EPFO APFC 2023 General Studies · Question 54 of 120

Correct — A, (a) A real and inverted image would be formed at the focal point. This is the first row of the standard table for a concave mirror, and it is the one every other row is built out from. An object at infinity sends out rays that arrive at the mirror parallel to one another. A concave mirror is a converging mirror, and its principal focus is defined as the point to which it brings a parallel beam after reflection — so the rays actually meet there, at the focus, and the image is formed at the focal point. Because the reflected rays genuinely cross, the image is real: it can be caught on a screen held at that place, and it is not merely something the eye traces backwards. In a mirror, a real image is always inverted, so the image here is inverted as well. It is also extremely small — reduced to a point for an object of ordinary size at a great distance — because the size of the image shrinks in proportion as the object recedes. The mirror formula says the same thing in one line. With 1/v + 1/u = 1/f, an object at infinity makes 1/u vanish, so 1/v = 1/f and v = f: the image distance equals the focal length. The magnification m = −v/u then tends to zero as u grows without limit, which is the algebraic statement that the image is highly diminished, and the minus sign carries the inversion. The result is not an examination curiosity but the working principle of several instruments. It is why a concave mirror concentrates sunlight — the sun is effectively at infinity, so its heat is gathered at the focus, which is what a solar cooker and a solar furnace exploit — and it is why the objective mirror of a reflecting telescope forms the image of a star in its focal plane. The item carries no ray diagram, so drawing two parallel rays and reflecting them is part of the work.

• (b)A virtual and inverted image would be formed at the focal point. — The position is right and the pairing is impossible. In a mirror a virtual image is always erect and a real image is always inverted; the two properties are not free to be combined at will. The reason is geometric: a real image is formed where reflected rays actually cross in front of the mirror, and crossing rays produce an inverted picture, while a virtual image is formed where the backward extensions of diverging rays appear to meet behind the mirror, which preserves the orientation. So virtual and inverted cannot occur together in any mirror, plane, concave or convex. Learning the two permitted pairings — real with inverted, virtual with erect — removes an entire option from most mirror questions before the geometry is even considered.
• (c)A real and erect image would be formed at infinity. — Two errors at once. First, real and erect is a combination a mirror cannot produce, for the reason just given. Second, an image at infinity is what a concave mirror forms when the object is placed at the focus — the exact reverse of the situation described, and a consequence of the principle of reversibility of light, which says that rays retrace their path when their direction is reversed. That reversibility is the reason the first and the fifth rows of the standard table mirror each other: object at infinity gives an image at the focus, and object at the focus gives an image at infinity. Reading the stem carefully enough to know which of the two is being described is the whole of the difficulty in this item.
• (d)A virtual and erect image would be formed at the focal point. — The pairing here is a permitted one, which makes this the most respectable of the three wrong options, but the configuration is not. A concave mirror forms a virtual and erect image only when the object lies between the pole and the principal focus, and that image is enlarged and lies behind the mirror — it is the arrangement used in a shaving mirror and in a dentist's mirror. It is never formed at the focal point, and it cannot arise from an object at infinity, whose rays converge in front of the mirror rather than diverging from it. Note how far apart the two configurations are: an object at infinity and an object nearer than the focus are the two extremes of the whole table.

A concave mirror is a converging mirror: a beam of rays parallel to the principal axis is reflected to meet at the principal focus, which lies midway between the pole and the centre of curvature, so that the focal length is half the radius of curvature. Every image the mirror can form is catalogued by the position of the object, and the table is short enough to be memorised as a whole. With the object at infinity the image is at the focus, real, inverted and highly diminished. With the object beyond the centre of curvature the image lies between the focus and the centre of curvature, real, inverted and diminished. With the object at the centre of curvature the image is at the same place, real, inverted and the same size. With the object between the centre of curvature and the focus the image lies beyond the centre of curvature, real, inverted and enlarged. With the object at the focus the image is at infinity, real, inverted and very greatly enlarged. And with the object between the pole and the focus the image is behind the mirror, virtual, erect and enlarged — the only virtual case, and the one used in shaving and dental mirrors. Two structural facts make the table easy to hold. The first and fifth entries are reverses of each other by the principle of reversibility of light, as are the second and fourth. And the character of the image is fixed by two permitted pairings only: real images are inverted and lie in front of the mirror, virtual images are erect and lie behind it. Quantitatively everything follows from the mirror formula 1/v + 1/u = 1/f, with f = R/2, and the magnification m = −v/u, applied under the new Cartesian sign convention in which distances are measured from the pole and a concave mirror has a negative focal length.

