माना `f_(k) (x)= (1)/(k) (sin ^(k) x + cos^(k) x)` है ,जहाँ `x epsilon R ` तथा ` k ge 1` है तो `f_(4) (x)- f_(6) (x)` बराबर है।

Options

A. `1/6`
B. `1/3`
C. `1/4`
D. `(1)/(12)`

Answer

D

Explanation

`f_(k) (x)= (1)/(k) (sin^(k) x + cos^(k) x)`
`:. f_(4) (x)- f_(6) (x) =(1)/(4) (sin^(4) x +cos^(4) x) -(1)/(6) (sin^(6) x + cos^(6) x)`
`=(1)/(4)(sin^(4) x + cos^(4) x+ 2 sin^(2) x cos^(2) x -2 sin^(2) x cos^(2) x) - (1)/(6) [(sin^(2) x)^(3) + (cos^(2) x)^(3)]`
`=(1)/(4) [(sin^(2) x + cos^(2) x)^(2) -2 sin^(2) x cos^(2) x] -(1)/(6) [(sin^(2) x + cos^(2)x) (sin^(4)x + cos^(4) x-sin^(2)x cos^(2) x)]`
`=(1)/(4)(1-2 sin^(2) x cos^(2) x)-(1)/(6) (sin^(4) x + cos^(4) x + 2 sin^(2) x cos^(2) x-2 sin^(2) x cos^(2) x - sin^(2) x cos^(2) x)`
`=(1)/(4)(1-2sin^(2) x cos^(2)x )- (1)/(6) [(sin^(2) x+ cos^(2) x)^(2) -3 sin^(2) x cos^(2) x]`
`=(1)/(4) (1-2sin^(2) x cos^(2) x)-(1)/(6) (1-3 sin^(2) x cos^(2) x)`
`=(1)/(4)-(1)/(6)= (3-2)/(12) =(1)/(12)`
अतः विकल्प (iv) सत्य है।