Questions — OCR MEI Further Extra Pure (38 questions)

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OCR MEI Further Extra Pure 2019 June Q5
15 marks Standard +0.8
A financial institution models the repayment of a loan to a client in the following way.
  • An amount, \(£C\), is loaned to the client at the start of the repayment period.
  • The amount owed \(n\) years after the start of the repayment period is \(£L_n\), so that \(L_0 = C\).
  • At the end of each year, interest of \(\alpha\%\) (\(\alpha > 0\)) of the amount owed at the start of that year is added to the amount owed.
  • Immediately after interest has been added to the amount owed a repayment of \(£R\) is made by the client.
  • Once \(L_n\) becomes negative the repayment is finished and the overpayment is refunded to the client.
  1. Show that during the repayment period, \(L_{n+1} = aL_n + b\), giving \(a\) and \(b\) in terms of \(\alpha\) and \(R\). [2]
  2. Find the solution of the recurrence relation \(L_{n+1} = aL_n + b\) with \(L_0 = C\), giving your solution in terms of \(a\), \(b\), \(C\) and \(n\). [5]
  3. Deduce from parts (a) and (b) that, for the repayment scheme to terminate, \(R > \frac{\alpha C}{100}\). [2]
A client takes out a £30000 loan at 8% interest and agrees to repay £3000 at the end of each year.
    1. Use an algebraic method to find the number of years it will take for the loan to be repaid. [3]
    2. Taking into account the refund of overpayment, find the total amount that the client repays over the lifetime of the loan. [3]
OCR MEI Further Extra Pure 2019 June Q6
13 marks Challenging +1.8
  1. Given that \(\sqrt{7}\) is an irrational number, prove that \(a^2 - 7b^2 \neq 0\) for all \(a, b \in \mathbb{Q}\) where \(a\) and \(b\) are not both 0. [2]
  2. A set \(G\) is defined by \(G = \{a + b\sqrt{7} : a, b \in \mathbb{Q}, a\) and \(b\) not both 0\(\}\). Prove that \(G\) is a group under multiplication. (You may assume that multiplication is associative.) [7]
  3. A subset \(H\) of \(G\) is defined by \(H = \{1 + c\sqrt{7} : c \in \mathbb{Q}\}\). Determine whether or not \(H\) is a subgroup of \((G, \times)\). [2]
  4. Using \((G, \times)\), prove by counter-example that the statement 'An infinite group cannot have a non-trivial subgroup of finite order' is false. [2]
OCR MEI Further Extra Pure 2021 November Q1
11 marks Challenging +1.2
In this question you must show detailed reasoning. A surface \(S\) is defined by \(z = \mathrm{f}(x, y)\) where \(\mathrm{f}(x, y) = x^3 + x^2 y - 2y^2\).
  1. On the coordinate axes in the Printed Answer Booklet, sketch the section \(z = \mathrm{f}(2, y)\) giving the coordinates of any turning points and any points of intersection with the axes. [4]
  2. Find the stationary points on \(S\). [7]
OCR MEI Further Extra Pure 2021 November Q2
7 marks Challenging +1.2
\(G\) is a group of order 8.
  1. Explain why there is no subgroup of \(G\) of order 6. [1]
You are now given that \(G\) is a cyclic group with the following features: • \(e\) is the identity element of \(G\), • \(g\) is a generator of \(G\), • \(H\) is the subgroup of \(G\) of order 4.
  1. Write down the possible generators of \(H\). [2]
\(M\) is the group \((\{0, 1, 2, 3, 4, 5, 6, 7\}, +_8)\) where \(+_8\) denotes the binary operation of addition modulo 8. You are given that \(M\) is isomorphic to \(G\).
  1. Specify all possible isomorphisms between \(M\) and \(G\). [4]
OCR MEI Further Extra Pure 2021 November Q3
14 marks Challenging +1.3
The matrix \(\mathbf{A}\) is given by \(\mathbf{A} = \begin{pmatrix} 3 & 3 & 0 \\ 0 & 2 & 2 \\ 1 & 3 & 4 \end{pmatrix}\).
  1. Determine the characteristic equation of \(\mathbf{A}\). [3]
  2. Hence verify that the eigenvalues of \(\mathbf{A}\) are 1, 2 and 6. [1]
  3. For each eigenvalue of \(\mathbf{A}\) determine an associated eigenvector. [4]
  4. Use the results of parts (b) and (c) to find \(\mathbf{A}^n\) as a single matrix, where \(n\) is a positive integer. [6]
OCR MEI Further Extra Pure 2021 November Q4
14 marks Challenging +1.3
The sequence \(u_0, u_1, u_2, \ldots\) satisfies the recurrence relation \(u_{n+2} - 3u_{n+1} - 10u_n = 24n - 10\).
  1. Determine the general solution of the recurrence relation. [6]
  2. Hence determine the particular solution of the recurrence relation for which \(u_0 = 6\) and \(u_1 = 10\). [3]
  3. Show, by direct calculation, that your solution in part (b) gives the correct value for \(u_2\). [1]
The sequence \(v_0, v_1, v_2, \ldots\) is defined by \(v_n = \frac{u_n}{p^n}\) for some constant \(p\), where \(u_n\) denotes the particular solution found in part (b). You are given that \(v_n\) converges to a finite non-zero limit, \(q\), as \(n \to \infty\).