Optics is a fixture of the general science block in this paper, and the Commission sets it as a pair — one item on lenses and one on mirrors — which is a deliberate test of whether the two sign conventions and the two tables have been kept apart. This item is the mirror half. Its options are built by mixing the three variables that describe an image: whether it is real or virtual, whether it is erect or inverted, and where it is formed. Because two of those three are not independent in a mirror, a candidate who knows the permitted pairings can discard two options immediately and then decide between the remaining two on position alone. That is a general strategy for the whole family, and it converts a question about ray diagrams into a question about two rules. The item carries no diagram, which is normal for this paper — none of the general science questions is illustrated — so the ray construction has to be done in the margin or in the head.

• A concave mirror is converging: rays arriving parallel to the principal axis are reflected through the principal focus, which lies midway between the pole and the centre of curvature, so f = R/2.
• An object at infinity gives an image at the focus that is real, inverted and highly diminished; the mirror formula gives v = f when 1/u vanishes.
• In a mirror, real images are inverted and formed in front of the mirror; virtual images are erect and formed behind it. No other pairing occurs.
• The magnification is m = −v/u, and it tends to zero as the object distance grows without limit, which is why a distant object gives a point-sized image.
• By the reversibility of light, an object at the focus gives an image at infinity — the exact reverse of the configuration in this question.
• A concave mirror forms a virtual, erect and enlarged image only when the object lies between the pole and the focus, as in a shaving or dentist's mirror.
• The convergence of parallel light at the focus is what makes a concave mirror the collector in a solar cooker or furnace and the objective of a reflecting telescope.

• Combining virtual with inverted or real with erect; in a mirror these pairings never occur and each wrong pairing kills an option outright
• Confusing the object at infinity with the object at the focus; the two are reverses of each other, and the second gives an image at infinity
• Applying the concave mirror table to a convex mirror, which forms only virtual, erect and diminished images wherever the object is placed
• Forgetting that the focal length is half the radius of curvature, which is where most numerical slips in this topic begin
• Assuming a real image must be large; the image of a distant object is real and yet reduced almost to a point

Mirror and lens questions in this paper are set as descriptive statements rather than as calculations, with each option combining a nature, an orientation and a position. The mark is won by knowing the standard table and the permitted pairings, not by manipulating the mirror formula, so the efficient revision is the six-row table for a concave mirror alongside the one-row summary for a convex one. Expect a lens item in the same block, and expect the two sign conventions to be tested against each other across the pair.

No directly related past PYQ was found.

• practice — not a real PYQ

  An object is placed between the pole and the principal focus of a concave mirror. The image formed is

  • (a)real, inverted and enlarged
  • (b)real, inverted and diminished
  • (c)virtual, erect and enlarged
  • (d)virtual, erect and diminished

  Answer(c) virtual, erect and enlarged — this is the only position of the object for which a concave mirror gives a virtual image, and the image lies behind the mirror. It is the arrangement used in a shaving mirror and a dentist's mirror, where a magnified upright view is wanted at close range.

• practice — not a real PYQ

  A concave mirror has a radius of curvature of 40 cm. Where will the image of an object placed at a very large distance from the mirror be formed?

  • (a)40 cm in front of the mirror
  • (b)20 cm in front of the mirror
  • (c)20 cm behind the mirror
  • (d)At the pole of the mirror

  Answer(b) 20 cm in front of the mirror — the focal length is half the radius of curvature, so f = 20 cm, and rays from a very distant object arrive parallel and converge at the focus. The image is real, so it lies in front of the mirror and can be caught on a screen placed there.