A smooth inclined plane is fixed inside a car that is accelerating...

In the frame of the accelerating car, a pseudo force $ma$ acts on the sphere in the horizontal direction opposite to the car's acceleration.

The forces acting on the sphere along the inclined plane are the component of weight down the incline, $mg \sin \theta$, and the component of pseudo force up the incline, $ma \cos \theta$.

The net force along the incline is:
$F_{\text{net}} = mg \sin \theta - ma \cos \theta$

Substituting $a = g \tan \theta$ into the equation gives:
$F_{\text{net}} = mg \sin \theta - m(g \tan \theta) \cos \theta$
$F_{\text{net}} = mg \sin \theta - mg \sin \theta = 0$

Since the net force along the incline is zero, the linear acceleration of the sphere relative to the incline is zero. Thus, its linear velocity $v$ remains constant.

The inclined plane is smooth, meaning there is no friction. The lines of action of gravity, normal reaction, and pseudo force all pass through the center of mass of the sphere. Therefore, the net torque about the center of mass is zero.

Since the net torque is zero, the angular acceleration is zero, and the angular velocity $\omega$ remains constant.

Because the sphere is initially set into pure rolling, the relation $v = \omega R$ holds initially. Since both $v$ and $\omega$ remain constant, this relation continues to hold at all times.

Therefore, the sphere will maintain pure rolling.