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Oxford Cambridge and RSA Examinations

GCE

Advanced GCE A2 7890 2

Mathematics

Advanced Subsidiary GCE AS 3890 2

Mark Schemes for the Units June 2009

3890-2/7890-2/MS/R/09

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, GCSEs, OCR Nationals, 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 syllabuses 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 todays 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 2009 Any enquiries about publications should be addressed to: OCR Publications PO Box 5050 Annesley NOTTINGHAM NG15 0DL Telephone: 0870 770 6622 Facsimile: 01223 552610 E-mail: [email protected]

CONTENTS

Advanced GCE Mathematics (7890) Advanced GCE Pure Mathematics (7891)

Advanced GCE Further Mathematics (7892)

Advanced Subsidiary GCE Mathematics (3890) Advanced Subsidiary GCE Pure Mathematics (3891)

Advanced Subsidiary GCE Further Mathematics (3892)

MARK SCHEMES FOR THE UNITS Unit/Content Page

4721 Core Mathematics 1 1

4722 Core Mathematics 2 5

4723 Core Mathematics 3 8

4724 Core Mathematics 4 12

4725 Further Pure Mathematics 1 17

4726 Further Pure Mathematics 2 20

4727 Further Pure Mathematics 3 24

4728 Mechanics 1 30

4729 Mechanics 2 33

4730 Mechanics 3 35

4731 Mechanics 4 39

4732 Probability & Statistics 1 45

4733 Probability & Statistics 2 50

4734 Probability & Statistics 3 54

4735 Probability & Statistics 4 57

4736 Decision Mathematics 1 60

4737 Decision Mathematics 2 64

Grade Thresholds 69

4721 Mark Scheme June 2009

1

4721 Core Mathematics 1

1 (i)

(ii)

4 3d 5 2dy x xx

23 4

2

d 20 6d

y x xx

B1 M1 A1 3

M1 A1 2 5

45x 2x before differentiation or in 3kx d

dyx

soi

32 x

Attempt to differentiate their (i) at least one term correct cao

2

72374

769

72727278

M1

A1 A1

A1 4 4

Multiply numerator and denominator by conjugate

Numerator correct and simplified Denominator correct and simplified

cao

3 (i) (ii) (iii)

23

31

3

40

3010

333

B1 1 B1 1 M1

A1 2 4

303 or soi 209

4

0)32)(12(038406168101616810)42(4

42

2

2

2

22

xxxxxxxxxx

xy

21

x , 23

x

y = -3, y = -1

M1*

A1 M1dep*

A1 A1 A1 6

6

Attempt to get an equation in 1 variable only

Obtain correct 3 term quadratic (aef) Correct method to solve quadratic of form ax2 + bx + c = 0 (b 0) Correct factorisation oe

Both x values correct

Both y values correct or one correct pair of values www B1 second correct pair of values B1

4721 Mark Scheme June 2009

5 (i) (ii)

2

3 2 2

3 2

(2 5 3)( 4)2 8 5 20 3 1

2 3 23 12

x x xx x x x x 2

x x x

4

44

972

xxx

9

M1

A1 A1 3

B1

B1 2

5

Attempt to multiply a quadratic by a linear factor or to expand all 3 brackets with an appropriate number of terms (including an x3 term)

Expansion with no more than one incorrect term

42x or soi www 47x

49x or 9

6 (i) (ii) (iii)

Translation Parallel to y-axis, 5 units

2xy

B1 B1 2 B1 B1 2 M1

A1 2 6

One to one graph only in bottom right hand quadrant Correct graph, passing through origin

2 or 2xx seen

cao

7 (i) (ii)

625

41

25

25

2

22

x

x

0625 2

2

yx

Centre

0,

25

Radius = 6

B1 M1 A1 3

B1 B1 B1 3 6

25

a

2

41 a

cao Correct x coordinate Correct y coordinate

2

4721 Mark Scheme June 2009

8 (i) (ii)

-42 < 6x < -6 -7 < x < -1

2x > 16 x > 4 or x < -4

M1 A1 A1 3 B1 B1 B1 3 6

2 equations or inequalities both dealing with all 3 terms -7 and -1 seen oe -7 < x < -1 (or x > -7 and x < -1) 4 oe seen x > 4 x < - 4 not wrapped, not and

