Some Basic doubts in mechanics

for m₁ , m₂ to be stationary wrt M , a_x of all three shud be equal and a_y of m₂ = 0

now F - N - T = Ma
T = m₁a
N = m₂a

so , a = Fm₁+m₂+M

for m₂ , T + f = m₂g

T = m₁a = m₁ Fm₁+m₂+M

so m₁Fm₁+m₂+M + f = m₂g

μN = μm₂Fm₁+m₂+M
when f = +μN , F = minimum , when f = - μN i.e opp direction, F = max

so F_max = m₂g(m₁+m₂+M)m₁ - μm₂
F_min= m₂g(m₁+m₂+M)m₁ +μm₂

so (m₁+m₂+M)gm₁ m₂+μ≤ F ≤ (m₁+m₂+M)gm₁ m₂-μ

now just check wich option doesnt lie in dis range

3)

Use the concept that work done by tension is zero .

assume that the bigger block moves x₂ horizontally.

and the smaller block x₁ vertically.

Now for bigger bolck (Tsin@+T)x₂-Tx₁=0

NOW (1+sin@)x₂=x₁

now differentiate both sides and hence you get d as answer.

Thanks.. it helps me too.!