Questions — OCR MEI (4455 questions)

Browse by module
All questions AEA AS Paper 1 AS Paper 2 AS Pure C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Extra Pure Further Mechanics Further Mechanics A AS Further Mechanics AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics Further Paper 4 Further Pure Core Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Pure with Technology Further Statistics Further Statistics A AS Further Statistics AS Further Statistics B AS Further Statistics Major Further Statistics Minor Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 H240/01 H240/02 H240/03 M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 S1 S2 S3 S4 Unit 1 Unit 2 Unit 3 Unit 4
Browse by board
AQA AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further AS Paper 1 Further AS Paper 2 Discrete Further AS Paper 2 Mechanics Further AS Paper 2 Statistics Further Paper 1 Further Paper 2 Further Paper 3 Discrete Further Paper 3 Mechanics Further Paper 3 Statistics M1 M2 M3 Paper 1 Paper 2 Paper 3 S1 S2 S3 CAIE FP1 FP2 Further Paper 1 Further Paper 2 Further Paper 3 Further Paper 4 M1 M2 P1 P2 P3 S1 S2 Edexcel AEA AS Paper 1 AS Paper 2 C1 C12 C2 C3 C34 C4 CP AS CP1 CP2 D1 D2 F1 F2 F3 FD1 FD1 AS FD2 FD2 AS FM1 FM1 AS FM2 FM2 AS FP1 FP1 AS FP2 FP2 AS FP3 FS1 FS1 AS FS2 FS2 AS M1 M2 M3 M4 M5 P1 P2 P3 P4 PMT Mocks PURE Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 OCR AS Pure C1 C2 C3 C4 D1 D2 FD1 AS FM1 AS FP1 FP1 AS FP2 FP3 FS1 AS Further Additional Pure Further Additional Pure AS Further Discrete Further Discrete AS Further Mechanics Further Mechanics AS Further Pure Core 1 Further Pure Core 2 Further Pure Core AS Further Statistics Further Statistics AS H240/01 H240/02 H240/03 M1 M2 M3 M4 PURE S1 S2 S3 S4 OCR MEI AS Paper 1 AS Paper 2 C1 C2 C3 C4 D1 D2 FP1 FP2 FP3 Further Extra Pure Further Mechanics A AS Further Mechanics B AS Further Mechanics Major Further Mechanics Minor Further Numerical Methods Further Pure Core Further Pure Core AS Further Pure with Technology Further Statistics A AS Further Statistics B AS Further Statistics Major Further Statistics Minor M1 M2 M3 M4 Paper 1 Paper 2 Paper 3 S1 S2 S3 S4 Pre-U Pre-U 9794/1 Pre-U 9794/2 Pre-U 9794/3 Pre-U 9795 Pre-U 9795/1 Pre-U 9795/2 WJEC Further Unit 1 Further Unit 2 Further Unit 3 Further Unit 4 Further Unit 5 Further Unit 6 Unit 1 Unit 2 Unit 3 Unit 4
Sort by: Default | Easiest first | Hardest first
OCR MEI S1 2008 June Q7
20 marks Moderate -0.8
7 The histogram shows the age distribution of people living in Inner London in 2001. \includegraphics[max width=\textwidth, alt={}, center]{be764df3-ff20-415d-9c5c-10edabf350de-5_814_1383_349_379} Data sourced from the 2001 Census, \href{http://www.statistics.gov.uk}{www.statistics.gov.uk}
  1. State the type of skewness shown by the distribution.
  2. Use the histogram to estimate the number of people aged under 25.
  3. The table below shows the cumulative frequency distribution.
    Age2030405065100
    Cumulative frequency (thousands)66012401810\(a\)24902770
    (A) Use the histogram to find the value of \(a\).
    (B) Use the table to calculate an estimate of the median age of these people. The ages of people living in Outer London in 2001 are summarised below.
