Question

An object kept on a smooth inclined plane of inclination theta with horizontal can be kept stationary relative to the incline by giving a horizontal acceleration to the inclined plane, equal to

Answer 1

mg sin θ = ma cosθ
cutting m from both sides gives us 

g sinθ= a cosθ
g sinθ/cosθ= a

as sinθ/cosθ = tanθ

so g tanθ= a

Answer 2

The object will remain stationary when acceleration is $g \tan \theta$

Explanation:

The object kept on smooth inclined plane with theta angle to horizontal be at rest until the horizontal acceleration to the inclined plane is given as below:

It follows the balance between the object acted upon by the force in both the condition.

Thereby, the equation be formed is,

$M g \sin \theta=m a \cos \theta$

$\Rightarrow g \sin \theta=a \cos \theta$

$\Rightarrow g \frac{\sin \theta}{\cos \theta}=a$

$\Rightarrow \frac{\sin \theta}{\cos \theta}=\tan \theta$

$\Rightarrow g \tan \theta=a$