  1. Determine \(p\) and \(q\). [4]
OCR MEI Further Extra Pure 2021 November Q5
6 marks Challenging +1.8
A surface \(S\) is defined for \(z \geqslant 0\) by \(x^2 + y^2 + 2z^2 = 126\). \(C\) is the set of points on \(S\) for which the tangent plane to \(S\) at that point intersects the \(x\)-\(y\) plane at an angle of \(\frac{1}{4}\pi\) radians. Show that \(C\) lies in a plane, \(\Pi\), whose equation should be determined. [6]
OCR MEI Further Extra Pure 2021 November Q6
8 marks Challenging +1.8
You are given that \(q \in \mathbb{Z}\) with \(q \geqslant 1\) and that $$S = \frac{1}{(q+1)} + \frac{1}{(q+1)(q+2)} + \frac{1}{(q+1)(q+2)(q+3)} + \cdots$$
  1. By considering a suitable geometric series show that \(S < \frac{1}{q}\). [3]
  2. Deduce that \(S \notin \mathbb{Z}\). [2]
You are also given that \(\mathrm{e} = \sum_{r=0}^{\infty} \frac{1}{r!}\).
  1. Assume that \(\mathrm{e} = \frac{p}{q}\), where \(p\) and \(q\) are positive integers. By writing the infinite series for \(\mathrm{e}\) in a form using \(q\) and \(S\) and using the result from part (b), prove by contradiction that \(\mathrm{e}\) is irrational. [3]
OCR MEI Further Extra Pure Specimen Q1
10 marks Challenging +1.8
The set \(G = \{1, 4, 5, 6, 7, 9, 11, 16, 17\}\) is a group of order 9 under the binary operation of multiplication modulo 19.
  1. Show that \(G\) is a cyclic group generated by the element 4. [3]
  2. Find another generator for \(G\). Justify your answer. [2]
  3. Specify two distinct isomorphisms from the group \(J = \{0, 1, 2, 3, 4, 5, 6, 7, 8\}\) under addition modulo 9 to \(G\). [5]
OCR MEI Further Extra Pure Specimen Q2
4 marks Challenging +1.2
A binary operation \(*\) is defined on the set \(S = \{p, q, r, s, t\}\) by the following composition table.
\(*\)\(p\)\(q\)\(r\)\(s\)\(t\)
\(p\)\(p\)\(q\)\(r\)\(s\)\(t\)
\(q\)\(q\)\(p\)\(s\)\(t\)\(r\)
\(r\)\(r\)\(t\)\(p\)\(q\)\(s\)
\(s\)\(s\)\(r\)\(t\)\(p\)\(q\)
\(t\)\(t\)\(s\)\(q\)\(r\)\(p\)
Determine whether \((S, *)\) is a group. [4]
OCR MEI Further Extra Pure Specimen Q3
12 marks Challenging +1.2
  1. Find the general solution of $$u_n = 8u_{n-1} - 16u_{n-2}, \quad n \geq 2. \quad (*)$$ [4]
A new sequence \(v_n\) is defined by \(v_n = \frac{u_n}{u_{n-1}}\) for \(n \geq 1\).
  1. (A) Use (*) to show that \(v_n = 8 - \frac{16}{v_{n-1}}\) for \(n \geq 2\). [2] (B) Deduce that if \(v_n\) tends to a limit then it must be 4. [2]
  2. Use your general solution in part (i) to show that \(\lim_{n \to \infty} v_n = 4\). [3]
  3. Deduce the value of \(\lim_{n \to \infty} \left(\frac{u_n}{u_{n-2}}\right)\). [1]
OCR MEI Further Extra Pure Specimen Q4
16 marks Challenging +1.8
A surface \(S\) has equation \(g(x, y, z) = 0\), where \(g(x, y, z) = (y - 2x)(y + z)^2 - 18\).
  1. Show that \(\frac{\partial g}{\partial y} = (y + z)(-4x + 3y + z)\). [2]
  2. Show that \(\frac{\partial g}{\partial x} + 2\frac{\partial g}{\partial y} - 2\frac{\partial g}{\partial z} = 0\). [4]
  3. Hence identify a vector which lies in the tangent plane of every point on \(S\), explaining your reasoning. [3]
  4. Find the cartesian equation of the tangent plane to the surface \(S\) at the point P\((1, 4, -7)\). [3]
The tangent plane to the surface \(S\) at the point Q\((0, 2, 1)\) has equation \(6x - 7y - 4z = -18\).
  1. Find a vector equation for the line of intersection of the tangent planes at P and Q. [4]
OCR MEI Further Extra Pure Specimen Q5
18 marks Challenging +1.8
In this question you must show detailed reasoning. You are given that the matrix $\mathbf{M} = \begin{pmatrix} \frac{1}{2} & -\frac{1}{\sqrt{2}} & \frac{1}{2}
\frac{1}{\sqrt{2}} & 0 & -\frac{1}{\sqrt{2}}
\frac{1}{2} & \frac{1}{\sqrt{2}} & \frac{1}{2} \end{pmatrix}$ represents a rotation in 3-D space.
  1. Explain why it follows that \(\mathbf{M}\) has 1 as an eigenvalue. [2]
  2. Find a vector equation for the axis of the rotation. [4]
  3. Show that the characteristic equation of \(\mathbf{M}\) can be written as $$\lambda^3 - \lambda^2 + \lambda - 1 = 0.$$ [5]
  4. Find the smallest positive integer \(n\) such that \(\mathbf{M}^n = \mathbf{I}\). [6]
  5. Find the magnitude of the angle of the rotation which \(\mathbf{M}\) represents. Give your reasoning. [1]