Q.For the study of any kind of motion, we never use Newton's first law of motion directly. Why should it be studied?

[!TLDR] The first law is not a calculation tool but the conceptual foundation -- it defines force, inertia, and the inertial frames that the (calculation-capable) second law depends on. [!ANSWER] It must be studied because it defines the very concepts (force, inertia, inertial frame, 'state of motion') that the second law's mathematics presupposes.

The second law, $\vec{F}=\frac{d\vec{p}}{dt}$, is what is actually used to solve numerical problems of motion -- so in that narrow, calculational sense, the first law is indeed never directly 'used'. But the first law does essential conceptual groundwork that the second law's formula silently assumes: it establishes what counts as an INERTIAL FRAME (one where a body with zero net force shows zero acceleration), it conceptually DEFINES FORCE as the cause of changes in the state of motion (before the second law makes this quantitative), and it DEFINES INERTIA/inertial mass as a body's resistance to having its state of motion changed. Without the first law establishing these foundational ideas first, the second law's equation would have no proper physical meaning or valid frame of applicability. This is exactly analogous to how a formal mathematical system needs its axioms stated explicitly even though later theorems (not the axioms themselves) are what get 'used' in day-to-day proofs. [!ANSWER] The first law is studied because it defines force, inertia, and the very notion of an inertial frame -- concepts the second law's equation depends on but does not itself define; only the second law is used for actual numerical calculation.

Distinguish between a law's CALCULATIONAL use (solving numericals) and its CONCEPTUAL role (defining the basic quantities/frames that other laws presuppose).

Concluding the first law is 'useless' or redundant simply because it is subsumed algebraically by the second law for constant mass, missing its distinct conceptual/definitional role (4.3.1).