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FalseThe term "arbitrary" refers to an unpredictable direction that cannot be measured or predicted precisely. Rising or decaying oscillations with a certain frequency are referred to as arbitrary motion. (Uniform and non uniform motion)

a) False

Average velocity v_{\text{average}} = \frac{\text{ Total displacement }}{\text{ time }}

The following relation cannot hold all the arbitrary relations since it is arbitrary motion.

v_{\text{average}} = \frac{1}{2} (v(t_1) + v(t_2))

b) True

v_{\text{average}} = \frac{(r(t_2) - r(t_1))}{t_2 - t_1}

This equation can reflect the arbitrary movement of the particle. Since it can be applicable in uniform and non uniform motion.

c) False

v(t) = v(0) + at

This motion equation is only valid for uniform movement. Therefore, this equation cannot represent the movement of the space particle.

d) False

r(t) = r(0) + v(0)t + \frac{1}{2} at^2

The provided equation is only valid for uniform acceleration motion. Since the particle's motion is arbitrary, and its acceleration can also be non-uniform. Therefore, this equation cannot describe particle motion in space.

e) True

a_{\text{average}} = \frac{(v(t_2) - v(t_1))}{t_2 - t_1}

This equation can describe the particle's arbitrary motion. This is the universal relationship between acceleration and velocity-time.

Acceleration: change in velocity with respect to time

a = \frac{\Delta v}{\Delta t}