# c4 marking schemes

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C41

SERIESa = 1 + (1)x + b = 1 + ( 1 )x + 2 =1+1 2( 1)( 2) 2 3

Answers - Worksheet Ax2 +( 1)( 2)( 3) 3 2

x3 +

= 1 x + x2 x + x

( 1 )( 1 ) 2 ( 1 )( 1 )( 3 ) 2 2 2 x + 2 32 2 2 1 2 1 x + 16 x3 + 8 ( 3)( 4) 2 2 3

x3 +

c = 2[1 + (3)x + d = 1 + ( 2 )x + 3 =1+2 3

x2 +

( 3)( 4)( 5) 3 2

x3 + ]

= 2 6x + 12x 20x + x1

( 2 )( 1 ) 2 ( 2 )( 1 )( 4 ) 3 3 3 x + 3 32 3 2 1 2 4 x + 81 x3 + 9 ( 1 )( 2 ) 3 3 2

x3 +

e = (1 x) 3 = 1 + ( 1 )(x) + 3 =11 3

(x)2 +

( 1 )( 2 )( 5 ) 3 3 3 3 2

(x)3 +

x

1 9

x2

5 81

x3 + ( 2)( 3) 2

f = (1 + x)2 = 1 + (2)x + = 1 2x + 3x2 4x3 + g = =1 4 1 4

x2 +

( 2)( 3)( 4) 32

x3 + ( 4)( 5)( 6) 3 2

(1 x)4 = +x+5 2

1 4

[1 + (4)(x) +

( 4)( 5) 2

(x)2 +

(x)3 + ]

x2 + 5x3 + = 3[1 + ( 1 )(x) + 29 8 ( 1 )( 3 ) 2 2 2

h = 3(1 x) =3+ 23 2

1 2

(x)2 +

( 1 )( 3 )( 5 ) 2 2 2 3 2

(x)3 + ]

x+

x2 +

15 16

x3 + (2x)3 + valid for | x | < (3x)3 + 1 3 1 2

a = 1 + ( 1 )(2x) + 2 =1+x1 2

x2 +

( 1 )( 1 ) ( 1 )( 1 )( 3 ) 2 2 2 (2x)2 + 2 32 2 2 1 3 x + , | 2x | < 1 2 ( 1)( 2) 2 3

b = 1 + (1)(3x) +

(3x)2 +

( 1)( 2)( 3) 3 2

= 1 + 3x + 9x2 + 27x + , | 3x | < 1 c = 1 + ( 1 )(4x) + 2 d = 1 + (3)( 1 x) + 2 =13 2 ( 1 )( 3 ) 2 2 2 3

valid for | x | < (4x)3 +

(4x)2 +

( 1 )( 3 )( 5 ) 2 2 2 3 2

= 1 + 2x + 6x2 + 20x + , | 4x | < 13 2

valid for | x | < ( 1 x)3 + 2

1 4

x+

x2

( 3)( 4) 2 5 3 x + 4

( 1 x)2 + 2

( 3)( 4)( 5) 3 2

, | 1 x | < 1 2( 1 )( 2 )( 5 ) 3 3 3 3 2

valid for | x | < 2 (6x)3 + valid for | x | < ( 1 x)3 + 41 6

e = 1 + ( 1 )(6x) + 3 = 1 2x 4x2 f = 1 + (4)( 1 x) + 4 =1x+5 8

( 1 )( 2 ) 3 3 2 40 3 x + 3

(6x)2 +

, | 6x | < 1 x)2 + |1 x| < 1 4

x2

( 4)( 5) 1 (4 2 5 3 x + , 16

( 4)( 5)( 6) 3 2

valid for | x | < 4 (2x)3 + valid for | x |

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