In a `DeltaABC ,angleB=(2pi)/3` and cos A + cos c = `lamda`. Then , the exhaustive set of value of `lamda` is

Options

A. `(1,3//2]`
B. `(3//2,sqrt3)`
C. `(1//2,sqrt3//2)`
D. none of these

Explanation

We have,
`lamda=cosA+cosC=2cos(A+C)/2cos(A-C)/2`
`rArrlamda=2sinB/2cos(A-C)/2`
`rArrlamda=2sin(pi)/3cos(A-C)/2rArrlamda=sqrt3cos((A-C)/2)`
Now `angleB=(2pi)/3`
`thereforeA+C=pi/3`
`rArr-(pi)/3ltA-Cltpi/3`
`rArr-pi/6lt(A-C)/2lt(pi)/6`
`rArr(sqrt3)/2ltcos((A-C)/2)lt1`
`rArr(sqrt3)/2lt(lamda)/(sqrt3)lt1rArr3/2ltlamdaltsqrt3rArrlamdain(3//2,sqrt3)`