# * * MATHEMATICS (MEI) 4757 Further Applications of Advanced Mathematics (FP3) ADVANCED GCE. Wednesday 9 June 2010 Afternoon

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2 Option 1: Vectors 1 Four points have coordinates where p is a constant. A (, 8, 7), B (5, 9, 5), C (8, 0, 1) and D (11, p, p), (i) Find the perpendicular distance from C to the line AB. [5] (ii) Find AB CD in terms of p, and show that the shortest distance between the lines AB and CD is 1 p 5 17p p + 6. [8] (iii) Find, in terms of p, the volume of the tetrahedron ABCD. [] (iv) 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. [7] Option : Multi-variable calculus In this question, L is the straight line with equation r =( 1) + λ( 1 ), and g(x, y, ) = (xy + )e x y. 1 (i) Find g x, g g and y. [] (ii) Show that the normal to the surface g(x, y, ) = at the point (, 1, 1) is the line L. [] On the line L, there are two points at which g(x, y, ) = 0. (iii) Show that one of these points is P (0,, 0), and find the coordinates of the other point Q. [] (iv) Show that, if x = µ, y = + µ, = µ, and µ is small, then g(x, y, ) 6µe 6. [] You are given that h is a small number. (v) There is a point on L, close to P, at which g(x, y, ) = h. Show that this point is approximately ( 1 e6 h, 1 e6 h, 1 6 e6 h). [] (vi) Find the approximate coordinates of the point on L, close to Q, at which g(x, y, ) = h. [7] OCR Jun10

3 Option : Differential geometry A curve C has equation y = x 1 1 x, for x 0. (i) Show that the arc of C for which 0 x a has length a a. [5] (ii) Find the area of the surface generated when the arc of C for which 0 x is rotated through π radians about the x-axis. [5] (iii) Find the coordinates of the centre of curvature corresponding to the point (, ) on C. [9] The curve C is one member of the family of curves defined by where p is a parameter (and p > 0). y = p x 1 1 p x (for x 0), (iv) Find the equation of the envelope of this family of curves. [5] OCR Jun10 Turn over

4 Option : Groups The group F = {p, q, r, s, t, u} consists of the six functions defined by p(x) = x q(x) = 1 x r(x) = 1 x s(x) = x 1 x the binary operation being composition of functions. t(x) = x x 1 u(x) = 1 1 x, (i) Show that st = r and find ts. [] (ii) Copy and complete the following composition table for F. [] p q r s t u p p q r s t u q q p s r u t r r u p t s q s s t q u r p t t s u u u r t (iii) Give the inverse of each element of F. [] (iv) List all the subgroups of F. [] The group M consists of {1, 1, e π j, e π j, e π j, e π j } with multiplication of complex numbers as its binary operation. (v) 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. (vi) Show that G is a cyclic group which can be generated by the element. [] (vii) Explain why G has no subgroup which is isomorphic to F. [1] (viii) Find a subgroup of G which is isomorphic to M. [] OCR Jun10

5 5 Option 5: Markov chains This question requires the use of a calculator with the ability to handle matrices. 5 In this question, give probabilities correct to decimal places. An electronic control unit on an aircraft is inspected weekly, replaced if necessary, and is labelled A, B, C or D according to whether it is in its first, second, third or fourth week of service. In Week 1, the unit is labelled A. At the start of each subsequent week, the following procedure is carried out. When the unit is labelled A, B or C, it is tested; if it passes the test it is relabelled B, C or D respectively; if it fails the test it is replaced by a new unit which is labelled A. When the unit is labelled D, it is replaced by a new unit which is labelled A. The probability that a unit fails the test is 0.16 when it is labelled A, 0.8 when it is labelled B, and 0. when it is labelled C. This situation is modelled as a Markov chain with four states. (i) Write down the transition matrix. [] (ii) In Week 10, find the probability that the unit is labelled C. [] (iii) Find the week (apart from Week 1) in which the probabilities that the unit is labelled A, B, C, D first form a decreasing sequence. Give the values of these probabilities. [] (iv) Find the probability that the unit is labelled B in Week 8 and is labelled C in Week 16. [] (v) Following a week in which the unit is labelled D, find the expected number of consecutive weeks in which the unit is labelled A. [] (vi) Find the equilibrium probabilities that the unit is labelled A, B, C or D. [] An airline has 15 of these units installed in its aircraft. They are all subjected to the inspection procedure described above, and may be assumed to behave independently. (vii) In the long run, find how many of these units are expected to be replaced each week. [] A different manufacturer has now been chosen to make the units. The inspection procedure remains the same as before, but the probabilities that the unit fails the test have changed. The equilibrium probabilities that the unit is labelled A, B, C or D are now found to be 0., 0.5, 0. and 0.15 respectively. (viii) Find the new probabilities that the unit fails the test when it is labelled A, B or C. [] OCR Jun10

