QJEEM14Mab993: JEE Main 2014 Mathematics - Class 11 JEEM14Mab993 Solution: Let f k ( x ) = 1 k ( sin k x + cos k x ) where x ∈ R and k ≥…

Question

Verified Solution • 2025–26 Session moderate

Let

f_{k} (x) = \frac{1}{k} (\sin^{k} x + \cos^{k} x)

where

x \in R

and

k \geq 1.

Then

f_{4} (x) - f_{6} (x)

equals :

Accepted Answer

Let

f_{k} (x) = \frac{1}{k} (\sin^{k} x + \cos^{k} . x)

Consider

f_{4} (x) - f_{6} (x)

=

\frac{1}{4} (\sin^{4} x + \cos^{4} x) - \frac{1}{6} (\sin^{6} x + \cos^{6} x)

= \frac{1}{4} [1 - 2 \sin^{2} x \cos^{2} x] - \frac{1}{6} [1 - 3 \sin^{2} x \cos^{2} x]

= \frac{1}{4} - \frac{1}{6} = \frac{1}{12}

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