If an object is placed symmetrically between two plane mirrors inclined at an angle of 36°, then how many images will be formed ?

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Correct — option (2), '9'. When two plane mirrors are set at an angle, light bounces repeatedly between them and each reflection of an image acts as the object for the next, so a whole family of images is produced. The geometry of that family is simple and is the quickest route to the answer. All the images, together with the object itself, lie on a circle whose centre is the line where the two mirrors meet and whose radius is the distance of the object from that line, and they are spaced evenly around the circle at intervals equal to the angle between the mirrors. A full turn is 360 degrees, so the number of positions on the circle is 360 divided by the angle. Here the angle is 36 degrees, and 360 divided by 36 is 10, which means ten equally spaced positions in all. One of those ten is occupied by the object, and every other one is an image, so the number of images is nine. That is option (2). The same result is usually quoted as a rule, and the rule is worth learning in full because it has a condition attached. Write n for 360 divided by the angle. If n comes out as an even whole number, the number of images is n minus one, and it makes no difference where between the mirrors the object is placed. If n comes out as an odd whole number, the answer depends on the position: an object standing on the bisector of the angle, that is, placed symmetrically, gives n minus one images, while an object placed off the bisector gives n. In this question n is 10, an even number, so the rule gives nine images and the phrase 'placed symmetrically' in the stem does not change anything — it is there because the same wording would be decisive if the angle had been, say, 72 degrees, where n is 5 and odd. Either way, by the circle or by the rule, the count is nine.

Why the other options are wrong — (a) Eight is two fewer than the ten positions the geometry gives, and it is generally reached by subtracting once for each mirror instead of once for the object. The subtraction in the rule is a single one and it has a definite reason behind it: of the ten evenly spaced positions on the circle, nine hold images and one holds the object, which is a real thing rather than a reflection and must not be counted among them. There is nothing further to remove. The other route to this answer is an arithmetic slip in the division, since a candidate who reads the angle as 40 degrees obtains n equal to 9 and then subtracts one. Both errors are worth guarding against by doing the division deliberately and writing down the value of n before applying any rule to it, because the rule is short enough that the mistakes almost always occur before it is reached.  (c) Ten is the value of n itself, and this option collects the candidate who divides 360 by 36, sees a whole number and stops. The division gives the number of equally spaced positions around the circle, not the number of images, and the two differ by exactly one because the object occupies one of those positions. There is a second way to see why the subtraction is needed: as the reflections proceed in each mirror, the last image formed by reflection in one mirror coincides with the last formed by reflection in the other, since both fall at the same point behind the pair, so counting the two sequences separately would count that shared image twice. Ten would be the correct answer only in the case where n is an odd number and the object is placed off the bisector, and neither condition holds here, since n is 10 and the object is placed symmetrically.  (d) Eleven is one more than the number of positions available, so it cannot be right under any reading of the geometry: there are only ten evenly spaced points on the circle, and even if the object were mistakenly counted as one of the images the total could not exceed ten. The answer arises from adding one to n instead of subtracting one, which usually happens when the rule has been memorised as a formula without the picture that justifies it — a candidate who remembers only that 'one' appears somewhere in the rule has an even chance of getting the sign wrong. This is the strongest argument for holding the circle of images in mind rather than the bare formula, because the picture makes the direction of the correction obvious and also makes clear why an odd value of n behaves differently from an even one.

Concept: A single plane mirror forms one image of an object, and that image is virtual, erect, the same size as the object, laterally inverted, and located as far behind the mirror as the object stands in front of it. Bring in a second mirror and the situation becomes richer, because the image formed in the first mirror can itself be reflected in the second, that second reflection can be reflected again in the first, and so on. Each bounce produces a further image, and the sequence continues until a reflection falls outside the reflecting surfaces. The resulting set has a neat geometry: object and images all lie on one circle centred on the line of intersection of the mirrors, spaced at angular intervals equal to the angle between the mirrors, which is why the count depends only on that angle and not on how far the object is from either surface. Two limiting cases are worth holding. If the mirrors are parallel, the angle is zero, the division is undefined and the reflections never terminate, so an object between two parallel mirrors produces an unlimited series of images receding into the distance, as in a barber's shop. If the angle is 90 degrees, the division gives four and the count is three, which is the familiar arrangement of two mirrors set in a corner. At 60 degrees the count is five, and that is the geometry of a kaleidoscope, whose three mirrors meeting at 60 degrees generate the symmetrical patterns the toy is made for.

State PCS 2023 Q37 — Science & Technology PYQ with Answer

State PCS 2023 Q37 — Science & Technology PYQ with Answer