Mass \(m_{1}\) moves on a slope making an angle \(\theta\) with the horizontal and is attached to mass \(m_{2}\) by a string passing over a frictionless pulley as shown in the figure. The coefficient of friction between \(m_{1}\) and the sloping surface is \(\mu\).
              

(a) If \(m_{2} > m_{1} \text{sin} ⁡ \theta \), the body will move up the plane.
(b) If  \(m_{2} > m_{1} (\text{sin} ⁡ \theta +\mu \text{cos} \theta)\), the body will move up the plane.
(c) If  \(m_{2} < m_{1} (\text{sin} ⁡ \theta +\mu \text{cos} \theta)\), the body will move up the plane.
(d) If  \(m_{2} < m_{1} (\text{sin} ⁡ \theta -\mu \text{cos} \theta)\), the body will move down the plane.
Which of the following statement/s is/are true?
1. (a), (d) 2. (a), (c)
3. (c), (d) 4. (b), (d)

(b, d) Hint: Apply Newton's laws of motion.
Let m1 moves up the plane, Different forces involved are shown in the diagram.
N= Normal reaction f= Frictional force T= Tension in the string f=μN=μm1gcosθ
Step 1: Find if the second body moves down.
 For the system (m1+m2) to move up m2g(m1gsinθ+f)>0   m2g(m1gsinθ+μm1gcosθ)>0m2>m1(sinθ+μcosθ)
Hence, option (b) is correct.
Step 2: Find if the first body moves down.
Let the body moves down the plane, in this case, f acts up the plane.
Hence,
m1gsinθf>m2g m1gsinθμm1gcosθ>m2g m1(sinθμcosθ)>m2 m2<m1(sinθμcosθ)
Hence, option (d) is correct.