Questions FP3 (539 questions)

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OCR MEI FP3 2010 June Q4
24 marks Challenging +1.8
4 The group \(F = \{ \mathrm { p } , \mathrm { q } , \mathrm { r } , \mathrm { s } , \mathrm { t } , \mathrm { u } \}\) consists of the six functions defined by $$\mathrm { p } ( x ) = x \quad \mathrm { q } ( x ) = 1 - x \quad \mathrm { r } ( x ) = \frac { 1 } { x } \quad \mathrm {~s} ( x ) = \frac { x - 1 } { x } \quad \mathrm { t } ( x ) = \frac { x } { x - 1 } \quad \mathrm { u } ( x ) = \frac { 1 } { 1 - x } ,$$ the binary operation being composition of functions.
  1. Show that st \(= \mathrm { r }\) and find ts.
  2. Copy and complete the following composition table for \(F\).
    pqrstu
    ppqrstu
    qqpsrut
    rruptsq
    sstqurp
    ttsu
    uurt
  3. Give the inverse of each element of \(F\).
  4. List all the subgroups of \(F\). The group \(M\) consists of \(\left\{ 1 , - 1 , e ^ { \frac { \pi } { 3 } \mathrm { j } } , e ^ { - \frac { \pi } { 3 } \mathrm { j } } , e ^ { \frac { 2 \pi } { 3 } \mathrm { j } } , e ^ { - \frac { 2 \pi } { 3 } \mathrm { j } } \right\}\) with multiplication of complex numbers as its binary operation.
  5. Find the order of each element of \(M\). The group \(G\) consists of the positive integers between 1 and 18 inclusive, under multiplication modulo 19.
  6. Show that \(G\) is a cyclic group which can be generated by the element 2 .
  7. Explain why \(G\) has no subgroup which is isomorphic to \(F\).
  8. Find a subgroup of \(G\) which is isomorphic to \(M\).
OCR MEI FP3 2012 June Q1
24 marks Challenging +1.2
1 A mine contains several underground tunnels beneath a hillside. The hillside is a plane, all the tunnels are straight and the width of the tunnels may be neglected. A coordinate system is chosen with the \(z\)-axis pointing vertically upwards and the units are metres. Three points on the hillside have coordinates \(\mathrm { A } ( 15 , - 60,20 )\), \(B ( - 75,100,40 )\) and \(C ( 18,138,35.6 )\).
  1. Find the vector product \(\overrightarrow { \mathrm { AB } } \times \overrightarrow { \mathrm { AC } }\) and hence show that the equation of the hillside is \(2 x - 2 y + 25 z = 650\). The tunnel \(T _ { \mathrm { A } }\) begins at A and goes in the direction of the vector \(15 \mathbf { i } + 14 \mathbf { j } - 2 \mathbf { k }\); the tunnel \(T _ { \mathrm { C } }\) begins at C and goes in the direction of the vector \(8 \mathbf { i } + 7 \mathbf { j } - 2 \mathbf { k }\). Both these tunnels extend a long way into the ground.
  2. Find the least possible length of a tunnel which connects B to a point in \(T _ { \mathrm { A } }\).
  3. Find the least possible length of a tunnel which connects a point in \(T _ { \mathrm { A } }\) to a point in \(T _ { \mathrm { C } }\).
  4. A tunnel starts at B , passes through the point ( \(18,138 , p\) ) vertically below C , and intersects \(T _ { \mathrm { A } }\) at the point Q . Find the value of \(p\) and the coordinates of Q .
OCR MEI FP3 2012 June Q2
24 marks Challenging +1.2
2 You are given that \(\mathrm { g } ( x , y , z ) = x ^ { 2 } + 2 y ^ { 2 } - z ^ { 2 } + 2 x z + 2 y z + 4 z - 3\).
  1. Find \(\frac { \partial \mathrm { g } } { \partial x } , \frac { \partial \mathrm {~g} } { \partial y }\) and \(\frac { \partial \mathrm { g } } { \partial z }\). The surface \(S\) has equation \(\mathrm { g } ( x , y , z ) = 0\), and \(\mathrm { P } ( - 2 , - 1,1 )\) is a point on \(S\).
  2. Find an equation for the normal line to the surface \(S\) at the point P .
  3. A point Q lies on this normal line and is close to P . At \(\mathrm { Q } , \mathrm { g } ( x , y , z ) = h\), where \(h\) is small. Find the constant \(c\) such that \(\mathrm { PQ } \approx c | h |\).
  4. Show that there is no point on \(S\) at which the normal line is parallel to the \(z\)-axis.
  5. Given that \(x + y + z = k\) is a tangent plane to the surface \(S\), find the two possible values of \(k\).
OCR MEI FP3 2012 June Q3
24 marks Challenging +1.8
3 A curve has parametric equations $$x = a \left( 1 - \cos ^ { 3 } \theta \right) , \quad y = a \sin ^ { 3 } \theta , \quad \text { for } 0 \leqslant \theta \leqslant \frac { \pi } { 3 }$$ where \(a\) is a positive constant.
