OCR MEI FP3 (Further Pure Mathematics 3) 2006 June

Question 1
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1 Four points have coordinates \(\mathrm { A } ( - 2 , - 3,2 ) , \mathrm { B } ( - 3,1,5 ) , \mathrm { C } ( k , 5 , - 2 )\) and \(\mathrm { D } ( 0,9 , k )\).
  1. Find the vector product \(\overrightarrow { \mathrm { AB } } \times \overrightarrow { \mathrm { CD } }\).
  2. For the case when AB is parallel to CD ,
    (A) state the value of \(k\),
    (B) find the shortest distance between the parallel lines AB and CD ,
    (C) find, in the form \(a x + b y + c z + d = 0\), the equation of the plane containing AB and CD .
  3. When AB is not parallel to CD , find the shortest distance between the lines AB and CD , in terms of \(k\).
  4. Find the value of \(k\) for which the line AB intersects the line CD , and find the coordinates of the point of intersection in this case.
Question 2
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2 A surface has equation \(x ^ { 2 } - 4 x y + 3 y ^ { 2 } - 2 z ^ { 2 } - 63 = 0\).
  1. Find a normal vector at the point \(( x , y , z )\) on the surface.
  2. Find the equation of the tangent plane to the surface at the point \(\mathrm { Q } ( 17,4,1 )\).
  3. The point \(( 17 + h , 4 + p , 1 - h )\), where \(h\) and \(p\) are small, is on the surface and is close to Q . Find an approximate expression for \(p\) in terms of \(h\).
  4. Show that there is no point on the surface where the normal line is parallel to the \(z\)-axis.
  5. Find the two values of \(k\) for which \(5 x - 6 y + 2 z = k\) is a tangent plane to the surface.
Question 3
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3 The curve \(C\) has parametric equations \(x = 2 t ^ { 3 } - 6 t , y = 6 t ^ { 2 }\).
  1. Find the length of the arc of \(C\) for which \(0 \leqslant t \leqslant 1\).
  2. Find the area of the surface generated when the arc of \(C\) for which \(0 \leqslant t \leqslant 1\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
  3. Show that the equation of the normal to \(C\) at the point with parameter \(t\) is $$y = \frac { 1 } { 2 } \left( \frac { 1 } { t } - t \right) x + 2 t ^ { 2 } + t ^ { 4 } + 3$$
  4. Find the cartesian equation of the envelope of the normals to \(C\).
  5. The point \(\mathrm { P } ( 64 , a )\) is the centre of curvature corresponding to a point on \(C\). Find \(a\).
Question 4
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\(\mathbf { 4 }\) The group \(G\) consists of the 8 complex matrices \(\{ \mathbf { I } , \mathbf { J } , \mathbf { K } , \mathbf { L } , - \mathbf { I } , - \mathbf { J } , - \mathbf { K } , - \mathbf { L } \}\) under matrix multiplication, where $$\mathbf { I } = \left( \begin{array} { l l } 1 & 0
0 & 1 \end{array} \right) , \quad \mathbf { J } = \left( \begin{array} { r r } \mathrm { j } & 0
0 & - \mathrm { j } \end{array} \right) , \quad \mathbf { K } = \left( \begin{array} { r r } 0 & 1
- 1 & 0 \end{array} \right) , \quad \mathbf { L } = \left( \begin{array} { c c } 0 & \mathrm { j }
\mathrm { j } & 0 \end{array} \right)$$
  1. Copy and complete the following composition table for \(G\).
    IJKL-I-J-K\(- \mathbf { L }\)
    IIJKL-I-J-K-L
    JJ-IL-K-JI-LK
    KK-L-I
    LLK
    -I-I-J
    -J-JI
    -K-KL
    -L-L-K
    (Note that \(\mathbf { J K } = \mathbf { L }\) and \(\mathbf { K J } = - \mathbf { L }\).)
  2. State the inverse of each element of \(G\).
  3. Find the order of each element of \(G\).
  4. Explain why, if \(G\) has a subgroup of order 4, that subgroup must be cyclic.
  5. Find all the proper subgroups of \(G\).
  6. Show that \(G\) is not isomorphic to the group of symmetries of a square.
Question 5
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5 A local hockey league has three divisions. Each team in the league plays in a division for a year. In the following year a team might play in the same division again, or it might move up or down one division. This question is about the progress of one particular team in the league. In 2007 this team will be playing in either Division 1 or Division 2. Because of its present position, the probability that it will be playing in Division 1 is 0.6 , and the probability that it will be playing in Division 2 is 0.4 . The following transition probabilities apply to this team from 2007 onwards.
  • When the team is playing in Division 1, the probability that it will play in Division 2 in the following year is 0.2 .
  • When the team is playing in Division 2, the probability that it will play in Division 1 in the following year is 0.1 , and the probability that it will play in Division 3 in the following year is 0.3 .
  • When the team is playing in Division 3, the probability that it will play in Division 2 in the following year is 0.15 .
This process is modelled as a Markov chain with three states corresponding to the three divisions.
  1. Write down the transition matrix.
  2. Determine in which division the team is most likely to be playing in 2014.
  3. Find the equilibrium probabilities for each division for this team. In 2015 the rules of the league are changed. A team playing in Division 3 might now be dropped from the league in the following year. Once dropped, a team does not play in the league again.
    -The transition probabilities from Divisions 1 and 2 remain the same as before.
    • When the team is playing in Division 3, the probability that it will play in Division 2 in the following year is 0.15 , and the probability that it will be dropped from the league is 0.1 .
    The team plays in Division 2 in 2015.
    The new situation is modelled as a Markov chain with four states: 'Division1', 'Division 2', 'Division 3' and 'Out of league'.
  4. Write down the transition matrix which applies from 2015.
  5. Find the probability that the team is still playing in the league in 2020.
  6. Find the first year for which the probability that the team is out of the league is greater than 0.5 .