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1.Relation and Function
hard
ધારો કે ${f_k}\left( x \right) = \frac{1}{k}\left( {{{\sin }^k}x + {{\cos }^k}x} \right)\;,x \in R$ અને $k \ge 1$, તો ${f_4}\left( x \right) - {f_6}\left( x \right)$ ની કિંમત મેળવો.
A
$\frac{1}{4}$
B
$\frac{1}{{12}}$
C
$\frac{1}{6}$
D
$\frac{1}{3}$
(JEE MAIN-2014)
Solution
$f_{4}(x)-f_{6}(x)=\frac{1}{4}\left(\sin ^{4} x+\cos ^{4} x\right)-\frac{1}{6}$
$\left(\sin ^{6} x+\cos ^{6} x\right)$
$=\frac{1}{4}\left(1-2 \sin ^{2} x \cos ^{2} x\right)-\frac{1}{6}\left(1-3 \sin ^{2} x \cos ^{2} x\right)$
$=\frac{1}{4}-\frac{1}{6}=\frac{3-2}{12}=\frac{1}{12}$
Standard 12
Mathematics