The arc length from the origin to a general point on the curve is denoted by \(s\), and \(\psi\) is the acute angle defined by \(\tan \psi = \frac { \mathrm { d } y } { \mathrm {~d} x }\).
  1. Express \(s\) and \(\psi\) in terms of \(\theta\), and hence show that the intrinsic equation of the curve is $$s = \frac { 3 } { 2 } a \sin ^ { 2 } \psi$$
  2. For the point on the curve given by \(\theta = \frac { \pi } { 6 }\), find the radius of curvature and the coordinates of the centre of curvature.
  3. Find the area of the curved surface generated when the curve is rotated through \(2 \pi\) radians about the \(y\)-axis.
OCR MEI FP3 2012 June Q4
24 marks Challenging +1.3
4
  1. Show that the set \(P = \{ 1,5,7,11 \}\), under the binary operation of multiplication modulo 12, is a group. You may assume associativity. A group \(Q\) has identity element \(e\). The result of applying the binary operation of \(Q\) to elements \(x\) and \(y\) is written \(x y\), and the inverse of \(x\) is written \(x ^ { - 1 }\).
  2. Verify that the inverse of \(x y\) is \(y ^ { - 1 } x ^ { - 1 }\). Three elements \(a , b\) and \(c\) of \(Q\) all have order 2, and \(a b = c\).
  3. By considering the inverse of \(c\), or otherwise, show that \(b a = c\).
  4. Show that \(b c = a\) and \(a c = b\). Find \(c b\) and \(c a\).
  5. Complete the composition table for \(R = \{ e , a , b , c \}\). Hence show that \(R\) is a subgroup of \(Q\) and that \(R\) is isomorphic to \(P\). The group \(T\) of symmetries of a square contains four reflections \(A , B , C , D\), the identity transformation \(E\) and three rotations \(F , G , H\). The binary operation is composition of transformations. The composition table for \(T\) is given below.
    A\(B\)\(C\)D\(E\)\(F\)\(G\)\(H\)
    AE\(G\)\(H\)\(F\)\(A\)D\(B\)\(C\)
    BGE\(F\)\(H\)\(B\)CAD
    C\(F\)HEGCAD\(B\)
    D\(H\)\(F\)\(G\)E\(D\)\(B\)C\(A\)
    EA\(B\)CD\(E\)\(F\)\(G\)\(H\)
    FCD\(B\)A\(F\)G\(H\)\(E\)
    \(G\)B\(A\)\(D\)C\(G\)HE\(F\)
    \(H\)DCAB\(H\)E\(F\)G
  6. Find the order of each element of \(T\).
  7. List all the proper subgroups of \(T\).
OCR MEI FP3 2013 June Q1
24 marks Standard +0.8
1 Three points have coordinates \(\mathrm { A } ( 3,2,10 ) , \mathrm { B } ( 11,0 , - 3 ) , \mathrm { C } ( 5,18,0 )\), and \(L\) is the straight line through A with equation $$\frac { x - 3 } { - 1 } = \frac { y - 2 } { 4 } = \frac { z - 10 } { 1 }$$
  1. Find the shortest distance between the lines \(L\) and BC .
  2. Find the shortest distance from A to the line BC . A straight line passes through B and the point \(\mathrm { P } ( 5,18 , k )\), and intersects the line \(L\).
  3. Find \(k\), and the point of intersection of the lines BP and \(L\). The point D is on the line \(L\), and AD has length 12 .
  4. Find the volume of the tetrahedron ABCD .
OCR MEI FP3 2013 June Q2
24 marks Challenging +1.8
2 A surface has equation \(z = 2 \left( x ^ { 3 } + y ^ { 3 } \right) + 3 \left( x ^ { 2 } + y ^ { 2 } \right) + 12 x y\).
  1. For a point on the surface at which \(\frac { \partial z } { \partial x } = \frac { \partial z } { \partial y }\), show that either \(y = x\) or \(y = 1 - x\).
  2. Show that there are exactly two stationary points on the surface, and find their coordinates.
  3. The point \(\mathrm { P } \left( \frac { 1 } { 2 } , \frac { 1 } { 2 } , 5 \right)\) is on the surface, and \(\mathrm { Q } \left( \frac { 1 } { 2 } + h , \frac { 1 } { 2 } + h , 5 + w \right)\) is a point on the surface close to P . Find an approximate expression for \(h\) in terms of \(w\).
  4. Find the four points on the surface at which the normal line is parallel to the vector \(24 \mathbf { i } + 24 \mathbf { j } - \mathbf { k }\).
