A cabin is moved up the inclined plane with constant acceleration g sin​​(theta). A particle is projected with some velocity with respect to the cabin in a direction perpendicular to the inclined plane. If maximum height attained by particle perpendicular to inclined plane is same as range of particle with respect to the cabin parallel to palne then calculate value of cot(theta).

Let the angle of the incline = A.  (instead of x  , as given)Let the cabin travel with an instantaneous velocity u1 up the incline at the instant the ball is thrown up at Origin (0,0) with a velocity u2 perpendicularly.Deceleration of ball in direction perpendicular to incline:  - g cos ATime to reach the max. height wrt incline = u2/(g cos A)Maximum height wrt incline:   u2² /(2 g cos A)For the range, we take the difference between the displacements of ball and origin when the ball meets the floor/incline.Displacement of the ball:           x = (u1 Cos A - u2 Sin A) t           y = (u1 sin A + u2 Cos A) t - 1/2 * g t^2For the ball to meet the incline/floor of cabin,  y = x * tan ASubstituting that we get:  t = (2 u2 Sec A) / gDisplacement along incline of the ball at t:  x sec A = (u1 - u2 tan A) tIn time t, the displacement of the cabin:   u1 t + 1/2 * g sin A t^2 Range of ball relative to cabin = difference of both displ.       =  (u2 tan A + 1/2 g t sin A) * tEquating relative max height and relative range:      u2^2 / (2g cos A) = (u2 tanA + g/2 SinA*2 u2 secA /g)*2 u2 secA /g                                  = 4 tan A u^2 sec A / g   =>  Cot A = 8