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Joint Entrance Examination
Mathematics
Functions
Question 90
Question 90 of 162
Medium
MCQ
Let fk(x) = \[{1 \over k}\left( {{{\sin }^k}x + {{\cos }^k}x} \right)\] for k = 1, 2, 3, ... Then for all x \[ \in \] R, the value of f4(x) - f6(x) is equal to
A
\[{1 \over 4}\]
B
\[{5 \over {12}}\]
C
\[{{ - 1} \over {12}}\]
D
\[{1 \over {12}}\]
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Official solution
f4(x) - f6(x) = \[{1 \over 4}\] (sin4 x + cos4 x) - \[{1 \over 6}\] (sin6 x + cos6 x) = \[{1 \over 4}\] (1- \[{1 \over 2}\] sin2 2x) - \[{1 \over 6}\] (1 - \[{3 \over 4}\] sin2 2x) = \[{1 \over 12}\]
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