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