EDEXCEL PURE MATHEMATICS C4 (6666) SPECIMEN PAPER MARK SCHEME

54

Questionnumber Scheme Marks

1. 21

)34(�

� x = 21

4� 2

1

431

�

��

���

�� x

= 21 � �� �� � � �� �� �� �

��

�

�

��

�

��

�����

����� .....

62831

343

25

23

212

43

23

21 xx

x B1 M1

= 21 + x

163 , + 2

25627 x , + 3

2048135 x ……. A1, A1, A1

(5 marks)

2. 26x + 26yy´ ; � 10xy´ – 10y = 0 M1A1; M1A1

y´(26y – 10x) = 10y – 26x

y´ = xyxy

10262610

�

�

= xyxy

513135�

�

M1 A1

(6 marks)

3. x = tan ��d

dx = sec2 � � I = ��

� dsecsec

4

2

��

� M1 A1

Limits 4� and 0 B1

I = ��

��� dcos2 = �

� d2

12cos��� �

M1 A1

= 4

0242sin

�

��

��

���

�� M1 A1

= 84

1 �

� (*) A1 cao

(8 marks)

EDEXCEL PURE MATHEMATICS C4 (6666) SPECIMEN PAPER MARK SCHEME

55

Questionnumber Scheme Marks

4. (a) tx

dd = sec2 t

ty

dd = 2 cos 2t, �

xy

dd =

tt

2sec2cos2

M1 A1, � M1

When t = 3� gradient is

41

� B1 (4)

(b) y � 23 =

m1

� (x – �3) P has coordinates (�3, 23 ) B1

y � 23 = 4 (x – �3) M1

y = 4x � 327

A1 (3)

(c)xy

dd = 0 � gradient of tan = 0, gradient of normal undefined M1

� x = tan 4� , i.e: x = 1 A1 (2)

(9 marks)

5. (a) 5 + � = 2 � 3�; 3 � 2� = �11 � 4� B1 B1

�� + 3� + 3 = 0

2� � 4� � 14 = 0

2� + 6� + 6 = 0

10� + 20 = 0 � � = �2 � � = 3 M1 A1

� point is (8, �3, 4) A1 (5)

(b) � a � 10 = 4 � a = 14 M1 A1 (2)

(c) cos � = 252591083�

���

M1 A1

= 253

15�

= 2

1

Angle = 45� M1 A1 (4)

(11 marks)

EDEXCEL PURE MATHEMATICS C4 (6666) SPECIMEN PAPER MARK SCHEME

56

Questionnumber Scheme Marks

6. (a) 11x � 1 � A(2 + 3x) + B(1 � x)(2 + 3x) + C(1 � x)2

Putting x = 1 � A = 2 B1

Putting x = 32

� � 325

� = C925

� C = � 3 B1

cf x2 0 = �3B + C � B = �1 M1A1 (4)

(b)� ��

�

�

�

21

021

2x

� )1(

1x�

� )32(

3x�

dx

= ��

���

�����

�xx

x32ln1ln

12 M1 A1ft A1ft

A1ft

= [4 + ln 21 � ln 3 2

1 � (2 � ln 2)] M1

= 2 + ln 21

21

32�

M1

= 2 + ln 72 A1 (7)

(11 marks)

7. (a) xu

dd = 2

1 � 21 cos 4x; = 2

1 � 21 (1 � 2 sin2 2x) = sin2 2x M1 A1; M1 A1

(4)

(b) V = � � x sin2 2x dx M1

= �4

0

d4sin81

24sin

81

2

�

��

���

��� �

�

��

� xxxxxx M1 A1 A1

= � 4

0

22

4cos321

44sin

82

�

��

���

����

�

� �� xxxxx M1 A1

= � ��

���

���

�

321

321

64

2 = � �

�

���

��

�

161

64

2

M1 A1 (8)

(12 marks)

EDEXCEL PURE MATHEMATICS C4 (6666) SPECIMEN PAPER MARK SCHEME

57

Questionnumber Scheme Marks

8. (a)tr

dd = 2r

kB1

A = �r2 �rA

dd = 2�r M1A1

� tA

dd = 2�r 2r

k = � �rk�2 ; = � �

21

2

��

���

�

�

�

A

k = A

k23

2�M1; M1

� tA

dd �

A1 (*) A1 (6)

(b) � S dS = � 2e2t dt M1

23

32 S = e2t + C M1A1

t = 0, S = 9 � C = 17 B1

� 23

32 S = e2t + 17 and use S = 16 M1

��

���

��17

3128 = e2t � t =

21 ln ��

���

�

377

M1

= 1.6 A1 (7)

(13 marks)