If `k in I` such that `lim_(nrarroo) (cos.(kpi)/(4))^(2n)-(cos.(kpi)/(6))^(2n)=0,` then

Options

A. k must not be divisible by 24
B. k is divisible by 24 or k is divisible neither by 4 nor by 6
C. k must be divisible by 12 but not necessarity by 24
D. none of these

Answer

B

Explanation

`underset(nrarroo)(lim)(cos.(kpi)/(4))^(2n)-(cos.(kpi)/(6))^(2n)=0`
holds good if
`"Case I": cos .(kpi)/(4)=cos.(kpi)/(6)=1`
i.e., `(kpi)/(4)=2mpi and (kpi)/(6)=2p pi,m,p in Z`
i.e., k = 8m and k = 12 p
i.e., k is divisible by both 8 and 12 i.e., k is divisible by 24.
Case II: `-1 lt cos.(kpi)/(4),cos.(kpi)/(6) lt 1`
i.e., k is not divisible by 4 and k is not divisible by 6.
Case III: `cos.(kpi)/(4)=(2m+1)pi and (kpi)/(6)=(2p+1)pi`
`k=4 (2m+1) and k=6(2p+1)`
This is not possible.