Analysis and Approaches for IBDP Maths Ebook 2 | Page 63

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Paper 1 Section A – Contradiction and Counter Examples
Example
(a)
(b)
If N is an odd number, prove that
By using a counter example, show that
where P , Q� .
2
(3N � 2) is also an odd number.
( P Q)
P Q
3 3 3
� � � is not always true,
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Solution
(a)
Suppose
2
(3N � 2) is an even number. M1
6
3N
2 (3 2)
2
� � N� is an even number A1
3N is an even number
N is an even number, which contradicts with the
statement N is an odd number.
Thus, if N is an odd number, then
an odd number.
2
(3N � 2) is also
AG
(b) Let P � 0.5 and Q �� 0.5 . A1
( P�Q) � (0.5 � ( � 0.5))
( P�Q) � 0
3 3
3 3
3
( P�Q) � 0
P
P
P
�Q
� 0.5 � ( � 0.5)
M1
3 3 3 3
�Q
� 0.125 � 0.125
3 3
�Q
� 0
3 3
�( P � Q)
� P � Q
Thus,
3 3 3
( P Q)
P Q
3 3 3
� � � is not always true. AG
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