Questions — OCR MEI FP3 (53 questions)

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OCR MEI FP3 2014 June Q3
3
  1. A curve has intrinsic equation \(s = 2 \ln \left( \frac { \pi } { \pi - 3 \psi } \right)\) for \(0 \leqslant \psi < \frac { 1 } { 3 } \pi\), where \(s\) is the arc length measured from a fixed point P and \(\tan \psi = \frac { \mathrm { d } y } { \mathrm {~d} x } . \mathrm { P }\) is in the third quadrant. The curve passes through the origin O , at which point \(\psi = \frac { 1 } { 6 } \pi . \mathrm { Q }\) is the point on the curve at which \(\psi = \frac { 3 } { 10 } \pi\).
    1. Express \(\psi\) in terms of \(s\), and sketch the curve, indicating the points \(\mathrm { O } , \mathrm { P }\) and Q .
    2. Find the arc length OQ .
    3. Find the radius of curvature at the point O .
    4. Find the coordinates of the centre of curvature corresponding to the point O .
    1. Find the surface area of revolution formed when the curve \(y = \frac { 1 } { 3 } \sqrt { x } ( x - 3 )\) for \(1 \leqslant x \leqslant 4\) is rotated through \(2 \pi\) radians about the \(y\)-axis.
    2. The curve in part (b)(i) is one member of the family \(y = \frac { 1 } { 9 } \lambda \sqrt { x } ( x - \lambda )\), where \(\lambda\) is a positive parameter. Find the equation of the envelope of this family of curves.
OCR MEI FP3 2014 June Q4
4 The twelve distinct elements of an abelian multiplicative group \(G\) are $$e , a , a ^ { 2 } , a ^ { 3 } , a ^ { 4 } , a ^ { 5 } , b , a b , a ^ { 2 } b , a ^ { 3 } b , a ^ { 4 } b , a ^ { 5 } b$$ where \(e\) is the identity element, \(a ^ { 6 } = e\) and \(b ^ { 2 } = e\).
  1. Show that the element \(a ^ { 2 } b\) has order 6 .
  2. Show that \(\left\{ e , a ^ { 3 } , b , a ^ { 3 } b \right\}\) is a subgroup of \(G\).
  3. List all the cyclic subgroups of \(G\). You are given that the set $$H = \{ 1,7,11,13,17,19,23,29,31,37,41,43,47,49,53,59,61,67,71,73,77,79,83,89 \}$$ with binary operation multiplication modulo 90 is a group.
  4. Determine the order of each of the elements 11, 17 and 19 .
  5. Give a cyclic subgroup of \(H\) with order 4.
  6. By identifying possible values for the elements \(a\) and \(b\) above, or otherwise, give one example of each of the following:
    (A) a non-cyclic subgroup of \(H\) with order 12,
    (B) a non-cyclic subgroup of \(H\) with order 4.
OCR MEI FP3 2014 June Q5
5 In this question, give probabilities correct to 4 decimal places.
The speeds of vehicles are measured on a busy stretch of road and are categorised as A (not more than 30 mph ), B (more than 30 mph but not more than 40 mph ) or C (more than 40 mph ).
  • Following a vehicle in category A , the probabilities that the next vehicle is in categories \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) are \(0.9,0.07,0.03\) respectively.
  • Following a vehicle in category B , the probabilities that the next vehicle is in categories \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) are \(0.3,0.6,0.1\) respectively.
  • Following a vehicle in category C , the probabilities that the next vehicle is in categories \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) are \(0.1,0.7,0.2\) respectively.
This is modelled as a Markov chain with three states corresponding to the categories A, B, C. The speed of the first vehicle is measured as 28 mph .
  1. Write down the transition matrix \(\mathbf { P }\).
  2. Find the probabilities that the 10th vehicle is in each of the three categories.
  3. Find the probability that the 12th and 13th vehicles are in the same category.
  4. Find the smallest value of \(n\) for which the probability that the \(n\)th and \(( n + 1 )\) th vehicles are in the same category is less than 0.8, and give the value of this probability.
  5. Find the expected number of vehicles (including the first vehicle) in category A before a vehicle in a different category.
  6. Find the limit of \(\mathbf { P } ^ { n }\) as \(n\) tends to infinity, and hence write down the equilibrium probabilities for the three categories.
  7. Find the probability that, after many vehicles have passed by, the next three vehicles are all in category A. On a new stretch of road, the same categories are used but some of the transition probabilities are different.