    Age ( \(x\) years)\(0 \leqslant x < 20\)\(20 \leqslant x < 30\)\(30 \leqslant x < 40\)\(40 \leqslant x < 50\)\(50 \leqslant x < 65\)\(65 \leqslant x < 100\)
    Frequency (thousands)1120650770590680610
  4. Illustrate these data by means of a histogram.
  5. Make two brief comments on the differences between the age distributions of the populations of Inner London and Outer London.
  6. The data given in the table for Outer London are used to calculate the following estimates. Mean 38.5, median 35.7, midrange 50, standard deviation 23.7, interquartile range 34.4.
    The final group in the table assumes that the maximum age of any resident is 100 years. These estimates are to be recalculated, based on a maximum age of 105, rather than 100. For each of the five estimates, state whether it would increase, decrease or be unchanged.
OCR MEI FP3 2006 June Q1
24 marks Challenging +1.2
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.
OCR MEI FP3 2006 June Q2
24 marks Challenging +1.2
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.
OCR MEI FP3 2006 June Q3
24 marks Challenging +1.8
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 \(a\).
OCR MEI FP3 2006 June Q4
24 marks Challenging +1.2
\(\mathbf { 4 }\) 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.
OCR MEI FP3 2006 June Q5
24 marks Challenging +1.2
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 .
OCR MEI FP3 2008 June Q1
24 marks Challenging +1.8
1 A tetrahedron ABCD has vertices \(\mathrm { A } ( - 3,5,2 ) , \mathrm { B } ( 3,13,7 ) , \mathrm { C } ( 7,0,3 )\) and \(\mathrm { D } ( 5,4,8 )\).
  1. Find the vector product \(\overrightarrow { \mathrm { AB } } \times \overrightarrow { \mathrm { AC } }\), and hence find the equation of the plane ABC .
  2. Find the shortest distance from \(D\) to the plane \(A B C\).
  3. Find the shortest distance between the lines AB and CD .
  4. Find the volume of the tetrahedron ABCD . The plane \(P\) with equation \(3 x - 2 z + 5 = 0\) contains the point B , and meets the lines AC and AD at E and F respectively.
  5. Find \(\lambda\) and \(\mu\) such that \(\overrightarrow { \mathrm { AE } } = \lambda \overrightarrow { \mathrm { AC } }\) and \(\overrightarrow { \mathrm { AF } } = \mu \overrightarrow { \mathrm { AD } }\). Deduce that E is between A and C , and that F is between A and D.
  6. Hence, or otherwise, show that \(P\) divides the tetrahedron ABCD into two parts having volumes in the ratio 4 to 17.
OCR MEI FP3 2008 June Q2
24 marks Challenging +1.2
2 You are given \(\mathrm { g } ( x , y , z ) = 6 x z - ( x + 2 y + 3 z ) ^ { 2 }\).
  1. Find \(\frac { \partial \mathrm { g } } { \partial x } , \frac { \partial \mathrm {~g} } { \partial y }\) and \(\frac { \partial \mathrm { g } } { \partial z }\). A surface \(S\) has equation \(\mathrm { g } ( x , y , z ) = 125\).
  2. Find the equation of the normal line to \(S\) at the point \(\mathrm { P } ( 7 , - 7.5,3 )\).
  3. The point Q is on this normal line and is close to P . At \(\mathrm { Q } , \mathrm { g } ( x , y , z ) = 125 + h\), where \(h\) is small. Find the vector \(\mathbf { n }\) such that \(\overrightarrow { \mathrm { PQ } } = h \mathbf { n }\) approximately.
  4. Show that there is no point on \(S\) at which the normal line is parallel to the \(z\)-axis.
  5. Find the two points on \(S\) at which the tangent plane is parallel to \(x + 5 y = 0\).
OCR MEI FP3 2008 June Q3
24 marks Challenging +1.2
3 The curve \(C\) has parametric equations \(x = 8 t ^ { 3 } , y = 9 t ^ { 2 } - 2 t ^ { 4 }\), for \(t \geqslant 0\).
  1. Show that \(\dot { x } ^ { 2 } + \dot { y } ^ { 2 } = \left( 18 t + 8 t ^ { 3 } \right) ^ { 2 }\). Find the length of the arc of \(C\) for which \(0 \leqslant t \leqslant 2\).
