BPSC 69th CCE Prelims General Studies · Question 15 of 144

Correct — B, at the centre of curvature. Of the six standard object positions for a concave mirror, exactly one returns an image the same size as the object, and it is the centre of curvature C. Two independent proofs settle it. The first is a ray argument: any ray leaving C travels along a radius, so it strikes the mirror surface along the normal and is reflected straight back along its own path. Every such ray returns to C, so the image must form at C, and since object and image are the same distance from the pole the two must be the same size — inverted, because the rays cross on the axis, and real, because they actually converge in front of the mirror. The second is arithmetic. For a spherical mirror of small aperture R = 2f, so an object at C sits at u = 2f. Substituting in the mirror formula 1/v + 1/u = 1/f gives 1/v = 1/f − 1/(2f) = 1/(2f), so v = 2f: the image lies at C as well. The magnification is m = −v/u = −2f/2f = −1, and the sign and the magnitude carry the two halves of the answer — the minus sign says inverted and real, the 1 says equal in size. This is also why laboratories use exactly this arrangement to measure a concave mirror's radius of curvature: slide the screen until the sharp image is the same height as the object, and the object distance you have found is R, whose half is the focal length.

• (a)at the focus — This is the inverse of the standard 'object at infinity' case, and it forms no image at any finite distance at all. Put u = f in the mirror formula and 1/v = 1/f − 1/f = 0, so v is infinite: the reflected rays leave the mirror parallel to one another. That is precisely how a torch, a searchlight and a vehicle headlight are built, with the bulb sitting at the focus of a concave reflector to throw a parallel beam. NCERT records the image here as at infinity and highly enlarged — the opposite of same-sized.
• (c)between the focus and centre of curvature — This gives a real, inverted image, which is why it survives the first half of the stem, but the image forms beyond C and is enlarged, not equal in size. It is the projector arrangement — a small object near the focus throwing a magnified image onto a distant screen. Candidates who remember only 'real and inverted' and not the magnification stop here, which is exactly what the option is for.
• (d)beyond the centre of curvature — The mirror image of option (c), and equally real and inverted, but the image now forms between F and C and is diminished. As the object recedes further the image shrinks towards a point at the focus, which is the limiting 'object at infinity' case used when a concave mirror collects light from a distant source. Diminished is not the same size, so this fails the stem's third condition.

A spherical mirror is described by five terms: the pole P (the centre of the reflecting surface), the centre of curvature C (the centre of the sphere the mirror was cut from), the radius of curvature R = PC, the principal focus F, and the focal length f = PF. For a mirror of small aperture these are tied together by R = 2f, so C always sits at twice the focal distance. Image construction uses any two of four standard rays: a ray parallel to the principal axis reflects through F; a ray through F reflects parallel to the axis; a ray through C returns along itself because it meets the surface along the normal; and a ray striking the pole reflects symmetrically about the axis. Images are real when the reflected rays actually meet in front of the mirror and can be caught on a screen, and virtual when they only appear to diverge from a point behind it. All of this is quantified by the mirror formula 1/v + 1/u = 1/f and the magnification m = h′/h = −v/u, read under the New Cartesian sign convention: the pole is the origin, the object is always placed to the left, distances measured to the right of the origin are positive and to the left negative — which makes a concave mirror's focal length negative and a convex mirror's positive.

The stem gives three conditions — real, inverted, same size — and only the third one is doing any work. A concave mirror produces a real, inverted image for every object position outside the focus, so options (c) and (d) both clear the first two tests and are eliminated only by magnification. Reason with |m| instead. Same size means |m| = 1, and since m = −v/u that means |v| = |u|: the object and its image must be equally far from the pole. On a concave mirror there is exactly one finite point where that happens, because as the object moves inward from infinity towards F the image moves outward from F towards infinity, and the two positions cross once — at C. Everything else follows from which side of C the object sits on: outside C the image is nearer the mirror and smaller, inside C (but outside F) it is farther and larger. If you would rather not compute at all, use the normal-incidence ray: a ray from the centre of curvature hits the mirror at right angles and comes straight back, so an object at C must be imaged onto itself. Option (a) is the one worth checking separately, because 'at the focus' produces no finite image at all — the reflected beam is parallel, which is the torch and headlight arrangement rather than an imaging one.

• The NCERT Class X image table for a concave mirror runs: object at infinity → image at F, point-sized, real and inverted; beyond C → between F and C, diminished, real and inverted; at C → at C, same size, real and inverted; between C and F → beyond C, enlarged, real and inverted; at F → at infinity, highly enlarged, real and inverted; between P and F → behind the mirror, enlarged, virtual and erect.
• For a spherical mirror of small aperture R = 2f, so an object at the centre of curvature is at u = 2f; the mirror formula 1/v + 1/u = 1/f then gives v = 2f and the magnification m = −v/u = −1, the negative sign meaning real and inverted and the magnitude meaning equal in size.
• Under the New Cartesian sign convention the pole is the origin and the object always lies to the left, with distances to the right taken positive; on this convention a concave mirror has a negative focal length and a convex mirror a positive one, and magnification is m = h′/h = −v/u.
• The centre of curvature is the only finite point at which object and image coincide on a concave mirror, because a ray leaving C travels along a radius, meets the surface along the normal and retraces its own path — which makes 'object and image the same size' the standard laboratory method for measuring the radius of curvature.
• The applications follow directly from the same table: a bulb at the focus of a concave reflector gives the parallel beam of a torch or headlight, an object between the pole and the focus gives the magnified erect virtual image of a shaving or dentist's mirror, and a distant source is concentrated near the focus in a solar cooker or a reflecting telescope.

• Stopping at 'real and inverted', which is true for every object position outside the focus on a concave mirror; the phrase that actually selects the answer is 'of the same size'
• Treating 'at the focus' as an imaging position — it produces no image at a finite distance, because the reflected rays emerge parallel
• Mixing up the two directions around C: an object beyond C gives a diminished image, an object between C and F gives an enlarged one, and only C itself gives equality

BPSC keeps its optics inside NCERT Class X and asks configuration recall in one line with nothing to compute — the object position for a same-sized image here, and in the 71st CCE of 2025 the focal distance of a plane mirror. UPSC almost never sets a ray-diagram drill; it asks what a phenomenon powers or explains, which is why its optics questions have been about total internal reflection in optical fibres and endoscopy, dispersion and refraction in a rainbow, what an air bubble in water behaves like, and how far a reflected ray turns when the mirror turns. Memorise the two mirror tables for BPSC, and learn the phenomena and their uses for UPSC.

• practice — not a real PYQ

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

  • (a)virtual, erect and enlarged, behind the mirror
  • (b)real, inverted and enlarged, beyond the centre of curvature
  • (c)real, inverted and diminished, between the focus and the centre of curvature
  • (d)real, inverted and of the same size, at the centre of curvature

  Answer(b) real, inverted and enlarged, beyond the centre of curvature — this is the projector arrangement; option (c) describes an object placed beyond C and option (d) an object placed at C.

• practice — not a real PYQ

  In a torch or a vehicle headlight fitted with a concave reflector, the bulb is placed

  • (a)at the centre of curvature
  • (b)at the principal focus
  • (c)between the pole and the principal focus
  • (d)beyond the centre of curvature

  Answer(b) at the principal focus — a source at F sends the reflected rays out parallel to the principal axis, which is what produces a directed beam rather than an image.