    • Following a vehicle in category A , the probability that the next vehicle is in category B is equal to the probability that it is in category C .
    • Following a vehicle in category B , the probability that the next vehicle is in category A is equal to the probability that it is in category C .
    • Following a vehicle in category C , the probabilities that the next vehicle is in categories \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) are \(0.1,0.7,0.2\) respectively.
    In the long run, the proportions of vehicles in categories A, B, C are 50\%, 40\%, 10\% respectively.
  8. Find the transition matrix for the new stretch of road.
OCR MEI FP3 2009 June Q1
1 The point \(\mathrm { A } ( - 1,12,5 )\) lies on the plane \(P\) with equation \(8 x - 3 y + 10 z = 6\). The point \(\mathrm { B } ( 6 , - 2,9 )\) lies on the plane \(Q\) with equation \(3 x - 4 y - 2 z = 8\). The planes \(P\) and \(Q\) intersect in the line \(L\).
  1. Find an equation for the line \(L\).
  2. Find the shortest distance between \(L\) and the line AB . The lines \(M\) and \(N\) are both parallel to \(L\), with \(M\) passing through A and \(N\) passing through B .
  3. Find the distance between the parallel lines \(M\) and \(N\). The point C has coordinates \(( k , 0,2 )\), and the line AC intersects the line \(N\) at the point D .
  4. Find the value of \(k\), and the coordinates of D .
OCR MEI FP3 2009 June Q2
2 A surface has equation \(z = 3 x ( x + y ) ^ { 3 } - 2 x ^ { 3 } + 24 x\).
  1. Find \(\frac { \partial z } { \partial x }\) and \(\frac { \partial z } { \partial y }\).
  2. Find the coordinates of the three stationary points on the surface.
  3. Find the equation of the normal line at the point \(\mathrm { P } ( 1 , - 2,19 )\) on the surface.
  4. The point \(\mathrm { Q } ( 1 + k , - 2 + h , 19 + 3 h )\) is on the surface and is close to P . Find an approximate expression for \(k\) in terms of \(h\).
  5. Show that there is only one point on the surface at which the tangent plane has an equation of the form \(27 x - z = d\). Find the coordinates of this point and the corresponding value of \(d\).
OCR MEI FP3 2009 June Q3
3 A curve has parametric equations \(x = a ( \theta + \sin \theta ) , y = a ( 1 - \cos \theta )\), for \(0 \leqslant \theta \leqslant \pi\), where \(a\) is a positive constant.
  1. Show that the arc length \(s\) from the origin to a general point on the curve is given by \(s = 4 a \sin \frac { 1 } { 2 } \theta\).
  2. Find the intrinsic equation of the curve giving \(s\) in terms of \(a\) and \(\psi\), where \(\tan \psi = \frac { \mathrm { d } y } { \mathrm {~d} x }\).
  3. Hence, or otherwise, show that the radius of curvature at a point on the curve is \(4 a \cos \frac { 1 } { 2 } \theta\).
  4. Find the coordinates of the centre of curvature corresponding to the point on the curve where \(\theta = \frac { 2 } { 3 } \pi\).
  5. Find the area of the surface generated when the curve is rotated through \(2 \pi\) radians about the \(x\)-axis.
OCR MEI FP3 2009 June Q4
4 The group \(G = \{ 1,2,3,4,5,6 \}\) has multiplication modulo 7 as its operation. The group \(H = \{ 1,5,7,11,13,17 \}\) has multiplication modulo 18 as its operation.
  1. Show that the groups \(G\) and \(H\) are both cyclic.
  2. List all the proper subgroups of \(G\).
  3. Specify an isomorphism between \(G\) and \(H\). The group \(S = \{ \mathrm { a } , \mathrm { b } , \mathrm { c } , \mathrm { d } , \mathrm { e } , \mathrm { f } \}\) consists of functions with domain \(\{ 1,2,3 \}\) given by $$\begin{array} { l l l } \mathrm { a } ( 1 ) = 2 & \mathrm { a } ( 2 ) = 3 & \mathrm { a } ( 3 ) = 1
    \mathrm {~b} ( 1 ) = 3 & \mathrm {~b} ( 2 ) = 1 & \mathrm {~b} ( 3 ) = 2
    \mathrm { c } ( 1 ) = 1 & \mathrm { c } ( 2 ) = 3 & \mathrm { c } ( 3 ) = 2
    \mathrm {~d} ( 1 ) = 3 & \mathrm {~d} ( 2 ) = 2 & \mathrm {~d} ( 3 ) = 1
    \mathrm { e } ( 1 ) = 1 & \mathrm { e } ( 2 ) = 2 & \mathrm { e } ( 3 ) = 3
    \mathrm { f } ( 1 ) = 2 & \mathrm { f } ( 2 ) = 1 & \mathrm { f } ( 3 ) = 3 \end{array}$$ and the group operation is composition of functions.
