Nov 2006 P1

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Ordinary Level

MARK SCHEME for the October/November 2006 question paper

4024 MATHEMATICS 4024/01

Paper 1, maximum raw mark 80

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 Examiners were instructed to award marks. It does not indicate the details of the discussions that took place at an Examiners’ meeting before marking began. 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 must be read in conjunction with the question papers and the report on the examination. The grade thresholds for various grades are published in the report on the examination for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses.

•

CIE will not enter into discussions or correspondence in connection with these mark schemes.

CIE is publishing the mark schemes for the October/November 2006 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses.

Page 2

1

7

(b) (a) (b) (a)

8

(b) (c) (a)

1 8 o.e. 15 1.77(0….) 147 5x6 1 1 or −2 2 80 1 62 2 7 7 0.72, , 0.7, 9 11 400 34 −9 13 o.e. 18 70 c.a.o. 8 c.a.o. 22 x 3 3

(b)

23 x 3 3 x 5

(c) (a) (b)

75 or 3 x 52 −1 (Yx <) 2 −1, 0, 1 √

2 3

4

5

6

9

10 11

(a) (b)

Mark Scheme GCE O LEVEL - OCT/NOV 2006

(a) (b) (a) (b) (a) (b) (a)

(a) (b) (a) (b)

12

13

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1* B1 + B1

NB: 0 must be included 5:2 c.a.o 2.1 x 108 4 o.e. 15  2 1  −  C = 1600 o.e. 5 3 ($)24 000

(a)

3a − 2c o.e.

(b)

Establishing k OP = l BA

(c)

3 o.e. 2 Correct, ruled, line (and no others) correct method to produce 900 (7 sided) or correct method to produce 720 (6 sided) or correct method to produce 540 (5 sided) or 360 − their 54 or 6x = 360 – 54 6 (ext < method) 129

(a) (b)

Syllabus 4024

1 2 1√ 1 2 1

M1 A1

Paper 01

Accept 0.53 or better

Accept

3 , 1.5 2

125 Accept 62.5 2 Accept any equivalents

Not

Accept –34, ±34 6.5 9 Accept –70, ±70, 7 x 10, 10 x 7 Accept –8, ±8 Not 8 x 1 Accept 2 x 2 etc. condone x1n throughout Answer 1080 look back. Give mark if correct prime factors seen

Not

Reversed answers – SC1 Given −p Y x < q in (a), allow √ if p and q are positive integers Inclusion of units ⇒ no marks SC1 for figs. 21; Condone –2.1 x 108 4x Allow 15 1 s.o.i. SC1 for 15

2* 1 1* 1

Accept 1.5, 3:2

1

Accept if line dotted. 3 mm tolerance

M1

A1

© UCLES 2006

Must be numerical

2*

Page 3

14

(a) (b)

15

16

(a) (b) (c)

Mark Scheme GCE O LEVEL - OCT/NOV 2006 16 27 2 1 1 4 x + x 3 9 3 9 6 27 9 3π π 60o or 3

M1 A1

* 1

One correct 2

(b)

 −  (i) (ii) 4 2 pairs of angles stated equal Reasons + conclusion (dep on 1st B1)

B1 B1

(i) (ii)

M1

18

(b) (a) (b)

2 5√ 3 their (a) −1 1  ,0  and (5,0 ) 2  2 1

1 1 1

12

x 2 = o.e. 14 − x 3 28 BX= o.e. 5

2* 1

Both correct 3

(i) (ii)

(a)

Paper 01

1

(a)

17

Syllabus 4024

1 1 2

Accept (y α ) x2 or (y =) kx2 Accept (y α ) x1 or (y =) kx1

2

Lost for wrong or irrelevant statements

1

A1

© UCLES 2006

In (b) condone 3, k = 3, 3kπ. If no marks:- M1 for circum = 18π or 2π x their (a) √ Must be numerical Condone 2p Condone 5q or 5x

2*

Page 4

14

15

(a) (b) (a) (b)

16

(a) (b)

17

18

(a) (b)

Mark Scheme GCE O LEVEL - OCT/NOV 2006 1550 (Y distance <) 1650 5.5 (Y speed <) 6.5 300 sec o.e.  − 4 2   o.e.  − 6 0

B1 B1

1  2 3   o.e. B1 + B1 2  4 5  (i) 4p + 7 c.a.o (ii) −1√ solution of (their 4p + 7) = 3 2 (a−1) −1 M1 a2 − 2a or a(a – 2) A1 y = 2x + 3 o.e. (i) Lines x = 1 and y = 3 drawn B1 Lines x + y = 2 drawn B1 (ii) Correct region identified dept. on all 3 lines correct condoning minor inacc.

–

(a)

Syllabus 4024

2 1 2

2*

Paper 01

SC1 for any 2 seen SC1 for 3 correct elements condone intrusive letters seen and isw

1 1 2* 1 2 1

Part of region below the x axis should be indicated

1

(b)

P ∩ Q/ o.e.

(c)

25 − x + x + 20 − x + 4 (= 36) 13

1 M1 A1

© UCLES 2006

2*

(P ′ ∪ Q )′ Diag. with x, 25 − x, 20 – x, 4 all marked earns the M1

Page 5

19

20

(a) (b) (c) (d) (a) (b) (c)

21

(a)

Mark Scheme GCE O LEVEL - OCT/NOV 2006 20 110 20 50 or 180 – [their (y + z)] √ 3 c.a.o 3:5 c.a.o. or 5 9:25 or (their (a))2

22

A1

8

95

(a)

Fully correct graph or √ to their 95 St. line (+ve gradient) from t = 0-6 correct curvature from t = 6-8 horiz line (not on axis) from t = 8-12 correct curvature from t = 12-20 (i) 15

(a) (b)

2* 1* 1 1

Graph from (0,0) to (20, 95) √

(10,9)

Accept fraction: condone inclusion of units Accept 9π: 25π 3

Allow 1.88 but not 1.9; Not

7.5 4

Graph must be continuous and non decending

2

If graph not fully correct:SC1 for 2 or 3 parts correct

1

Not –15 [but allow √ mark in (c) for –30]

1 1

30 √ 2 x their 15

6 o.e. 10 k 2 − 111 5 − or k 10k Arc of circle, centre L, radius 2 cm St lines, parallel to AB and BC, 2 cm distance Fully correct locus

( ) o

1 1 1 B1 +B1 dep B1 + B1

2 2

(c)

25 (and) 48 or 29 and 48 ± 2

(a) (b)

∆ drawn (4,4), (8,4) and (10,2) Rotation

B1

90 o CW, centre (0,0)

B1

2

∆ drawn (−2, 2), (−4, 2), (−5, 1)

B2

2*

(c)

Paper 01

5 NB.   is M1 3

M1

(b)

(b)

24

1

3 Idea of   5 27:98 c.a.o (i) 15 o.e. seen

(ii) (iii) (iv)

23

1 1 1 1 1

3

(ii)

Syllabus 4024

Accept −

5

61 Allow within 2 mm

Correct locus range 23 → 50 incl. If sharp loci range 27 → 50 incl. SC1 if one angle in range or for reversed angles √ from their loci (arc or point) dept. on relevant locus

1

© UCLES 2006

Not turn: extra transf. seen loses both marks 0 Condone –90°; Allow   or 0. 0 SC1 for 2 points plotted or for 3 pts stated