Questions Further Extra Pure (35 questions)

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OCR MEI Further Extra Pure 2021 November Q1
1 In this question you must show detailed reasoning.
A surface \(S\) is defined by \(z = f ( x , y )\) where \(f ( x , y ) = x ^ { 3 } + x ^ { 2 } y - 2 y ^ { 2 }\).
  1. On the coordinate axes in the Printed Answer Booklet, sketch the section \(z = f ( 2 , y )\) giving the coordinates of any turning points and any points of intersection with the axes.
  2. Find the stationary points on \(S\).
    \(2 G\) is a group of order 8.
  3. Explain why there is no subgroup of \(G\) of order 6 . 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.
    • Write down the possible generators of \(H\).
      \(M\) 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 \(G\).
    • Specify all possible isomorphisms between \(M\) and \(G\).
OCR MEI Further Extra Pure 2021 November Q3
3 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { l l l } 3 & 3 & 0
0 & 2 & 2
1 & 3 & 4 \end{array} \right)\).
  1. Determine the characteristic equation of \(\mathbf { A }\).
  2. Hence verify that the eigenvalues of \(\mathbf { A }\) are 1, 2 and 6 .
  3. For each eigenvalue of \(\mathbf { A }\) determine an associated eigenvector.
  4. Use the results of parts (b) and (c) to find \(\mathbf { A } ^ { n }\) as a single matrix, where \(n\) is a positive integer.
OCR MEI Further Extra Pure 2021 November Q4
4 The sequence \(u _ { 0 } , u _ { 1 } , u _ { 2 } , \ldots\) satisfies the recurrence relation \(u _ { n + 2 } - 3 u _ { n + 1 } - 10 u _ { n } = 24 n - 10\).
  1. Determine the general solution of the recurrence relation.
  2. Hence determine the particular solution of the recurrence relation for which \(u _ { 0 } = 6\) and \(u _ { 1 } = 10\).
  3. Show, by direct calculation, that your solution in part (b) gives the correct value for \(u _ { 2 }\). 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). particular solution found in part (b). You are given that \(\mathrm { v } _ { \mathrm { n } }\) converges to a finite non-zero limit, \(q\), as \(n \rightarrow \infty\).
  4. Determine \(p\) and \(q\).
OCR MEI Further Extra Pure 2021 November Q5
5 A surface \(S\) is defined for \(z \geqslant 0\) by \(x ^ { 2 } + y ^ { 2 } + 2 z ^ { 2 } = 126\). \(C\) 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 \(\frac { 1 } { 3 } \pi\) radians. Show that \(C\) lies in a plane, \(\Pi\), whose equation should be determined.
OCR MEI Further Extra Pure 2021 November Q6
6 You are given that \(q \in \mathbb { Z }\) with \(q \geqslant 1\) and that
\(\mathrm { S } = \frac { 1 } { ( \mathrm { q } + 1 ) } + \frac { 1 } { ( \mathrm { q } + 1 ) ( \mathrm { q } + 2 ) } + \frac { 1 } { ( \mathrm { q } + 1 ) ( \mathrm { q } + 2 ) ( \mathrm { q } + 3 ) } + \ldots\).
  1. By considering a suitable geometric series show that \(\mathrm { S } < \frac { 1 } { \mathrm { q } }\).
  2. Deduce that \(S \notin \mathbb { Z }\). You are also given that \(\mathrm { e } = \sum _ { r = 0 } ^ { \infty } \frac { 1 } { r ! }\).
  3. Assume that \(\mathrm { e } = \frac { \mathrm { p } } { \mathrm { q } }\), where \(p\) and \(q\) are positive integers. By writing the infinite series for e in a form using \(q\) and \(S\) and using the result from part (b), prove by contradiction that e is irrational.
OCR MEI Further Extra Pure Specimen Q1
1 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 .
  2. Find another generator for \(G\). Justify your answer.
  3. Specify two distinct isomorphisms from the group \(J = \{ 0,1,2,3,4,5,6,7,8 \}\) under addition modulo 9 to \(G\).
OCR MEI Further Extra Pure Specimen Q2
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.
  1. Find the general solution of $$u _ { n } = 8 u _ { n - 1 } - 16 u _ { n - 2 } , n \geq 2 .$$ A new sequence \(v _ { n }\) is defined by \(v _ { n } = \frac { u _ { n } } { u _ { n - 1 } }\) for \(n \geq 1\).
  2. (A) Use \(( * )\) to show that \(v _ { n } = 8 - \frac { 16 } { v _ { n - 1 } }\). for \(n \geq 2\).
    (B) Deduce that if \(v _ { n }\) tends to a limit then it must be 4 .
  3. Use your general solution in part (i) to show that \(\lim _ { n \rightarrow \infty } v _ { n } = 4\).
  4. Deduce the value of \(\lim _ { n \rightarrow \infty } \left( \frac { u _ { n } } { u _ { n - 2 } } \right)\).
OCR MEI Further Extra Pure Specimen Q4
4 A surface \(S\) has equation \(\mathrm { g } ( x , y , z ) = 0\), where \(\mathrm { g } ( x , y , z ) = ( y - 2 x ) ( y + z ) ^ { 2 } - 18\).
  1. Show that \(\frac { \partial \mathrm { g } } { \partial y } = ( y + z ) ( - 4 x + 3 y + z )\).
  2. Show that \(\frac { \partial \mathrm { g } } { \partial x } + 2 \frac { \partial \mathrm {~g} } { \partial y } - 2 \frac { \partial \mathrm {~g} } { \partial \mathrm { z } } = 0\).
  3. Hence identify a vector which lies in the tangent plane of every point on \(S\), explaining your reasoning.
  4. Find the cartesian equation of the tangent plane to the surface \(S\) at the point \(\mathrm { P } ( 1,4 , - 7 )\). The tangent plane to the surface \(S\) at the point \(\mathrm { Q } ( 0,2,1 )\) has equation \(6 x - 7 y - 4 z = - 18\).
  5. Find a vector equation for the line of intersection of the tangent planes at P and Q .
OCR MEI Further Extra Pure Specimen Q5
5 In this question you must show detailed reasoning. You are given that the matrix \(\mathbf { M } = \left( \begin{array} { c c c } \frac { 1 } { 2 } & - \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { 2 }
\frac { 1 } { \sqrt { 2 } } & 0 & - \frac { 1 } { \sqrt { 2 } }
\frac { 1 } { 2 } & \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { 2 } \end{array} \right)\) represents a rotation in 3-D space.
  1. Explain why it follows that \(\mathbf { M }\) has 1 as an eigenvalue.
  2. Find a vector equation for the axis of the rotation.
  3. Show that the characteristic equation of \(\mathbf { M }\) can be written as $$\lambda ^ { 3 } - \lambda ^ { 2 } + \lambda - 1 = 0 .$$
  4. Find the smallest positive integer \(n\) such that \(\mathbf { M } ^ { n } = \mathbf { I }\).
  5. Find the magnitude of the angle of the rotation which \(\mathbf { M }\) represents. Give your reasoning. {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.
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OCR MEI Further Extra Pure 2020 November Q5
  1. Show that \(\mathbf { f }\) is also an eigenvector of \(\mathbf { A }\).
  2. State the eigenvalue associated with \(\mathbf { f }\). You are now given that \(\mathbf { A }\) represents a reflection in 3-D space.
  3. Explain the significance of \(\mathbf { e }\) and \(\mathbf { f }\) in relation to the transformation that \(\mathbf { A }\) represents.
  4. State the cartesian equation of the plane of reflection of the transformation represented by \(\mathbf { A }\).