Fhmm1014_april2013_5_.docx
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Q1.
(a)
z = 2 - 2i
(i)
z = 4+ 4 = 2 2
(ii)
arg( z ) = q =
(iii) = (b)
= (i)
3log x
(ii) (c)
-p 4
x log 3
2 3log x
3log x = 81
�
3log x = 34
log x = 4
�
x = 104
x =1
P( x) = A( x - 2)( x - (2 - i ))( x - (2 + i))
P ( x ) = A( x - 2)( x 2 - 4 x + 5) P ( x ) = A( x3 - 6 x 2 + 13x - 10) P ( x) =
(d)
1 3 ( x - 6 x 2 + 13 x - 10) 2
( A - ( B I C ) ) �=
( A I ( B I C ) �) �
= A� U( B I C ) = ( A� U B ) I ( A� UC ) = ( A I B� ) �I ( A I C � )� = ( A - B)� I ( A-C)�
Q2.
(a)
x3 - 2 x 2 + 3x - 4 - x2 + 2x - 3 = 1 + x3 - x 2 + x - 1 x3 - x 2 + x - 1
x3 - 2 x 2 + 3x - 4 1 2 = 1+ 2 3 2 x - x + x -1 x - 1 ( x + 1)
162
2 FHMM1014 MATHEMATICS I (b)
(c)
(i)
x>
1 4
1< x < 2
(ii)
x �0
(i)
2n - 1 an = n 3 `
(ii)
2n - 1 �2 � �1 � lim an = lim n = lim � �- lim � �= 0 n �� n �� 3 n �� 3 � � n���3 �
and
n
n
The series is convergent.
¿
Q3
(a)
(i)
r= 6 ¿ ⇒ ¿ a = ¿
5 21
¿ r=−6 ¿ ⇒ ¿ a=− ¿
(ii)
S = 319
1 2
(b)
(i)
1 3
(1 - x) = 1 -
1 1 1 x - x 2 - x 3 + ... 2 8 16
1
(ii)
(iii)
(c)
Q4.
(a)
2 � 1 � 50 � 1 - 2 � = 7 51 � 50 �
51 �
4999 700
(i)
�QPR = 123.03�
(ii)
�PQR = 24.83�
(iii)
PS = 16.96 cm
(i)
1 1 1 + cos A + 1 - cos A + = 1 - cos A 1 + cos A (1 - cos A)(1 + cos A)
3 FHMM1014 MATHEMATICS I =
2 1 - cos 2 A
=
2 sin 2 A
= 2 csc 2 A
(ii) (b)
q = 0.4637 r , 1.107r , 3.605r , 4.249 r
(i)
12 sin q - 5cos q �13sin(q - 22.62� )
(ii)
q = 55.2�, 170.04�
(iii)
1 1 1 is = -13 + 16 3 The maximum value of 12sin q - 5cos q + 16
(i)
� 3� �- 5 � a=� � ~ �4 � � � �5 � ^
(c)
(ii)
p=-
q=
(iii)
109 13
89 13
a� b = -3(1) + 4(3) = 9 ~ ~
q = 55.31�
(a)
z = 2 - 2i
(i)
z = 4+ 4 = 2 2
(ii)
arg( z ) = q =
(iii) = (b)
= (i)
3log x
(ii) (c)
-p 4
x log 3
2 3log x
3log x = 81
�
3log x = 34
log x = 4
�
x = 104
x =1
P( x) = A( x - 2)( x - (2 - i ))( x - (2 + i))
P ( x ) = A( x - 2)( x 2 - 4 x + 5) P ( x ) = A( x3 - 6 x 2 + 13x - 10) P ( x) =
(d)
1 3 ( x - 6 x 2 + 13 x - 10) 2
( A - ( B I C ) ) �=
( A I ( B I C ) �) �
= A� U( B I C ) = ( A� U B ) I ( A� UC ) = ( A I B� ) �I ( A I C � )� = ( A - B)� I ( A-C)�
Q2.
(a)
x3 - 2 x 2 + 3x - 4 - x2 + 2x - 3 = 1 + x3 - x 2 + x - 1 x3 - x 2 + x - 1
x3 - 2 x 2 + 3x - 4 1 2 = 1+ 2 3 2 x - x + x -1 x - 1 ( x + 1)
162
2 FHMM1014 MATHEMATICS I (b)
(c)
(i)
x>
1 4
1< x < 2
(ii)
x �0
(i)
2n - 1 an = n 3 `
(ii)
2n - 1 �2 � �1 � lim an = lim n = lim � �- lim � �= 0 n �� n �� 3 n �� 3 � � n���3 �
and
n
n
The series is convergent.
¿
Q3
(a)
(i)
r= 6 ¿ ⇒ ¿ a = ¿
5 21
¿ r=−6 ¿ ⇒ ¿ a=− ¿
(ii)
S = 319
1 2
(b)
(i)
1 3
(1 - x) = 1 -
1 1 1 x - x 2 - x 3 + ... 2 8 16
1
(ii)
(iii)
(c)
Q4.
(a)
2 � 1 � 50 � 1 - 2 � = 7 51 � 50 �
51 �
4999 700
(i)
�QPR = 123.03�
(ii)
�PQR = 24.83�
(iii)
PS = 16.96 cm
(i)
1 1 1 + cos A + 1 - cos A + = 1 - cos A 1 + cos A (1 - cos A)(1 + cos A)
3 FHMM1014 MATHEMATICS I =
2 1 - cos 2 A
=
2 sin 2 A
= 2 csc 2 A
(ii) (b)
q = 0.4637 r , 1.107r , 3.605r , 4.249 r
(i)
12 sin q - 5cos q �13sin(q - 22.62� )
(ii)
q = 55.2�, 170.04�
(iii)
1 1 1 is = -13 + 16 3 The maximum value of 12sin q - 5cos q + 16
(i)
� 3� �- 5 � a=� � ~ �4 � � � �5 � ^
(c)
(ii)
p=-
q=
(iii)
109 13
89 13
a� b = -3(1) + 4(3) = 9 ~ ~
q = 55.31�
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