• Optical power represents the degree of convergence or divergence a lens introduces to the incident light beam $\rightarrow$ Statement (A) is correct.
• Power $P = 1/f$ is measured in dioptres ($1\,\text{D} = 1\,\text{m}^{-1}$) when the focal length $f$ is expressed in metres, not centimetres $\rightarrow$ Statement (B) is incorrect.
• Since power is inversely proportional to focal length ($P \propto 1/f$), a lens with a larger focal length has smaller power $\rightarrow$ Statement (C) is correct.
• For thin lenses placed in contact, the net power of the combination is the algebraic sum $P = P_1 + P_2 + \dots$ $\rightarrow$ Statement (D) is incorrect.

• For a real inverted image formed by a convex lens:
  $$m = -\frac{h_i}{h_o} = -\frac{1\,\text{cm}}{5\,\text{cm}} = -0.2$$
• By the magnification formula for thin lenses:
  $$m = \frac{v}{u} \implies -0.2 = \frac{v}{-40\,\text{cm}} \implies v = +8\,\text{cm}$$
• Using the thin lens equation:
  $$\frac{1}{f} = \frac{1}{v} - \frac{1}{u} = \frac{1}{8} - \frac{1}{-40} = \frac{1}{8} + \frac{1}{40} = \frac{5 + 1}{40} = \frac{6}{40} = \frac{3}{20}\,\text{cm}^{-1}$$
  $$f = \frac{20}{3}\,\text{cm} \approx 6.67\,\text{cm}$$

• When light travels from an optically denser medium of refractive index $n_1$ into a rarer medium of refractive index $n_2$, it bends away from the normal.
• By Snell's Law: $n_1 \sin i = n_2 \sin r$.
• The critical angle $i_c$ is defined as the angle of incidence in the denser medium for which the angle of refraction in the rarer medium becomes exactly $90^\circ$ ($r = 90^\circ$):
  $$n_1 \sin i_c = n_2 \sin 90^\circ = n_2 \implies \sin i_c = \frac{n_2}{n_1}$$

Evaluating each statement regarding Total Internal Reflection (TIR):

• Statement (A): TIR occurs strictly when light travels from an optically denser medium to an optically rarer medium at an angle exceeding the critical angle. Hence (A) is incorrect.
• Statement (B): In TIR, $100\%$ of the incident optical energy is reflected back into the denser medium; none propagates into the second medium $\rightarrow$ (B) is correct.
• Statement (C): TIR obeys the fundamental law of reflection ($i = r$) within the denser medium $\rightarrow$ (C) is correct.
• Statement (D): Because no refracted ray enters the second medium, there is no refracted ray and hence no angle of refraction $\rightarrow$ (D) is correct.