  4. Show that ad \(= \mathrm { c }\) and find da.
  5. Give a reason why \(S\) is not isomorphic to \(G\).
  6. Find the order of each element of \(S\).
  7. List all the proper subgroups of \(S\).
OCR MEI FP3 2009 June Q5
5 Each level of a fantasy computer game is set in a single location, Alphaworld, Betaworld, Chiworld or Deltaworld. After completing a level, a player goes on to the next level, which could be set in the same location as the previous level, or in a different location. In the first version of the game, the initial and transition probabilities are as follows.
Level 1 is set in Alphaworld or Betaworld, with probabilities 0.6, 0.4 respectively.
After a level set in Alphaworld, the next level will be set in Betaworld, Chiworld or Deltaworld, with probabilities \(0.7,0.1,0.2\) respectively.
After a level set in Betaworld, the next level will be set in Alphaworld, Betaworld or Deltaworld, with probabilities \(0.1,0.8,0.1\) respectively.
After a level set in Chiworld, the next level will also be set in Chiworld.
After a level set in Deltaworld, the next level will be set in Alphaworld, Betaworld or Chiworld, with probabilities \(0.3,0.6,0.1\) respectively. The situation is modelled as a Markov chain with four states.
  1. Write down the transition matrix.
  2. Find the probabilities that level 14 is set in each location.
  3. Find the probability that level 15 is set in the same location as level 14 .
  4. Find the level at which the probability of being set in Chiworld first exceeds 0.5.
  5. Following a level set in Betaworld, find the expected number of further levels which will be set in Betaworld before changing to a different location. In the second version of the game, the initial probabilities and the transition probabilities after Alphaworld, Betaworld and Deltaworld are all the same as in the first version; but after a level set in Chiworld, the next level will be set in Chiworld or Deltaworld, with probabilities \(0.9,0.1\) respectively.
  6. By considering powers of the new transition matrix, or otherwise, find the equilibrium probabilities for the four locations. In the third version of the game, the initial probabilities and the transition probabilities after Alphaworld, Betaworld and Deltaworld are again all the same as in the first version; but the transition probabilities after Chiworld have changed again. The equilibrium probabilities for Alphaworld, Betaworld, Chiworld and Deltaworld are now 0.11, 0.75, 0.04, 0.1 respectively.
  7. Find the new transition probabilities after a level set in Chiworld. OCR is committed to seeking permission to reproduce all third-party content that it uses in its assessment materials. OCR has attempted to identify and contact all copyright holders whose work is used in this paper. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced in the OCR Copyright Acknowledgements Booklet. This is produced for each series of examinations, is given to all schools that receive assessment material and is freely available to download from our public website (\href{http://www.ocr.org.uk}{www.ocr.org.uk}) after the live examination series.
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OCR MEI FP3 2011 June Q1
1 The points \(\mathrm { A } ( 2 , - 1,3 ) , \mathrm { B } ( - 2 , - 7,7 )\) and \(\mathrm { C } ( 7,5,1 )\) are three vertices of a tetrahedron ABCD .
The plane ABD has equation \(x + 4 y + 7 z = 19\).
The plane ACD has equation \(2 x - y + 2 z = 11\).
  1. Find the shortest distance from \(B\) to the plane \(A C D\).
  2. Find an equation for the line AD .
  3. Find the shortest distance from C to the line AD .
  4. Find the shortest distance between the lines \(A D\) and \(B C\).
  5. Given that the tetrahedron ABCD has volume 20, find the coordinates of the two possible positions for the vertex \(D\).
OCR MEI FP3 2011 June Q2
2 A surface \(S\) has equation \(z = 8 y ^ { 3 } - 6 x ^ { 2 } y - 15 x ^ { 2 } + 36 x\).
  1. Sketch the section of \(S\) given by \(y = - 3\), and sketch the section of \(S\) given by \(x = - 6\). Your sketches should include the coordinates of any stationary points but need not include the coordinates of the points where the sections cross the axes.
