Function `x=Asin^2omegat+Bcos^2omegat+Csinomegatcosomegat` represents SHM
(i) For any value of A, B and C(except `C=0`)
(ii) If `A=-B`, `C=2B`, amplitude `=|Bsqrt2|`
(iii) If `A=B`, `C=0`
(iv) If `A=B`, `C=2B`, amplitude `=|B|`

Options

A. (i), (ii)
B. (ii), (iii)
C. (i), (ii), (iii)
D. (i), (ii), (iv)

Answer

D

Explanation

`x=Asin^2omegat+Bcos^2omegat+Csinomegatcosomegat`
`=[A/2*2sin^2omegat+B/2*2cos^2omegat+(C*2)/(2)sinomegatcosomegat]`
`=[A/2(1-cos2omegat)+B/2(1+cos2omegat)+C/2sin2omegat]`
`=(A+B)/(2)+(B-A)/(2)(cos2omegat)+C/2sin2omegat`
(i) For `A=B=0`, `x=C/2sin2omegat` (SHM)
(ii) If `A=-B`, `C=2B`, `x=Bcos2omegat+Bsin2omegat`
`=sqrt2Bsin(2omegat+pi//4)`(SHM)
Resultant amplitude `=|sqrt2B|`
(iii) `A=B`, `C=0`, `x=(A+B)/(2)` (No SHM)
(iv) `A=B`, `C=2B`
`x=B+Bsin2omegat` (SHM of amplitude `|B|`)