OCR MEI FP3 2013 June Q3
24 marks Challenging +1.2
3
  1. Find the length of the arc of the polar curve \(r = a ( 1 + \cos \theta )\) for which \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\).
  2. A curve \(C\) has cartesian equation \(y = \frac { x ^ { 3 } } { 6 } + \frac { 1 } { 2 x }\).
    1. The arc of \(C\) for which \(1 \leqslant x \leqslant 2\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a surface of revolution. Find the area of this surface. For the point on \(C\) at which \(x = 2\),
    2. show that the radius of curvature is \(\frac { 289 } { 64 }\),
    3. find the coordinates of the centre of curvature.
OCR MEI FP3 2013 June Q4
24 marks Challenging +1.3
4
  1. The composition table for a group \(G\) of order 8 is given below.
    \(a\)\(b\)\(c\)\(d\)\(e\)\(f\)\(g\)\(h\)
    \(a\)\(c\)\(e\)\(b\)\(f\)\(a\)\(h\)\(d\)\(g\)
    \(b\)\(e\)\(c\)\(a\)\(g\)\(b\)\(d\)h\(f\)
    \(c\)\(b\)\(a\)\(e\)\(h\)\(c\)\(g\)\(f\)\(d\)
    \(d\)\(f\)\(g\)\(h\)\(a\)\(d\)\(c\)\(e\)\(b\)
    \(e\)\(a\)\(b\)\(c\)\(d\)\(e\)\(f\)\(g\)\(h\)
    \(f\)\(h\)\(d\)\(g\)\(c\)\(f\)\(b\)\(a\)\(e\)
    \(g\)\(d\)\(h\)\(f\)\(e\)\(g\)\(a\)\(b\)\(c\)
    \(h\)\(g\)\(f\)\(d\)\(b\)\(h\)\(e\)\(c\)\(a\)
    1. State which is the identity element, and give the inverse of each element of \(G\).
    2. Show that \(G\) is cyclic.
    3. Specify an isomorphism between \(G\) and the group \(H\) consisting of \(\{ 0,2,4,6,8,10,12,14 \}\) under addition modulo 16 .
    4. Show that \(G\) is not isomorphic to the group of symmetries of a square.
  2. The set \(S\) consists of the functions \(\mathrm { f } _ { n } ( x ) = \frac { x } { 1 + n x }\), for all integers \(n\), and the binary operation is composition of functions.
    1. Show that \(\mathrm { f } _ { m } \mathrm { f } _ { n } = \mathrm { f } _ { m + n }\).
    2. Hence show that the binary operation is associative.
    3. Prove that \(S\) is a group.
    4. Describe one subgroup of \(S\) which contains more than one element, but which is not the whole of \(S\).
OCR MEI FP3 2013 June Q5
24 marks Challenging +1.8
5 In this question, give probabilities correct to 4 decimal places.
A contestant in a game-show starts with one, two or three 'lives', and then performs a series of tasks. After each task, the number of lives either decreases by one, or remains the same, or increases by one. The game ends when the number of lives becomes either four or zero. If the number of lives is four, the contestant wins a prize; if the number of lives is zero, the contestant loses and leaves with nothing. At the start, the number of lives is decided at random, so that the contestant is equally likely to start with one, two or three lives. The tasks do not involve any skill, and after every task:
  • the probability that the number of lives decreases by one is 0.5 ,
  • the probability that the number of lives remains the same is 0.05 ,
  • the probability that the number of lives increases by one is 0.45 .
This is modelled as a Markov chain with five states corresponding to the possible numbers of lives. The states corresponding to zero lives and four lives are absorbing states.
  1. Write down the transition matrix \(\mathbf { P }\).
  2. Show that, after 8 tasks, the probability that the contestant has three lives is 0.0207 , correct to 4 decimal places.
  3. Find the probability that, after 10 tasks, the game has not yet ended.
  4. Find the probability that the game ends after exactly 10 tasks.
  5. Find the smallest value of \(N\) for which the probability that the game has not yet ended after \(N\) tasks is less than 0.01 .
  6. Find the limit of \(\mathbf { P } ^ { n }\) as \(n\) tends to infinity.
  7. Find the probability that the contestant wins a prize. The beginning of the game is now changed, so that the probabilities of starting with one, two or three lives can be adjusted.
  8. State the maximum possible probability that the contestant wins a prize, and how this can be achieved.
  9. Given that the probability of starting with one life is 0.1 , and the probability of winning a prize is 0.6 , find the probabilities of starting with two lives and starting with three lives. }{www.ocr.org.uk}) after the live examination series.
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OCR MEI FP3 2014 June Q1
24 marks Challenging +1.2
1 Three points have coordinates \(\mathrm { A } ( - 3,12 , - 7 ) , \mathrm { B } ( - 2,6,9 ) , \mathrm { C } ( 6,0 , - 10 )\). The plane \(P\) passes through the points \(\mathrm { A } , \mathrm { B }\) and C .
