GCE. Mathematics. Mark Scheme for June Advanced GCE 4725 Further Pure Mathematics 1

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1 GCE Mathematics Advanced GCE 475 Further Pure Mathematics Mark Scheme for June 00

2 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 00 Any enquiries about publications should be addressed to: OCR Publications PO Box 5050 Annesley NOTTINGHAM NG5 0DL Telephone: Facsimile:

3 475 Mark Scheme June Establish result true for n = or n = Add next term to given sum formula Attempt to factorise or expand and simplify to correct expression Correct expression obtained Specific statement of induction conclusion (i) ( 7) Obtain a single value Obtain correct answer as a matrix BA = 5 0 Obtain a matrix All elements correct C seen or implied by correct answer ft 4 Obtain correct answer, ft for a slip in BA Either Express as a sum of terms Use standard sum results n ( n + )(n + ) n( n + ) + n Correct unsimplified answer n (n )(n + ) Or n r = r 4 n r = r n (n + )(4n + ) 4 n( n + )(n + ) n (n )(n + ) Attempt to factorise Obtain at least factor of n and a quadratic Obtain correct answer a.e.f. Express as difference of Use standard result Correct unsimplified answer Attempt to factorise Obtain at least factor of n Obtain correct answer r series

4 475 Mark Scheme June 00 4 (i) 5 + i 7.4 o or.8 ft ft 4 Correct real and imaginary parts Correct modulus Correct argument i 7 Multiply by conjugate Obtain correct numerator Obtain correct denominator 5 (a) 0 Each column correct 0 SC B use correct matrix from MF Can be trig form (b) (i) Stretch, in x-direction sf 5 Rotation, 0 o clockwise (i) (a) (b) Circle centre (, 4 ), through origin Vertical line, clearly x = ft ft Inside their circle And to right of their line, if vertical

5 475 Mark Scheme June 00 7 Either α + β = k αβ = k y 4ky + 4k = 0 ft 7 State or use correct results Attempt to find sum of new roots Attempt to find product of new roots Correct quadratic equation a.e.f. Or α + β = k k α k y = x y 4ky + 4k = 0 State or use correct result State or imply form of new roots State correct substitution Rearrange and substitute for x Correct unsimplified equation Attempt to clear fractions Correct quadratic equation a.e.f. Or k ± α + β = α k k k + k α + β, = k β k k k k k Find roots of original equation Express both new roots in terms of k y 4ky + 4k = 0 ft Attempt to find sum of new roots Attempt to find product of new roots Correct quadratic equation a.e.f. 7

6 475 Mark Scheme June 00 8 (i) Attempt to rationalise denominator or cross multiply Obtain given answer correctly ( n + + n + ) Express terms as differences using (i) Attempt this for at least st three terms st three terms all correct Last two terms all correct Show pairs cancelling Obtain correct answer, in terms of n (iii) 9 Sensible statement for divergence 9 (i) det A = a a Show correct expansion process for x Correct evaluation of any x Obtain correct answer (a) (b) (c) 9 Find a pair of inconsistent equations State inconsistent or no solutions Find a repeated equation State non unique solutions State that det A is non-zero or find correct solution State unique solution SC if deta incorrect, can score marks for correct deduction of a unique solution, but only once 0 (i) Attempt to equate real and imaginary parts x y = xy = Obtain both results Eliminate to obtain quadratic in x or y Solve to obtain x or y value z = + i 5 Obtain correct answer as a complex no. Obtain given answer correctly (iii) w = ± i w = i 5 Attempt to solve quadratic equation Obtain correct answers Choose negative sign Relate required value to conjugate of (i) Obtain correct answer 4

7 OCR (Oxford Cambridge and RSA Examinations) Hills Road Cambridge C EU OCR Customer Contact Centre 4 9 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; Hills Road, Cambridge, C EU Registered Company Number: 4844 OCR is an exempt Charity OCR (Oxford Cambridge and RSA Examinations) Head office Telephone: Facsimile: OCR 00

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