1. Recall the basic facts about self-inductance

   For a coil, self-inductance is defined by $L = \frac{N\Phi}{I}$ and for a solenoid, $L = \mu \frac{N^2 A}{l}$ where:

   • $\mu$ = permeability of the medium,
   • $N$ = number of turns,
   • $A$ = cross-sectional area,
   • $l$ = length of the coil.

   Also, the self-induced emf is $e = -L\frac{dI}{dt}$ where the negative sign is due to Lenz's law.

2. Check each statement

   Statement A: The self-inductance of the coil depends on its geometry.

   From $L = \mu \frac{N^2 A}{l},$ we see that $L$ depends on $A$, $l$, and $N$, i.e. the shape/size/geometry of the coil.

   So, A is true.

   Statement B: Self-inductance does not depend on the permeability of the medium.

   From the same formula, $L \propto \mu.$ Therefore self-inductance does depend on permeability.

   So, B is false.

   Statement C: Self-induced e.m.f. opposes any change in the current in a circuit.

   Since $e = -L\frac{dI}{dt},$ the induced emf always opposes the change in current causing it.

   So, C is true.

   Statement D: Self-inductance is electromagnetic analogue of mass in mechanics.

   In mechanics, mass resists change in motion; similarly, inductance resists change in current.

   So, D is true.

   Statement E: Work needs to be done against self-induced e.m.f. in establishing the current.

   While building up current in an inductor, the self-induced emf opposes the rise of current. Hence external work must be done, and this energy is stored in the magnetic field: $U = \frac{1}{2}LI^2.$

   So, E is true.

3. Collect the correct statements

   True statements are: $A, C, D, E$

4. Match with the options

   • Option A: $A, B, C, E$ only → includes false statement $B$ ❌
   • Option B: $A, B, C, D$ only → includes false statement $B$ ❌
   • Option C: $A, C, D, E$ only ✅
   • Option D: $B, C, D, E$ only → includes false statement $B$ ❌

Therefore, the correct option is C.