9 (i) (ii)

(iii)

2 2( 1 4) (9 3)

=13

3 ,23

293,

214

Gradient of AB = 5

12

027512

15

123

yx

xy

M1 A1 2 M1 A1 2 B1

M1 A1 A1 4 8

Correct method to find line length using Pythagoras theorem cao Correct method to find midpoint

Correct equation for line, any gradient, through (1, 3) Correct equation in any form with gradient simplified

027512 yx

10 (i) (ii) (iii) (iv)

31,

37

0)13)(73(

xx

xx

d 18 18d18 18 0

116

y xxx

xy

x > -1

M1 A1 A1 3 M1 M1

A1 A1 ft 4 B1 B1 B1 3

B1 1 11

Correct method to find roots Correct factorisation oe Correct roots Attempt to differentiate y

Uses d 0dyx

Positive quadratic curve y intercept (0, -7) Good graph, with correct roots indicated and minimum point in correct quadrant

3

4721 Mark Scheme June 2009

11 (i) (ii)

Gradient of normal = 32

12d 1

d 2y kxx

When x = 4, dd 4y kx

23

4

k

6k P is point (4, 12) Q is point (22, 0)

Area of triangle = 221221

= 132 sq. units

B1

M1* A1 M1dep* M1dep* A1 6 B1 ft M1 A1 M1 A1 5 11

Attempt to differentiate equation of curve

21

21 kx

Attempt to substitute x = 4 into their d

dyx

soi

Equate their gradient expression to negative reciprocal of their gradient of normal cao Correct method to find coordinates of Q Correct x coordinate

Must use y coordinate of P and x coordinate of Q

4

4722 Mark Scheme June 2009

5

4722 Core Mathematics 2

1 (i) cos = 0.74.623.110.74.6 222

M1 Attempt use of cosine rule (any angle)

= 0.4211 A1 Obtain one of 115o, 34.2o, 30.9o, 2.01, 0.597, 0.539 = 115o or 2.01 rads A1 3 Obtain 115o or 2.01 rads, or better (ii) area = 115sin4.672

1 M1 Attempt triangle area using ()absinC, or equiv

= 20.3 cm2 A1 2 Obtain 20.3 (cao) 5

2 (i) a + 9d = 2(a + 3d) M1* Attempt use of a + (n 1)d or a + nd at least once for u4, u10 or u20

a = 3d A1 Obtain a = 3d (or unsimplified equiv) and a + 19d = 44 a + 19d = 44 22d = 44 M1dep* Attempt to eliminate one variable from two simultaneous

equations in a and d, from u4, u10, u20 and no others d = 2, a = 6 A1 4 Obtain d = 2, a = 6 (ii) S50 = 50/2 (2x6 + 49x2) M1 Attempt S50 of AP, using correct formula, with n = 50, allow 25(2a + 24d) = 2750 A1 2 Obtain 2750

6

3 M1 Introduce logarithms throughout, or equiv with base 7 or 2 12log7log xx

M1 Drop power on at least one side 2log17log xx A1 Obtain correct linear equation (allow with no brackets) M1 Either expand bracket and attempt to gather x terms, 2log2log7log x or deal correctly with algebraic fraction x = 0.553 A1 5 Obtain x = 0.55, or rounding to this, with no errors seen 5

4 (i)(x2 5)3 = M1* Attempt expansion, with product of powers of x 3222232 55353 xxx 2 and + 5, at least 3 terms

= M1* Use at least 3 of binomial coeffs of 1, 3, 3, 1 1257515 246 xxx A1dep* Obtain at least two correct terms, coeffs simplified A1 4 Obtain fully correct expansion, coeffs simplified OR (x2 5)3 = (x2 5)(x4 10x2 + 25) M2 Attempt full expansion of all 3 brackets = x6 15x4 + 75x2 125 A1 Obtain at least two correct terms A1 Obtain full correct expansion

(ii) cxxxxxx 125253d5 3577132 M1 Attempt integration of terms of form kxn

A1 Obtain at least two correct terms, allow unsimplified coeffs A1 Obtain xxxx 125253 3577

1