Analysis and Approaches for IBDP Maths Ebook 2 | Page 186

Your Practice Set – Analysis and Approaches for IBDP Mathematics
Solution
(a)
1
1 � cosec� �
� �
�
�4
�
�
1 � cosec 4
1 sin 4
�
�
A1
� �
�
4 2
A1
� � 2�
AG
(b) f ( �x) � cosec2 �( � x)
M1
(c)
f ( �x) � cosec( � 2 � x)
1
f( �x)
�
sin( �2 � x)
1
f( �x)
� � A1
sin(2 � x)
f ( �x) � � cosec(2 � x)
f ( �x) � � f ( x)
Thus, f is an odd function.
� 1 1�
�x : � � x � 0 or 0 � x � �
� 4 4�
(d) �y : y 1or y 1�
(e)
AG
A2
� � � A1
1
4 2 1
�cosec 2�xx
d �
a
�
2
(A1) for correct approach
Let u� 2�
x. (M1) for substitution
du
1
� 2�
� du�
�dx
dx
2
1 1 �
x� � u� 2� �
�
��
4 �4 � 2
x � a � u � 2�
a
�
1 1
2 2
�
2
2
� cosec uu d �
2�
a
(A2) for correct working
�
2 2
cosec uu d �1
2�
a
�
[2]
[2]
[2]
[1]
� � 2 2
�cot � 1
A1
u � � a
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