calculating \(\mathbf{M}^2\), show that Student Q's suggestion does not work. [2]
   3. Find a different function \(f\) that will satisfy the requirements of the task. [4]

The members of the family of the sequences \(\{u_n\}\) satisfy the recurrence relation $$u_{n+1} = 10u_n - u_{n-1} \text{ for } n \geq 1. \quad (*)$$

1. Determine the general solution of (*). [3]
2. The sequences \(\{a_n\}\) and \(\{b_n\}\) are members of this family of sequences, corresponding to the initial terms \(a_0 = 1\), \(a_1 = 5\) and \(b_0 = 0\), \(b_1 = 2\) respectively.
   1. Find the next two terms of each sequence. [1]
   2. Prove that, for all non-negative integers \(n\), \((a_n)^2 - 6(b_n)^2 = 1\). [8]
   3. Determine \(\lim_{n \to \infty} \frac{a_n}{b_n}\). [2]