Function `x=Asin^(2)omegat+Bcos^(2)omegat=Csinomegat cos omegat` represents simple harmonic motion,

Options

A. for any value of `A,B` and `C` (except `C=0)`
B. If `A=-B,C=2B,` amplitude `=|Bsqrt(2)|`
C. if `A=B,C=0`
D. if `A=B,C=2B,` amplitude `=|B|`

Answer

A, B, D

Explanation

`x=Asin^(2)omegat+Bcos^(2)omegat+Csinomegat cos omegat`
`=A((1-cos2omegat)/(2))+B((1+cos2omegat)/(2))+C((sin2omegat)/(2))`
When `A=0` and `B=0` , then
`x=(C)/(2)sin 2 omegat`
It represents a SHM. Hence, option (a) is correct.
When, `A=-B,C=2B,` then
`x=-B((1-cos2omegat)/(2))+B((1+cos3omegat)/(2))+2B((sin2omegat)/(2))`
`=Bcos 2omegat+Bsn 2 omegat` ...(i)
Let, `B=Rsin pi//4=Rcos pi//4, ` then form (i)
`x=Rsin pi//4 cos 2 omegat +Rcos pi//4sin2 omegat`
`=Rsin(2omegat+pi//4)` ...(ii)
which represents a S.H.M. of amplitude `R`, where
`R=(B)/(sinpi//4)=(b)/(1//sqrt(2))=Bsqrt(2)`
Hence, option (b) is correct.
When, `A=B,C=0,` then
`x=Asin^(2)omegat+Acos^(2)omegat=A`
It does not represent S.H.M. Hence option (c) is incorrect.
When `A=B,C=2B` then
`x=Bsin^(2)omegat+Bcos^(2)omegat+2Bsinomegatcosomegat` ltbr. `=B+Bsin2omegat`
This represents a S.H.M. with amplitude `B`.Hence, option (d) in correct.