  2. From your sketches in part (i), deduce that \(( - 6 , - 3 , - 324 )\) is a stationary point on \(S\), and state the nature of this stationary point.
  3. Find \(\frac { \partial z } { \partial x }\) and \(\frac { \partial z } { \partial y }\), and hence find the coordinates of the other three stationary points on \(S\).
  4. Show that there are exactly two values of \(k\) for which the plane with equation $$120 x - z = k$$ is a tangent plane to \(S\), and find these values of \(k\).
OCR MEI FP3 2011 June Q3
3
    1. Given that \(y = \mathrm { e } ^ { \frac { 1 } { 2 } x } + \mathrm { e } ^ { - \frac { 1 } { 2 } x }\), show that \(1 + \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) ^ { 2 } = \left( \frac { 1 } { 2 } \mathrm { e } ^ { \frac { 1 } { 2 } x } + \frac { 1 } { 2 } \mathrm { e } ^ { - \frac { 1 } { 2 } x } \right) ^ { 2 }\). The arc of the curve \(y = \mathrm { e } ^ { \frac { 1 } { 2 } x } + \mathrm { e } ^ { - \frac { 1 } { 2 } x }\) for \(0 \leqslant x \leqslant \ln a\) (where \(a > 1\) ) is denoted by \(C\).
    2. Show that the length of \(C\) is \(\frac { a - 1 } { \sqrt { a } }\).
    3. Find the area of the surface formed when \(C\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
  2. An ellipse has parametric equations \(x = 2 \cos \theta , y = \sin \theta\) for \(0 \leqslant \theta < 2 \pi\).
    1. Show that the normal to the ellipse at the point with parameter \(\theta\) has equation $$y = 2 x \tan \theta - 3 \sin \theta$$
    2. Find parametric equations for the evolute of the ellipse, and show that the evolute has cartesian equation $$( 2 x ) ^ { \frac { 2 } { 3 } } + y ^ { \frac { 2 } { 3 } } = 3 ^ { \frac { 2 } { 3 } }$$
    3. Using the evolute found in part (ii), or otherwise, find the radius of curvature of the ellipse
      (A) at the point \(( 2,0 )\),
      (B) at the point \(( 0,1 )\).
OCR MEI FP3 2011 June Q4
4
  1. Show that the set \(G = \{ 1,3,4,5,9 \}\), under the binary operation of multiplication modulo 11 , is a group. You may assume associativity.
  2. Explain why any two groups of order 5 must be isomorphic to each other. The set \(H = \left\{ 1 , \mathrm { e } ^ { \frac { 2 } { 5 } \pi \mathrm { j } } , \mathrm { e } ^ { \frac { 4 } { 5 } \pi \mathrm { j } } , \mathrm { e } ^ { \frac { 6 } { 5 } \pi \mathrm { j } } , \mathrm { e } ^ { \frac { 8 } { 5 } \pi \mathrm { j } } \right\}\) is a group under the binary operation of multiplication of complex numbers.
  3. Specify an isomorphism between the groups \(G\) and \(H\). The set \(K\) consists of the 25 ordered pairs \(( x , y )\), where \(x\) and \(y\) are elements of \(G\). The set \(K\) is a group under the binary operation defined by $$\left( x _ { 1 } , y _ { 1 } \right) \left( x _ { 2 } , y _ { 2 } \right) = \left( x _ { 1 } x _ { 2 } , y _ { 1 } y _ { 2 } \right)$$ where the multiplications are carried out modulo 11 ; for example, \(( 3,5 ) ( 4,4 ) = ( 1,9 )\).
  4. Write down the identity element of \(K\), and find the inverse of the element \(( 9,3 )\).
  5. Explain why \(( x , y ) ^ { 5 } = ( 1,1 )\) for every element \(( x , y )\) in \(K\).
  6. Deduce that all the elements of \(K\), except for one, have order 5. State which is the exceptional element.
  7. A subgroup of \(K\) has order 5 and contains the element (9, 3). List the elements of this subgroup.
  8. Determine how many subgroups of \(K\) there are with order 5 .
OCR MEI FP3 2011 June Q5
5 In this question, give probabilities correct to 4 decimal places.
Alpha and Delta are two companies which compete for the ownership of insurance bonds. Boyles and Cayleys are companies which trade in these bonds. When a new bond becomes available, it is first acquired by either Boyles or Cayleys. After a certain amount of trading it is eventually owned by either Alpha or Delta. Change of ownership always takes place overnight, so that on any particular day the bond is owned by one of the four companies. The trading process is modelled as a Markov chain with four states, as follows. On the first day, the bond is owned by Boyles or Cayleys, with probabilities \(0.4,0.6\) respectively.
