NDA (II) 2022 General Ability Test · Question 131 of 150

Correct — B, +6 dioptre. Power in dioptres is the reciprocal of the focal length measured in metres, so convert first. A focal length of 50 cm is 0.5 m, giving a power of 1 ÷ 0.5 = +2 D. A focal length of 25 cm is 0.25 m, giving 1 ÷ 0.25 = +4 D. For thin lenses in contact the powers simply add, so the combination has a power of 2 + 4 = +6 D. The sign is positive because both lenses are convex and therefore converging, and the combination is equivalent to a single convex lens of focal length 1 ÷ 6 m, about 16.7 cm.

• (a)+2 dioptre — This is the power of the 50 cm lens alone. It ignores the second lens entirely, so it answers a question that was not asked.
• (c)-6 dioptre — The magnitude is right but the sign is wrong. A negative power belongs to a diverging, concave lens; both lenses here are stated to be convex, so their focal lengths and powers are positive.
• (d)+3 dioptre — This is the average of the two powers, 2 and 4, rather than their sum. Powers of lenses in contact add; they are not averaged, and they are not obtained by adding focal lengths either.

The power of a lens measures how strongly it bends light, and it is defined as the reciprocal of the focal length in metres, P = 1/f, with the unit called the dioptre. A short focal length means strong bending and therefore high power. Converging convex lenses have positive focal length and positive power; diverging concave lenses have negative focal length and negative power. When thin lenses are placed in contact the combination behaves like a single lens whose power is the algebraic sum of the individual powers, P = P₁ + P₂, which is equivalent to 1/f = 1/f₁ + 1/f₂.

Two habits decide this question. The first is to convert centimetres to metres before taking the reciprocal — forgetting that step turns 25 cm into 1/25 = 0.04 D and wrecks the arithmetic. The second is to add powers rather than focal lengths, because the additive quantity for lenses in contact is the power, not the focal length. A quick check on the answer is that the combination must be stronger than either lens alone, so the result has to exceed +4 D, which rules out the +2 D and +3 D options at a glance, and the positive sign rules out the negative one.

• Power in dioptres equals 1 divided by the focal length expressed in metres.
• A convex lens has positive power and a concave lens negative power.
• For thin lenses in contact the powers add algebraically, P = P₁ + P₂.
• Here 50 cm gives +2 D, 25 cm gives +4 D, and the combination is +6 D, equivalent to a single lens of focal length about 16.7 cm.

• Taking the reciprocal of the focal length in centimetres instead of metres.
• Adding the focal lengths, or averaging the powers, instead of adding the powers.
• Attaching a negative sign to a convex lens.

As a numerical like this, as a single-lens power calculation, or as a spectacle-prescription item asking what lens power corrects a stated defect.

• practice — not a real PYQ

  The power of a convex lens of focal length 20 cm is

  • (a)+0.05 dioptre
  • (b)+2 dioptre
  • (c)+5 dioptre
  • (d)+20 dioptre

  Answer(c) +5 dioptre — 20 cm is 0.2 m, and 1 ÷ 0.2 = 5.

• practice — not a real PYQ

  A convex lens of power +5 D is placed in contact with a concave lens of power -2 D. The power of the combination is

  • (a)+7 D
  • (b)+3 D
  • (c)-3 D
  • (d)+2.5 D

  Answer(b) +3 D — powers of lenses in contact add algebraically, so 5 + (-2) = 3.