Questions Further Additional Pure (91 questions)

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OCR Further Additional Pure 2018 September Q1
5 marks Standard +0.8
  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]
OCR Further Additional Pure 2018 September Q2
10 marks Challenging +1.8
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]
OCR Further Additional Pure 2018 September Q3
11 marks Standard +0.8
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. 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]
OCR Further Additional Pure 2018 September Q4
12 marks Challenging +1.2
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. 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]
    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]
OCR Further Additional Pure 2018 September Q5
11 marks Hard +2.3
  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]
OCR Further Additional Pure 2018 September Q6
12 marks Hard +2.3
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. 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]
OCR Further Additional Pure 2018 September Q7
14 marks Challenging +1.8
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]
OCR Further Additional Pure 2017 Specimen Q1
4 marks Challenging +1.2
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]
OCR Further Additional Pure 2017 Specimen Q2
3 marks Moderate -0.5
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]
OCR Further Additional Pure 2017 Specimen Q3
5 marks Standard +0.8
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]
OCR Further Additional Pure 2017 Specimen Q4
6 marks Challenging +1.2
  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]
OCR Further Additional Pure 2017 Specimen Q5
9 marks Challenging +1.2
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]
OCR Further Additional Pure 2017 Specimen Q6
10 marks Challenging +1.2
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]
OCR Further Additional Pure 2017 Specimen Q7
11 marks Standard +0.3
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. 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]
OCR Further Additional Pure 2017 Specimen Q8
13 marks Challenging +1.8
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. 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]
OCR Further Additional Pure 2017 Specimen Q9
14 marks Hard +2.3
    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]