If the bond is owned by Boyles, then on the next day it could be owned by Alpha, Boyles or Cayleys, with probabilities \(0.07,0.8,0.13\) respectively. If the bond is owned by Cayleys, then on the next day it could be owned by Boyles, Cayleys or Delta, with probabilities \(0.15,0.75,0.1\) respectively. If the bond is owned by Alpha or Delta, then no further trading takes place, so on the next day it is owned by the same company.
  1. Write down the transition matrix \(\mathbf { P }\).
  2. Explain what is meant by an absorbing state of a Markov chain. Identify any absorbing states in this situation.
  3. Find the probability that the bond is owned by Boyles on the 10th day.
  4. Find the probability that on the 14th day the bond is owned by the same company as on the 10th day.
  5. Find the day on which the probability that the bond is owned by Alpha or Delta exceeds 0.9 for the first time.
  6. Find the limit of \(\mathbf { P } ^ { n }\) as \(n\) tends to infinity.
  7. Find the probability that the bond is eventually owned by Alpha. The probabilities that Boyles and Cayleys own the bond on the first day are changed (but all the transition probabilities remain the same as before). The bond is now equally likely to be owned by Alpha or Delta at the end of the trading process.
  8. Find the new probabilities for the ownership of the bond on the first day.
OCR MEI FP3 2007 June Q1
1 Three planes \(P , Q\) and \(R\) have the following equations. $$\begin{array} { l l } \text { Plane } P : & 8 x - y - 14 z = 20
\text { Plane } Q : & 6 x + 2 y - 5 z = 26
\text { Plane } R : & 2 x + y - z = 40 \end{array}$$ The line of intersection of the planes \(P\) and \(Q\) is \(K\).
The line of intersection of the planes \(P\) and \(R\) is \(L\).
  1. Show that \(K\) and \(L\) are parallel lines, and find the shortest distance between them.
  2. Show that the shortest distance between the line \(K\) and the plane \(R\) is \(5 \sqrt { 6 }\). The line \(M\) has equation \(\mathbf { r } = ( \mathbf { i } - 4 \mathbf { j } ) + \lambda ( 5 \mathbf { i } - 4 \mathbf { j } + 3 \mathbf { k } )\).
  3. Show that the lines \(K\) and \(M\) intersect, and find the coordinates of the point of intersection.
  4. Find the shortest distance between the lines \(L\) and \(M\).
OCR MEI FP3 2007 June Q2
2 A surface has equation \(z = x y ^ { 2 } - 4 x ^ { 2 } y - 2 x ^ { 3 } + 27 x ^ { 2 } - 36 x + 20\).
  1. Find \(\frac { \partial z } { \partial x }\) and \(\frac { \partial z } { \partial y }\).
  2. Find the coordinates of the four stationary points on the surface, showing that one of them is \(( 2,4,8 )\).
  3. Sketch, on separate diagrams, the sections of the surface defined by \(x = 2\) and by \(y = 4\). Indicate the point \(( 2,4,8 )\) on these sections, and deduce that it is neither a maximum nor a minimum.
  4. Show that there are just two points on the surface where the normal line is parallel to the vector \(36 \mathbf { i } + \mathbf { k }\), and find the coordinates of these points.
OCR MEI FP3 2007 June Q3
3 The curve \(C\) has equation \(y = \frac { 1 } { 2 } x ^ { 2 } - \frac { 1 } { 4 } \ln x\), and \(a\) is a constant with \(a \geqslant 1\).
  1. Show that the length of the arc of \(C\) for which \(1 \leqslant x \leqslant a\) is \(\frac { 1 } { 2 } a ^ { 2 } + \frac { 1 } { 4 } \ln a - \frac { 1 } { 2 }\).
  2. Find the area of the surface generated when the arc of \(C\) for which \(1 \leqslant x \leqslant 4\) is rotated through \(2 \pi\) radians about the \(\boldsymbol { y }\)-axis.
