A book with many printing errors contains four different formulas for the displacement y of a particle undergoing a certain peroidic motion :
(a) `y = asin2pi t//T`
(b) y=asinvt
`("c")` `y=(a//T) sin t//a`
(d) `y=(a//sqrt2) (sin 2pi t//T + cos 2pit//T)`
(a = maximum displacement of the particle, v=speed of the particle. T=time-period of motion). Rule out the wrong formulas on dimensional grounds)

Explanation

(ii) `y=asinvt`
Here, angle = vt. Its dimensional formula should be `M^(0)L^(0)T^(0)`
But `[vt]=(M^(0)L^(1)T^(-1))(T^(1))=M^(0)L^(1)T^(0)`
Hence, (ii) is dimensionally incorrect.
(iii) `y=(a)/(T)"sin"(t)/(a)`
Here `(a)/(T)` should have same dimensional formula as of y(Displacement).
`[(a)/(t)]=(M^(0)L^(1)T^(-2))/(T^(1))=M^(0)L^(1)T^(-1)`
Hence, (iii) is dimensionally incorrect.
(i) and (iv) are dimensionally correct.