  1. Find the vector product \(\overrightarrow { \mathrm { AB } } \times \overrightarrow { \mathrm { AC } }\). Hence or otherwise find an equation for the plane \(P\) in the form \(a x + b y + c z = d\). The plane \(Q\) has equation \(6 x + 3 y + 2 z = 32\). The perpendicular from A to the plane \(Q\) meets \(Q\) at the point D. The planes \(P\) and \(Q\) intersect in the line \(L\).
  2. Find the distance AD .
  3. Find an equation for the line \(L\).
  4. Find the shortest distance from A to the line \(L\).
  5. Find the volume of the tetrahedron ABCD .
OCR MEI FP3 2014 June Q2
24 marks Challenging +1.2
2 A surface \(S\) has equation \(\mathrm { g } ( x , y , z ) = 0\), where \(\mathrm { g } ( x , y , z ) = x ^ { 2 } + 3 y ^ { 2 } + 2 z ^ { 2 } + 2 y z + 6 x z - 4 x y - 24\). \(\mathrm { P } ( 2,6 , - 2 )\) is a point on the surface \(S\).
  1. Find \(\frac { \partial \mathrm { g } } { \partial x } , \frac { \partial \mathrm {~g} } { \partial y }\) and \(\frac { \partial \mathrm { g } } { \partial z }\).
  2. Find the equation of the normal line to the surface \(S\) at the point P .
  3. The point Q is on this normal line and close to P . At \(\mathrm { Q } , \mathrm { g } ( x , y , z ) = h\), where \(h\) is small. Find, in terms of \(h\), the approximate perpendicular distance from Q to the surface \(S\).
  4. Find the coordinates of the two points on the surface at which the normal line is parallel to the \(y\)-axis.
  5. Given that \(10 x - y + 2 z = 6\) is the equation of a tangent plane to the surface \(S\), find the coordinates of the point of contact.
OCR MEI FP3 2014 June Q3
24 marks Hard +2.3
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
24 marks Challenging +1.8
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
24 marks Easy -2.5
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
24 marks Challenging +1.8
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
24 marks Challenging +1.8
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
24 marks Challenging +1.8
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
24 marks Challenging +1.2
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
24 marks Moderate -0.5
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. }{www.ocr.org.uk}) after the live examination series.
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OCR FP3 2009 January Q1
5 marks Standard +0.8
1 In this question \(G\) is a group of order \(n\), where \(3 \leqslant n < 8\).
  1. In each case, write down the smallest possible value of \(n\) :
    1. if \(G\) is cyclic,
    2. if \(G\) has a proper subgroup of order 3,
    3. if \(G\) has at least two elements of order 2 .
    4. Another group has the same order as \(G\), but is not isomorphic to \(G\). Write down the possible value(s) of \(n\).
OCR FP3 2009 January Q2
5 marks Standard +0.3
2
  1. Express \(\frac { \sqrt { 3 } + \mathrm { i } } { \sqrt { 3 } - \mathrm { i } }\) in the form \(r \mathrm { e } ^ { \mathrm { i } \theta }\), where \(r > 0\) and \(0 \leqslant \theta < 2 \pi\).
  2. Hence find the smallest positive value of \(n\) for which \(\left( \frac { \sqrt { 3 } + \mathrm { i } } { \sqrt { 3 } - \mathrm { i } } \right) ^ { n }\) is real and positive.
OCR FP3 2009 January Q3
6 marks Challenging +1.2
3 Two skew lines have equations $$\frac { x } { 2 } = \frac { y + 3 } { 1 } = \frac { z - 6 } { 3 } \quad \text { and } \quad \frac { x - 5 } { 3 } = \frac { y + 1 } { 1 } = \frac { z - 7 } { 5 } .$$
  1. Find the direction of the common perpendicular to the lines.
  2. Find the shortest distance between the lines.
OCR FP3 2009 January Q4
9 marks Standard +0.8
4 Find the general solution of the differential equation $$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 \frac { \mathrm {~d} y } { \mathrm {~d} x } + 5 y = 65 \sin 2 x$$
OCR FP3 2009 January Q5
9 marks Standard +0.8
5 The variables \(x\) and \(y\) are related by the differential equation $$x ^ { 3 } \frac { \mathrm {~d} y } { \mathrm {~d} x } = x y + x + 1 .$$
  1. Use the substitution \(y = u - \frac { 1 } { x }\), where \(u\) is a function of \(x\), to show that the differential equation may be written as $$x ^ { 2 } \frac { \mathrm {~d} u } { \mathrm {~d} x } = u .$$
  2. Hence find the general solution of the differential equation (A), giving your answer in the form \(y = \mathrm { f } ( x )\).