2014 2 sar tun abdul razak
DESCRIPTION
mathsTRANSCRIPT
20
14
-2-S
AR
-TU
N A
BD
UL
RA
ZA
K M
AT
HM
EM
AT
ICS
T M
AR
KIN
G S
CH
EM
E
1.
(a) lim
��� ��2��2,
lim��� ���11��2� ,� lim
��� �����2
� �1 ���2. �lim��� �����2�� �1 �, h
ence f is
contin
uous a
t x =
1.
(b) f is
contin
uous in
the in
terv
al [-3
, 5] , �� �2 �� 2 �2��5� �� ,6��
�� ,������ .
2. �� ���ln�� ��2ln,� !"!� �2 #$%�
%&'� �2cot. ��� ��� + � ,
1+ � , � -�- � 2+ � ,�1+ ���+ � ,�+ ���1+ �� �
� + � ,�2+ ��2�1���1+ �� �
3.
(a) L
et .��1, !/!� �1. 01+ �2,.�1; �3,.�2
4� ,
��5�� 6��
-�4�/7�� ,/ 6
�� -.�4
/ ,7�/7�/ 6
-.��
� 4�/
�/ , �/ 6 -.� 8ln.�
�/ ���/ , 9� �
���
:ln2�1��; <�8ln1�2�
�� 9�ln2��;
y
(b
) ����2� 6, ,� � ����2� �
2
4
x
V
olu
me g
enera
ted =
4=� � -
>�� = 4 ��2� �
>�-�=8 �> ��2� > 9� > �
?> 82 ��0 9�4=
4. B�2� , !C!� �2
!"!� ; !"!� � !C!� �2
!"!� �
��7"7���7"5� ; �
!C!� �2�C7�C5� ; !C!� �
C7�C5� 2�
�CC5�
D B�1B -B�D3 -; �B�lnB�3E ; �
�2� ��ln |2� |�3G.
5.
(a) HIJ ,��
��5#$%��
���
�5��5 �,K� ,,7 �,K� LL! 5 �,K� NN! 7O���
��� 6�
�� P>Q �O
D�� �
RLS-� D
� �3 2 Q45 �ORLS
- � 8 �2 � >
�3��4� 2 T
�45��6� 9S RL�0.2785
(b)
�Q7� �W'��7��X K5�
� �Q7����5 K ,, 7 K 66 5O�Y�7�7 K ,,! 7O Z5�
�Q�5 PK ,, 7� ,7O
�7 K ,,! 7O�
Q5 6K ,, 7O�7 K, 7O
lim��S � �Q7� �W'��7��
X K5���lim
��S Q5 6K ,, 7O�7 K, 7O
�5.
6.
.(a
)
X
0
0.4
0,8
1.2
1.6
��+ � , 0
0.4
69
1.5
17
5.0
65
20.6
97
U
sin
g tra
pe
ziu
m ru
le: 4
+ � ,�.TS
-��� �0.4 � 820.6972 �0.4691.5175.065 � 9�6.96.
(b) T
here
is o
nly
1 p
oin
t of in
ters
ectio
n b
etw
een
G
raphs ����4
and
��ln, h
ence o
nly
1 re
al ro
ot.
� � ��4ln
� \� ��1 1
S �0.02 ; � �0.017883 ; � �0.017988; � �0.017989; �]+�^ ]__`�0.0180
7.
(a) !"!� � "� � �+E � ;a.b.�+ 4 5 cK !��+ 5W'��
��
Multip
lyin
g w
ith I.F
. �� !"!� �"� , ��+E �
; �!!� d "� e��+E �
� �D�+E � -; � � �tanG ; � ��tanG
(b) !�!g h�10���20��; !�!g �i�10���20�� . G
iven
x=
0, !�!g �2.�2�i �10 ��20 �; �
i�0.01
� --` �0.01�10���20�� 4
���S5����S5�� -�4 0.01 -` ; � 4
j�S5k l�S5k -�0.01`G ;�
��S 4��S5� �
��S5� -�0.01`G
110 8ln �20� ��ln�10��9�0.01`G; `�0,�0 ,�G�110 ln 2
��S lnm�S5�
���S5�� m�0.01 ` ; �S5�
���S5�� �+ S.�g ; ���S�X n.co5���X n.co5�
8.
(i) ��cosqln�1�r,; !"!� ��sin �ln �1 ��.��7� ; � �1 � !"!� ��sin �ln �1 ��.
(i) D
iffere
ntia
ting w
rt x : �1 � ! ,"!� ,
!"!� ��cos�ln�1�����7� � ; � �1 � ! ,"!� ,
!"!� ������7� �
Multip
lyin
g b
y (1
+x) : � �1 � � ! ,"!� , �1 � !"!� ��0
.
Diffe
rentia
ting w
r t x: �1 � � ! 6"!� 6 2 �1 � ! ,"!� , �1 � ! ,"!� ,
!"!� + !"!� =
0
�1 � � - ��- � 3 �1 � - ��- � 2 -�- �0
W
hen �0, ��1, !"!� �0, ! ,"!� , � �1, ! 6"!� 6 �3
.
U
sin
g M
acla
urin
’s th
eore
m: ��cosqln�1�r�1�
� ,� �� 6�! O � 1�
� ,� � 6� O
Usin
g th
e s
tandard
serie
s:
ln �1 ��� �2 �3 �O��1� s7� s] O,
cos�1� �2! >4! O ��1 � s �s�2] �! O
��cosqln�1�r�1� ��� ,� � �2
O�1� 12 � �� �O ��1� 12 � 12 �O