Answer:
 - 8.33 x 10⁻³ rad /s ( anticlockwise)
Explanation:
The rotational movement of beetle and turntable is caused by torque generated by internal forces , we can apply conservation of  angular momentum. 
That is , 
I₁ ω₁ = I₂ω₂ ,   ω₁ and ω₂ are angular velocity of beetle and turntable respectively. 
ω₁ + ω₂ = .05 radian /s ( given )
Momentum of inertia of beetle I₁ = mass x (distance from axis)² 
= 15 x 10⁻³ x R² ( R is radius of the turntable )
Momentum of inertia of turntable  I₂ =1/2 mass x (distance from axis)² 
= 75/2 x 10⁻³ x R² ( R is radius of the turntable )
I₁ ω₁ = I₂ω₂ ,
15 x 10⁻³ x R² x ( .05 -  ω₂ ) = 75/2 x 10⁻³ x R² ω₂
15 x  ( .05 -  ω₂ ) = 75/2 x  ω₂
.75 - 15ω₂ = 37.5ω₂
.75 = 52.5  ω₂
ω₂ = - 14.3  x 10⁻³ rad /s ( anticlockwise)