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JEE MainPhysicsModerateError analysis

The velocity v (in cm/sec) of a particle is given in terms of
time t (in sec) by the equation:v=at+bt + cv = at + \frac{b}{t\ + \ c}
The dimensions of a, b and c are:
------------------------------------
a b c

Show answer and solution
[v]=[at]; [a]=[vt]=[LT−1T]=[LT−2]\lbrack v\rbrack = \lbrack at\rbrack;\ \lbrack a\rbrack = \left\lbrack \frac{v}{t} \right\rbrack = \left\lbrack \frac{{LT}^{- 1}}{T} \right\rbrack = \left\lbrack {LT}^{- 2} \right\rbrack
[t]=[c]=[T]\lbrack t\rbrack = \lbrack c\rbrack = \lbrack T\rbrack
[bt+c]=[v] ⇒ [b][t]=[v]⇒[b]=[vt]=[LT−1T]=[L]\left\lbrack \frac{b}{t + c} \right\rbrack = \lbrack v\rbrack\ \Rightarrow \ \frac{\lbrack b\rbrack}{\lbrack t\rbrack} = \lbrack v\rbrack \Rightarrow \lbrack b\rbrack = \lbrack vt\rbrack = \left\lbrack {LT}^{- 1}T \right\rbrack = \lbrack L\rbrack

Source: Grass-G / P-05 Physics paper 10 October.docx

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