The positive valu of the parameter 'k' for which the area of the figure bounded by the curve `y=sin(kx),x=(2pi)/(3k),x=(5pi)/(3k)` and x-axis is less than 2 can be

Options

A. `(1)/(8)ltklt(3)/(8)`
B. `0ltklt(1)/(8)`
C. `1ltklt2`
D. `(3)/(8)ltklt(5)/(8)`

Answer

C

Explanation

`A=overset((pi)/(k))underset((2pi)/(3k))int(sin kx)dx-overset((5pi)/(3k))underset((pi)/(k))int(sin kx ) dx`
`=-((cos kx)/(k))_(2pi//3k)^(pi//k)+((cos kx)/(k))_(pi//k)^(5pi//3k)`
`=-(1)/(k)[-1cos""(2pi)/(3)]+(1)/(k)[cos""(5pi)/(3)-cos pi]`
`=-(1)/(k)[-1+(1)/(2)]+(1)/(k)[(1)/(2)+1]`
`=(1)/(2k)+(3)/(2k)`
`=(2)/(k)lt2`