An object moves with constant acceleration a → . Which of the following expressions is/are also…

Let $\overrightarrow{a}=a_x\hat{\mathrm{i}}+a_y\hat{\mathrm{j}}+a_z\hat{\mathrm{k}}=\frac{d{v}_x}{dt}\hat{\mathrm{i}}+\frac{d{v}_y}{dt}\hat{\mathrm{j}}+\frac{d{v}_z}{dt}\hat{\mathrm{k}}$

As $\overrightarrow{a}$ is constant, so its magnitude as well as direction is not changing, but $v_x, v_y$ and $v_z$ all can be varying (any combination of this if $a_x, a_y$ or $a_z=0$, then corresponding velocity component may be constant).

$\left|\overrightarrow{v}\right|=v=\sqrt{v_x^2+v_y^2+v_z^2}$

$\frac{d\left|\overrightarrow{v}\right|}{dt}=\frac{1}{2\sqrt{v_x^2+v_y^2+v_z^2}}\times \left[2v_x\frac{d{v}_x}{dt}+2v_y\frac{d{v}_y}{dt}+2v_z\frac{d{v}_z}{dt}\right]=\frac{\overrightarrow{v}·\overrightarrow{a}}{v}$

which is a variable quantity.
$\left|\frac{d\overrightarrow{v}}{dt}\right|=\left|\overrightarrow{a}\right|$, which would be constant.

$\frac{d{v}^2}{dt}=\frac{d\left(v_x^2+v_y^2+v_z^2\right)}{dt}=2\overrightarrow{v}·\overrightarrow{a}$ (a variable quantity)

$\frac{d\left({\displaystyle \frac{\overrightarrow{v}}{v}}\right)}{dt}=\frac{d}{dt}\left[\frac{v_x\hat{\mathrm{i}}+v_y\hat{\mathrm{j}}+v_z\hat{\mathrm{k}}}{\sqrt{v_x^2+v_y^2+v_z^2}}\right]$ (a variable quantity)