The image formed by concave mirror is real, inverted and of the same size as that of the object. The position of the object should be

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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.

Why the other options are wrong — (a) 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) 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) 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.

Concept: 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.

State PCS 2023 Q15 — Science & Technology PYQ with Answer

State PCS 2023 Q15 — Science & Technology PYQ with Answer