6 6 BLANK PAGE OCR Jun10

7 7 BLANK PAGE OCR Jun10

9 GCE Mathematics (MEI) Advanced GCE 757 Further Applications of Advanced Mathematics (FP) Mark Scheme for June 010 Oxford Cambridge and RSA Examinations

10 OCR (Oxford Cambridge and RSA) is a leading UK awarding body, providing a wide range of qualifications to meet the needs of pupils of all ages and abilities. OCR qualifications include AS/A Levels, Diplomas, GCSEs, OCR Nationals, Functional Skills, Key Skills, Entry Level qualifications, NVQs and vocational qualifications in areas such as IT, business, languages, teaching/training, administration and secretarial skills. It is also responsible for developing new specifications to meet national requirements and the needs of students and teachers. OCR is a not-for-profit organisation; any surplus made is invested back into the establishment to help towards the development of qualifications and support which keep pace with the changing needs of today s society. This mark scheme is published as an aid to teachers and students, to indicate the requirements of the examination. It shows the basis on which marks were awarded by Examiners. It does not indicate the details of the discussions which took place at an Examiners meeting before marking commenced. All Examiners are instructed that alternative correct answers and unexpected approaches in candidates scripts must be given marks that fairly reflect the relevant knowledge and skills demonstrated. Mark schemes should be read in conjunction with the published question papers and the Report on the Examination. OCR will not enter into any discussion or correspondence in connection with this mark scheme. OCR 010 Any enquiries about publications should be addressed to: OCR Publications PO Box 5050 Annesley NOTTINGHAM NG15 0DL Telephone: Facsimile:

11 757 Mark Scheme June (i) 5 ACAB AC AB Perpendicular distance is AB OR ( 5) ( 8) ( 6) 0 ft 6 [F is (15, 1, 15)] CF (ii) p 1 ABCD 1 p p p1 p 5 p 1 AC. (ABCD) 8. p 6 p 5( p1) 8( p) 6( p) [ 1p105 ] ABCD (p1) ( p) ( p) B ft ft 5 Give for one component correct Calculating magnitude of a vector product www Appropriate scalar product Distance is 17 p p6 AC. (AB CD) 1 p 5 AB CD 17 p p6 (ag) 8 Correctly obtained (iii) V ( ) (ACAB).AD ( ) p 1 p 7 7 ( ) 56 7( p8) ( p7) 5 7 ( ) p p 5 7 (iv) Intersect when p [8 p ] 7 1 [ 7 1 ( p 1) ] 7, Point of intersection is (17, 15, 1) ft ft ft ft 7 Appropriate scalar triple product In any form Evaluation of scalar triple product Dependent on previous (105 1 p ) or better 1 6 Equations of both lines (may involve p) Equation for intersection (must have different parameters) Equation involving and Second equation involving and or Two equations in,, p Obtaining or 1

12 757 Mark Scheme June 010 (i) g x ( y xy z )e x g x ( x xy z )e y g x y z e z (ii) (iii) (iv) (v) (vi) At (,1, 1), y y g g g,, x y z Normal has direction L passes through (,1, 1) and has this direction When g 0, xy z 0 ( )(1 ) ( 1 ) givesP(0,,0) 1givesQ(, 1, ) g 6 g g At P, e, 0, 0 x y z g g g g x y z x y z 6 6 e ( ) 0 0 6e When 6e h, e h 6 Point (,, ) is approximately ( e h, e h, e h) g 6 g 6 g 6 At Q, e, e, e x y z When x, y 1, z g ( e )( ) (e )( ) ( e )( ) 6 6 e If 6 e h, then 6 e h Point is approximately ( e h, 1 e h, e h) 6 (ag) (ag) (ag) (ag) A 7 Partial differentiation Obtaining a value of Or for verifying g(0,, 0) 0 and showing that P is on L OR give M www for g(,, ) ( 6 )e 6e OR give M www for g(, 1, ) ( 6 )e 6e Give for one coordinate correct If partial derivatives are not evaluated at Q, max mark is M0M0M0