  3. Show that the radius of curvature of \(C\) at the point where \(x = a\) is \(a \left( a + \frac { 1 } { 4 a } \right) ^ { 2 }\).
  4. Find the centre of curvature corresponding to the point \(\left( 1 , \frac { 1 } { 2 } \right)\) on \(C\).
    \(C\) is one member of the family of curves defined by \(y = p x ^ { 2 } - p ^ { 2 } \ln x\), where \(p\) is a parameter.
  5. Find the envelope of this family of curves.
OCR MEI FP3 2007 June Q4
4
  1. Prove that, for a group of order 10, every proper subgroup must be cyclic. The set \(M = \{ 1,2,3,4,5,6,7,8,9,10 \}\) is a group under the binary operation of multiplication modulo 11.
  2. Show that \(M\) is cyclic.
  3. List all the proper subgroups of \(M\). The group \(P\) of symmetries of a regular pentagon consists of 10 transformations $$\{ \mathrm { A } , \mathrm {~B} , \mathrm { C } , \mathrm { D } , \mathrm { E } , \mathrm {~F} , \mathrm { G } , \mathrm { H } , \mathrm { I } , \mathrm {~J} \}$$ and the binary operation is composition of transformations. The composition table for \(P\) is given below.
    ABCDEFGHIJ
    ACJGHABIFED
    BFEHGBADCJI
    CGDIFCJEBAH
    DJCBEDGFIHA
    EABCDEFGHIJ
    FHIDCFEJABG
    GIHEBGDAJCF
    HDGJAHIBEFC
    IEFAJIHCDGB
    JBAFIJCHGDE
    One of these transformations is the identity transformation, some are rotations and the rest are reflections.
  4. Identify which transformation is the identity, which are rotations and which are reflections.
  5. State, giving a reason, whether \(P\) is isomorphic to \(M\).
  6. Find the order of each element of \(P\).
  7. List all the proper subgroups of \(P\).
OCR MEI FP3 2007 June Q5
5 A computer is programmed to generate a sequence of letters. The process is represented by a Markov chain with four states, as follows. The first letter is \(A , B , C\) or \(D\), with probabilities \(0.4,0.3,0.2\) and 0.1 respectively.
After \(A\), the next letter is either \(C\) or \(D\), with probabilities 0.8 and 0.2 respectively.
After \(B\), the next letter is either \(C\) or \(D\), with probabilities 0.1 and 0.9 respectively.
After \(C\), the next letter is either \(A\) or \(B\), with probabilities 0.4 and 0.6 respectively.
After \(D\), the next letter is either \(A\) or \(B\), with probabilities 0.3 and 0.7 respectively.
  1. Write down the transition matrix \(\mathbf { P }\).
  2. Use your calculator to find \(\mathbf { P } ^ { 4 }\) and \(\mathbf { P } ^ { 7 }\). (Give elements correct to 4 decimal places.)
  3. Find the probability that the 8th letter is \(C\).
  4. Find the probability that the 12th letter is the same as the 8th letter.
  5. By investigating the behaviour of \(\mathbf { P } ^ { n }\) when \(n\) is large, find the probability that the ( \(n + 1\) )th letter is \(A\) when
    (A) \(n\) is a large even number,
    (B) \(n\) is a large odd number. The program is now changed. The initial probabilities and the transition probabilities are the same as before, except for the following. After \(D\), the next letter is \(A , B\) or \(D\), with probabilities \(0.3,0.6\) and 0.1 respectively.
  6. Write down the new transition matrix \(\mathbf { Q }\).
  7. Verify that \(\mathbf { Q } ^ { n }\) approaches a limit as \(n\) becomes large, and hence write down the equilibrium probabilities for \(A , B , C\) and \(D\).
  8. When \(n\) is large, find the probability that the \(( n + 1 )\) th, \(( n + 2 )\) th and \(( n + 3 )\) th letters are DDD.
OCR MEI FP3 2016 June Q1
1 Positions in space around an aerodrome are modelled by a coordinate system with a point on the runway as the origin, O . The \(x\)-axis is east, the \(y\)-axis is north and the \(z\)-axis is vertically upwards. Units of distance are kilometres. Units of time are hours.
