Assertion (A): If (x ) and (y ) are the distance along (x ) and (y ) axes respectively then the dimensions of ( d^3 y d x^3 ) is ( M ^0 ~L ⁻² ~T ^0 ). Reason (R): Dimensions of ( _a^b y d x ) is ( M ^0 ~L ^2 ~T ^0 ).

Step-by-step solution

Assertion (A): "If (x ) and (y ) are distances along (x ) and (y ) axes respectively, then the dimensions of ( d^3 y d x^3 ) is (M^0 L⁻² T^ 0 . " ) - Here, (y ) has dimensions of length (L). - (x ) also has dimensions of length (L). - Differentiating once: ( d y d x L / L= ) dimensionless. - Differentiating twice: ( d^2 y d x^2 ) dimension (=1 / L ). - Differentiating thrice: ( d^3 y d x^3 ) dimension (=1 / L^2=L⁻² ). So, dimensions = (M^0 L⁻² T^0 ). (1) Assertion (A) is true.