  2. Find the area of the surface generated when the arc of \(C\) for which \(0 \leqslant t \leqslant 2\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
  3. Show that the curvature at a general point on \(C\) is \(\frac { - 6 } { t \left( 4 t ^ { 2 } + 9 \right) ^ { 2 } }\).
  4. Find the coordinates of the centre of curvature corresponding to the point on \(C\) where \(t = 1\).
OCR MEI FP3 2008 June Q4
24 marks Standard +0.8
4 A binary operation * is defined on real numbers \(x\) and \(y\) by $$x * y = 2 x y + x + y$$ You may assume that the operation \(*\) is commutative and associative.
  1. Explain briefly the meanings of the terms 'commutative' and 'associative'.
  2. Show that \(x * y = 2 \left( x + \frac { 1 } { 2 } \right) \left( y + \frac { 1 } { 2 } \right) - \frac { 1 } { 2 }\). The set \(S\) consists of all real numbers greater than \(- \frac { 1 } { 2 }\).
  3. (A) Use the result in part (ii) to show that \(S\) is closed under the operation *.
    (B) Show that \(S\), with the operation \(*\), is a group.
  4. Show that \(S\) contains no element of order 2 . The group \(G = \{ 0,1,2,4,5,6 \}\) has binary operation ∘ defined by $$x \circ y \text { is the remainder when } x * y \text { is divided by } 7 \text {. }$$
  5. Show that \(4 \circ 6 = 2\). The composition table for \(G\) is as follows.
    \(\circ\)012456
    0012456
    1140625
    2205164
    4461502
    5526041
    6654210
  6. Find the order of each element of \(G\).
  7. List all the subgroups of \(G\).
OCR MEI FP3 2008 June Q5
24 marks Challenging +1.2
5 Every day, a security firm transports a large sum of money from one bank to another. There are three possible routes \(A , B\) and \(C\). The route to be used is decided just before the journey begins, by a computer programmed as follows. On the first day, each of the three routes is equally likely to be used.
If route \(A\) was used on the previous day, route \(A\), \(B\) or \(C\) will be used, with probabilities \(0.1,0.4,0.5\) respectively.
If route \(B\) was used on the previous day, route \(A , B\) or \(C\) will be used, with probabilities \(0.7,0.2,0.1\) respectively.
If route \(C\) was used on the previous day, route \(A , B\) or \(C\) will be used, with probabilities \(0.1,0.6,0.3\) respectively. The situation is modelled as a Markov chain with three states.
  1. Write down the transition matrix \(\mathbf { P }\).
  2. Find the probability that route \(B\) is used on the 7th day.
  3. Find the probability that the same route is used on the 7th and 8th days.
  4. Find the probability that the route used on the 10th day is the same as that used on the 7th day.
  5. Given that \(\mathbf { P } ^ { n } \rightarrow \mathbf { Q }\) as \(n \rightarrow \infty\), find the matrix \(\mathbf { Q }\) (give the elements to 4 decimal places). Interpret the probabilities which occur in the matrix \(\mathbf { Q }\). The computer program is now to be changed, so that the long-run probabilities for routes \(A , B\) and \(C\) will become \(0.4,0.2\) and 0.4 respectively. The transition probabilities after routes \(A\) and \(B\) remain the same as before.
  6. Find the new transition probabilities after route \(C\).
  7. A long time after the change of program, a day is chosen at random. Find the probability that the route used on that day is the same as on the previous day.
OCR MEI FP3 2010 June Q1
24 marks Challenging +1.2
1 Four points have coordinates $$\mathrm { A } ( 3,8,27 ) , \quad \mathrm { B } ( 5,9,25 ) , \quad \mathrm { C } ( 8,0,1 ) \quad \text { and } \quad \mathrm { D } ( 11 , p , p ) ,$$ where \(p\) is a constant.
