Understanding Lens Power and Properties

This question tests the understanding of the concept of lens power, its units, and its relationship with focal length and combinations of lenses.

Analysis of Statements:

• Statement (A): The power of a lens is the ability of the lens to converge or diverge the incident rays.

  This statement accurately defines the power of a lens. The power quantifies how strongly a lens converges or diverges light. A lens with higher power bends light more sharply. Thus, statement (A) is correct.

• Statement (B): S.I. unit of the power of a lens is dioptre while focal length is in centimetres

  The S.I. unit of the power of a lens is indeed the dioptre (D). However, the formula relating power ($P$) and focal length ($f$) is $P = \frac{1}{f}$, where $f$ must be in meters (m) for the power to be in dioptres. While focal length is often measured in centimetres, the S.I. unit is meters. Therefore, stating that the S.I. unit for focal length is centimetres makes this statement technically incorrect in the context of the standard definition and calculation.

  For example, if focal length $f = 1 \text{ m}$, then $P = \frac{1}{1 \text{ m}} = 1 \text{ D}$. If $f = 20 \text{ cm} = 0.2 \text{ m}$, then $P = \frac{1}{0.2 \text{ m}} = 5 \text{ D}$.

  Thus, statement (B) is incorrect.

• Statement (C): For a lens of larger focal length, power is smaller

  The power ($P$) of a lens is inversely proportional to its focal length ($f$). The relationship is given by the formula $P = \frac{1}{f}$, where $f$ is measured in meters. This inverse relationship means that as the focal length increases, the power decreases, and vice versa. For instance, a lens with a focal length of 2 meters has a power of $0.5$ D ($\frac{1}{2}$), while a lens with a focal length of 0.5 meters has a power of $2$ D ($\frac{1}{0.5}$).

  Thus, statement (C) is correct.

• Statement (D): In any combination of lenses, the power of combination is not algebraic addition of power of combined lenses

  When multiple lenses are placed in contact or in close proximity, the total power of the combination is found by the algebraic sum of the powers of the individual lenses. If $P_1, P_2, P_3, \dots$ are the powers of the individual lenses, the total power $P_{total}$ is given by $P_{total} = P_1 + P_2 + P_3 + \dots$. This principle applies regardless of whether the lenses are converging or diverging (positive or negative power).

  Thus, statement (D) is incorrect.

Conclusion:

Based on the analysis, statements (A) and (C) are correct, while statements (B) and (D) are incorrect.

Therefore, the correct option is the one that includes only statements (A) and (C).