13 757 Mark Scheme June 010 (i) dy x x dx dy ( x x ) dx x x x x Arc length is x x a ( ) ( x x )dx x a (ii) Curved surface area is yds (iii) (iv) x 1 1 a ( )( x x x x )dx ( x x )dx x x x 6 18 dy When x, dx Unit normal vector is 5 5 d y x x ( ) dx 1 ( ) 5 ( ) ( ) c Differentiating partially w.r.t. p 1 0px p x p x 1 8 Envelope is y x x x x y x 0 1 a 0 (ag) ft ft For yds Correct integral form including limits For x x x 6 18 Finding a normal vector Correct unit normal (either direction) Applying formula for or

14 757 Mark Scheme June 010 (i) x 1 x st( x) s x 1 x 1 x x 1 x( x1) 1 r( x) x x (ag) x 1 x 1 ts( x) t x x x 1 1 x x 1 1x q( x) ( x1) x (ii) p q r s t u p p q r s t u q q p s r u t r r u p t s q s s t q u r p t t s u q p r u u r t p q s B Give B for correct, for correct (iii) Element p q r s t u Inverse p q r u t s B Give B for correct, for correct (iv) {p}, F { p, q }, { p, r }, { p, t } {p, s, u} Ignore these in the marking Deduct one mark for each non-trivial subgroup in excess of four (v) Element 1 1 e j e j e j Order e j B Give B for correct, B for correct for correct (vi) 1 5 6,, 8, 16, 1, 7 (vii) , 9, 18, 17, 15, , 6, 1, 5, 10, 1 Hence has order 18 G is abelian (so all its subgroups are abelian) F is not abelian (viii) Subgroup of order 6 is {1,,,,, } i.e. {1,7,8,11,1,18} 1 Finding (at least two) powers of For and 18 Correctly shown All powers listed implies final Can have cyclic instead of abelian or B

15 757 Mark Scheme June 010 Pre-multiplication by transition matrix 5 (i) P (ii) P (iii) Week P (iv) (v) Prob( C) P P Probability is Expected run length is ( sf) (vi) n P A: B: C: D: 0.16 (vii) Expected number is (viii) a b c a b c a0.5b0.c (1 a) (1 b) 0. 0.(1 c) 0.15 a 0.75, b 0., c 0.5 B M A ft Allow tolerance of in probabilities throughout this question Give for two columns correct 9 P Using Give for using 10 P First column of a power of P SC Give B0 for Week 9 and Elements from P 7 8 and Multiplying appropriate probabilities Allow 1. n Evaluating P with n 10 or Obtaining (at least) equations from Pp p Limiting matrix with equal columns or Solving to obtain one equilib prob Give for two correct P Transition matrix and Forming at least one equation Dependent on previous 5

16 757 Mark Scheme June 010 Post-multiplication by transition matrix 5 (i) P (ii) P Prob( C) 0.1 (iii) Week P (iv) (v) P P Probability is Expected run length is ( sf) B M Allow tolerance of in probabilities throughout this question Give for two rows correct 9 P Using Give for using 10 P First row of a power of P SC Give B0 for Week 9 and Elements from P 7 8 and Multiplying appropriate probabilities Allow 1. P (vi) n P A: B: C: D: 0.16 (vii) Expected number is (viii) 5 a 1 a b b c c a0.5b0.c (1 a) (1 b) 0. 0.(1 c) 0.15 a 0.75, b 0., c 0.5 A ft n Evaluating P with n 10 or Obtaining (at least) equations from pp p Limiting matrix with equal rows or Solving to obtain one equilib prob Give for two correct Transition matrix and Forming at least one equation Dependent on previous 6

17 OCR (Oxford Cambridge and RSA Examinations) 1 Hills Road Cambridge C EU OCR Customer Contact Centre 1 19 Qualifications (General) Telephone: Facsimile: For staff training purposes and as part of our quality assurance programme your call may be recorded or monitored Oxford Cambridge and RSA Examinations is a Company Limited by Guarantee Registered in England Registered Office; 1 Hills Road, Cambridge, C EU Registered Company Number: 866 OCR is an exempt Charity OCR (Oxford Cambridge and RSA Examinations) Head office Telephone: Facsimile: OCR 010

19 Reports on the Units taken in June 010 5) (Markov chains) This was found to be more difficult than the corresponding question last year. The techniques were generally well understood and calculators were used competently; parts (i), (ii), (vi), (vii) and (viii) were all answered very well. Many candidates were not sufficiently careful in part (iii), for example having found that P is the appropriate power, giving the answer as week instead of week 5. Most could not answer part (iv) correctly; the usual error was to use elements from P and P instead of from P and P. In part (v), many candidates used the formula p /(1 p) forgetting that, in this case, the first day of the run is to be included in the expected run length.

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