At time \(t = 0\), an aeroplane, P , is at \(( 3,4,8 )\) and is travelling in a direction \(\left( \begin{array} { l } 2
1
0 \end{array} \right)\) at a constant speed of
\(900 \mathrm { kmh } ^ { - 1 }\).
  1. Find the least distance of the path of P from the point O . At time \(t = 0\), a second aeroplane, Q , is at \(( 80,40,10 )\). It is travelling in a straight line towards the point O . Its speed is constant at \(270 \mathrm { kmh } ^ { - 1 }\).
  2. Show that the shortest distance between the paths of the two aeroplanes is 2.24 km correct to three significant figures.
  3. By finding the points on the paths where the shortest distance occurs and the times at which the aeroplanes are at these points, show that in fact the aeroplanes are never this close.
  4. A third aeroplane, R , is at position \(( 29,19,5.5 )\) at time \(t = 0\) and is travelling at \(285 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) in a direction \(\left( \begin{array} { c } 18
    6
    1 \end{array} \right)\). Given that Q is in the process of landing and cannot change course, show that R needs to be instructed to alter course or change speed.
OCR MEI FP3 2016 June Q2
2 A surface, S , has equation \(z = 3 x ^ { 2 } + 6 x y + y ^ { 3 }\).
  1. Find the equation of the section where \(y = 1\) in the form \(z = \mathrm { f } ( x )\). Sketch this section. Find in three-dimensional vector form the equation of the line of symmetry of this section.
  2. Show that there are two stationary points on S , at \(\mathrm { O } ( 0,0,0 )\) and at \(\mathrm { P } ( - 2,2 , - 4 )\).
  3. Given that the point ( \(- 2 + h , 2 + k , \lambda\) ) lies on the surface, show that $$\lambda = - 4 + 3 ( h + k ) ^ { 2 } + k ^ { 2 } ( k + 3 ) .$$ By considering small values of \(h\) and \(k\), deduce that there is a local minimum at P .
  4. By considering small values of \(x\) and \(y\), show that the stationary point at O is neither a maximum nor a minimum.
  5. Given that \(18 x + 18 y - z = d\) is a tangent plane to S , find the two possible values of \(d\).
OCR MEI FP3 2016 June Q3
3 Fig. 3 shows the curve with parametric equations \(x = t - 3 t ^ { 3 } , y = 1 + 3 t ^ { 2 }\). \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{07eaad51-dc00-44d2-8bff-8652d62902ec-4_634_1294_388_386} \captionsetup{labelformat=empty} \caption{Fig. 3}
\end{figure}
  1. Show that the values of \(t\) where the curve cuts the \(y\)-axis are \(t = 0 , \pm \frac { 1 } { \sqrt { 3 } }\). Write down the corresponding values of \(y\).
  2. Find the radius and centre of curvature when \(t = \frac { 1 } { \sqrt { 3 } }\). The arc of the curve given by \(0 \leqslant t \leqslant \frac { 1 } { \sqrt { 3 } }\) is denoted by \(C\).
  3. Find the length of \(C\).
  4. Show that the area of the curved surface generated when \(C\) is rotated about the \(y\)-axis through \(2 \pi\) radians is \(\frac { \pi } { 3 }\).
OCR MEI FP3 2016 June Q4
4
  1. The elements of the set \(P = \{ 1,3,9,11 \}\) are combined under the binary operation, *, defined as multiplication modulo 16.
    1. Demonstrate associativity for the elements \(3,9,11\) in that order. Assuming associativity holds in general, show that \(P\) forms a group under the binary operation *.
    2. Write down the order of each element.
    3. Write down all subgroups of \(P\).
    4. Show that the group in part (i) is cyclic.
  2. Now consider a group of order 4 containing the identity element \(e\) and the two distinct elements, \(a\) and \(b\), where \(a ^ { 2 } = b ^ { 2 } = e\). Construct the composition table. Show that the group is non-cyclic.
  3. Now consider the four matrices \(\mathbf { I } , \mathbf { X } , \mathbf { Y }\) and \(\mathbf { Z }\) where $$\mathbf { I } = \left( \begin{array} { l l } 1 & 0
    0 & 1 \end{array} \right) , \mathbf { X } = \left( \begin{array} { r r } 1 & 0
    0 & - 1 \end{array} \right) , \mathbf { Y } = \left( \begin{array} { r r } - 1 & 0
    0 & 1 \end{array} \right) , \mathbf { Z } = \left( \begin{array} { r r } - 1 & 0
    0 & - 1 \end{array} \right) .$$ The group G consists of the set \(\{ \mathbf { I } , \mathbf { X } , \mathbf { Y } , \mathbf { Z } \}\) with binary operation matrix multiplication. Determine which of the groups in parts (a) and (b) is isomorphic to G, and specify the isomorphism.
