Which equation are dimensionally valid out of following equations
(i) Pressure `P= rho gh` where `rho`= density of matter, g= acceleration due to gravity. H= height.
(ii) F.S `=(1)/(2) mv^(2)-(1)/(2) mv_(0)^(2)` where F= force `rho`= displacement m= mass, v= final velocity and `v_(0)=` initial velocity
(iii) `s=v_(0)t+(1)/(2) (at)^(2)`
s= dispacement `v_(0)=` initial velocity,
a= accelration and t= time
(iv) `F=(mxx a xx s)/(t)`
Where m= mass, a= acceleration, s= distance and t= time

Explanation

As per dimensional formula.
(1) Pressure `P= rho gh` where,
`[P]=M^(1)L^(-1)T^(-2)`
`[rho]=M^(1)L^(-3)T^(0)`
`[g]=M^(0)L^(1)T^(-2)`
`[h]=L^(1)`
`:.M^(1)L^(-1)T^(-2)=M^(1)L^(-3)T^(0)] (L^(1)T^(-2)) (L^(1))`
`:. LHS= RHS`
`:.` This equation is dimensionally valid.
(2) `F*s =(1)/(2) mv^(2)-(1)/(2) mv_(0)^(2)`
`LHS=FS`
`M^(1)L^(1)T^(-2)xxT^(1)`
`=M^(1)L^(2)T^(-2)`
`RHS=(1)(1)/(2)mv^(2)`
`(1)/(2)=` constant= dimensionless
`=mv^(2)` `=M^(1)L^(2)T^(-2)`
(2) `(1)/(2) mv_(0)^(2)`
`(1)/(2)`= constant = dimensionless
`=mv_(0)^(2)`
`=M^(1)xx(L^(1)T^(-1))^(2)`
`=M^(1)L^(2)T^(-2)`
All three terms have same dimension hence by principle of homogeinity equation is dimensionally correct.
`:.` Option (A) is correct.