  1. Find the perpendicular distance from C to the line AB .
  2. Find \(\overrightarrow { \mathrm { AB } } \times \overrightarrow { \mathrm { CD } }\) in terms of \(p\), and show that the shortest distance between the lines AB and CD is $$\frac { 21 | p - 5 | } { \sqrt { 17 p ^ { 2 } - 2 p + 26 } }$$
  3. Find, in terms of \(p\), the volume of the tetrahedron ABCD .
  4. State the value of \(p\) for which the lines AB and CD intersect, and find the coordinates of the point of intersection in this case. Option 2: Multi-variable calculus
OCR MEI FP3 2010 June Q2
24 marks Challenging +1.2
2 In this question, \(L\) is the straight line with equation \(\mathbf { r } = \left( \begin{array} { r } 2 \\ 1 \\ - 1 \end{array} \right) + \lambda \left( \begin{array} { r } - 2 \\ 2 \\ 1 \end{array} \right)\), and \(\mathrm { g } ( x , y , z ) = \left( x y + z ^ { 2 } \right) \mathrm { e } ^ { x - 2 y }\).
  1. Find \(\frac { \partial \mathrm { g } } { \partial x } , \frac { \partial \mathrm {~g} } { \partial y }\) and \(\frac { \partial \mathrm { g } } { \partial z }\).
  2. Show that the normal to the surface \(\mathrm { g } ( x , y , z ) = 3\) at the point \(( 2,1 , - 1 )\) is the line \(L\). On the line \(L\), there are two points at which \(\mathrm { g } ( x , y , z ) = 0\).
  3. Show that one of these points is \(\mathrm { P } ( 0,3,0 )\), and find the coordinates of the other point Q .
  4. Show that, if \(x = - 2 \mu , y = 3 + 2 \mu , z = \mu\), and \(\mu\) is small, then $$\mathrm { g } ( x , y , z ) \approx - 6 \mu \mathrm { e } ^ { - 6 }$$ You are given that \(h\) is a small number.
  5. There is a point on \(L\), close to P , at which \(\mathrm { g } ( x , y , z ) = h\). Show that this point is approximately $$\left( \frac { 1 } { 3 } \mathrm { e } ^ { 6 } h , 3 - \frac { 1 } { 3 } \mathrm { e } ^ { 6 } h , - \frac { 1 } { 6 } \mathrm { e } ^ { 6 } h \right)$$
  6. Find the approximate coordinates of the point on \(L\), close to Q , at which \(\mathrm { g } ( x , y , z ) = h\).
OCR MEI FP3 2010 June Q3
24 marks Challenging +1.8
3 A curve \(C\) has equation \(y = x ^ { \frac { 1 } { 2 } } - \frac { 1 } { 3 } x ^ { \frac { 3 } { 2 } }\), for \(x \geqslant 0\).
  1. Show that the arc of \(C\) for which \(0 \leqslant x \leqslant a\) has length \(a ^ { \frac { 1 } { 2 } } + \frac { 1 } { 3 } a ^ { \frac { 3 } { 2 } }\).
  2. Find the area of the surface generated when the arc of \(C\) for which \(0 \leqslant x \leqslant 3\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
  3. Find the coordinates of the centre of curvature corresponding to the point \(\left( 4 , - \frac { 2 } { 3 } \right)\) on \(C\). The curve \(C\) is one member of the family of curves defined by $$y = p ^ { 2 } x ^ { \frac { 1 } { 2 } } - \frac { 1 } { 3 } p ^ { 3 } x ^ { \frac { 3 } { 2 } } \quad ( \text { for } x \geqslant 0 )$$ where \(p\) is a parameter (and \(p > 0\) ).
  4. Find the equation of the envelope of this family of curves.
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.
    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.
    OCR is part of the
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 .