  4. The distinct elements \(\{ p , q , r , s \}\) are combined under the binary operation \({ } ^ { \circ }\). You are given that \(p ^ { \circ } q = r\) and \(q ^ { \circ } p = s\). By reference to the group axioms, prove that \(\{ p , q , r , s \}\) is not a group under \({ } ^ { \circ }\). Option 5: Markov chains \section*{This question requires the use of a calculator with the ability to handle matrices.}
OCR MEI FP3 2016 June Q5
5 Each day that Adam is at work he carries out one of three tasks A, B or C. Each task takes a whole day. Adam chooses the task to carry out on each day according to the following set of three rules.
  1. If, on any given day, he has worked on task A then the next day he will choose task A with probability 0.75 , and tasks B and C with equal probability.
  2. If, on any given day, he has worked on task B then the next day he will choose task B or task C with equal probability but will never choose task A .
  3. If, on any given day, he has worked on task C then the next day he will choose task A with probability \(p\) and tasks B and C with equal probability.
    1. Write down the transition matrix.
    2. Over a long period Adam carries out the tasks \(\mathrm { A } , \mathrm { B }\) and C with equal frequency. Find the value of \(p\).
    3. On day 1 Adam chooses task A . Find the probability that he also chooses task A on day 5 .
    Adam decides to change rule 3 as follows.
    If, on any given day, he has worked on task C then the next day he will choose tasks \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) with probabilities \(0.4,0.3,0.3\) respectively.
  4. On day 1 Adam chooses task A. Find the probability that he chooses the same task on day 7 as he did on day 4 .
  5. On a particular day, Adam chooses task A. Find the expected number of consecutive further days on which he will choose A. Adam changes all three rules again as follows.
    • If he works on A one day then on the next day he chooses C .
    • If he works on B one day then on the next day he chooses A or C each with probability 0.5.
    • If he works on C one day then on the next day he chooses A or B each with probability 0.5 .
    • Find the long term probabilities for each task.
OCR MEI FP3 2015 June Q1
1 The point A has coordinates \(( 2,5,4 )\) and the line BC has equation $$\mathbf { r } = \left( \begin{array} { c } 8
25
43 \end{array} \right) + \lambda \left( \begin{array} { c } 4
15
25 \end{array} \right)$$ You are given that \(\mathrm { AB } = \mathrm { AC } = 15\).
  1. Show that the coordinates of one of the points B and C are (4, 10, 18). Find the coordinates of the other point. These points are B and C respectively.
  2. Find the equation of the plane ABC in cartesian form.
  3. Show that the plane containing the line BC and perpendicular to the plane ABC has equation \(5 y - 3 z + 4 = 0\). The point D has coordinates \(( 1,1,3 )\).
  4. Show that \(| \overrightarrow { B C } \times \overrightarrow { A D } | = \sqrt { 7667 }\) and hence find the shortest distance between the lines \(B C\) and \(A D\).
  5. Find the volume of the tetrahedron ABCD .
OCR MEI FP3 2015 June Q2
2 A surface has equation \(z = 3 x ^ { 2 } - 12 x y + 2 y ^ { 3 } + 60\).
  1. Show that the point \(\mathrm { A } ( 8,4 , - 4 )\) is a stationary point on the surface. Find the coordinates of the other stationary point, B , on this surface.
  2. A point P with coordinates \(( 8 + h , 4 + k , p )\) lies on the surface.
    (A) Show that \(p = - 4 + 3 ( h - 2 k ) ^ { 2 } + 2 k ^ { 2 } ( 6 + k )\).
    (B) Deduce that the stationary point A is a local minimum.
    (C) By considering sections of the surface near to B in each of the planes \(x = 0\) and \(y = 0\), investigate the nature of the stationary point B .
  3. The point Q with coordinates \(( 1,1,53 )\) lies on the surface. Show that the equation of the tangent plane at Q is $$6 x + 6 y + z = 65$$
  4. The tangent plane at the point R has equation \(6 x + 6 y + z = \lambda\) where \(\lambda \neq 65\). Find the coordinates of R .