A cat is sitting in between two plane mirrors. If the angle between two plane mirrors is 30 degrees, how many images of the cat will appear ?

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Correct — option (3), 11. Two plane mirrors placed at an angle to one another do not give one image each. Light that leaves the cat and strikes the first mirror is reflected onto the second, reflected back onto the first, and so on, so each mirror keeps producing images of the images formed by the other. The series does not run on for ever, because every reflection carries the image further round the line where the two mirrors meet, and once the images have gone right round the circle there is nowhere left to place a new one. The working rule that follows from this is to divide the full turn by the angle between the mirrors and then decide what to do with the last image. Write n for 360 divided by the angle. When n comes out as an even whole number, the number of images is n minus one, and this holds wherever the object is placed between the mirrors. When n is an odd whole number the count depends on where the object stands: it is n minus one if the object lies exactly on the bisector of the angle, and n if it lies anywhere else. Here the angle is 30 degrees, so n is 360 divided by 30, which is 12. Twelve is even, so the number of images is 12 minus 1, that is 11, and the position of the cat between the mirrors does not matter. The reason one image is lost when n is even is worth holding on to rather than memorising the formula blind. The images formed by the two mirrors all lie on a circle whose centre is the line of intersection of the mirrors and whose radius is the distance of the object from that line. The images are laid out around this circle by successive reflection, one series running clockwise from the first mirror and the other running anticlockwise from the second. When n is even the two series meet at the same point on the far side of the mirrors, so the last image of one series and the last image of the other are the same image and are counted once instead of twice — which is exactly why the total is one fewer than the number of sectors the mirrors would cut the circle into. Option (3) is therefore the answer.

Why the other options are wrong — (a) There is no arrangement of two plane mirrors facing each other in which an object placed between them produces no image at all. A single plane mirror already produces one image of anything placed in front of it, formed by the reflection of the light that reaches the surface, and putting a second mirror in the way can only add to that count, never remove it. The option is in the set as a test of whether the candidate is willing to give a reflex answer to a question about mirrors without thinking about what a mirror does; it can be struck out before any arithmetic is attempted. Note also the difference between an image being formed and an image being seen: with mirrors inclined at a sharp angle, some of the eleven images lie behind the mirror surfaces and an observer at a particular spot may not see all of them at once, but they are formed nevertheless and the question asks how many appear in the system, not how many one eye catches from one position.  (b) One image is what a single plane mirror gives, and it is also what the formula gives for two mirrors placed in the same plane, that is at 180 degrees to each other, since 360 divided by 180 is 2 and 2 minus 1 is 1. Choosing it here means treating the two mirrors as though they acted independently, or overlooking that they are inclined and therefore able to reflect each other's images. The whole point of the arrangement in the question is multiple reflection: the first mirror forms an image, the second mirror forms an image of that image, the first mirror forms an image of that one, and the process continues until the images have gone round the full circle. Any answer of one, or of two, misses that the count rises sharply as the angle between the mirrors is made smaller — 90 degrees gives three images, 60 degrees gives five, 45 degrees gives seven, and 30 degrees gives eleven.  (d) Twelve is the trap that catches candidates who remember the division but not the subtraction. It is the value of n itself, 360 divided by 30, and it counts the number of sectors that the two mirrors and their extensions cut the surrounding circle into, not the number of distinct images. One of those twelve positions is occupied by the cat itself, and it is the position diametrically opposite that gives the trouble: the last image reached by reflecting round one way and the last image reached by reflecting round the other way land on the same spot, so they are a single image and must not be counted twice. That is why the even case always gives n minus one. A candidate who has time can check the rule on a case that is easy to picture — two mirrors at right angles give 360 divided by 90, that is 4, and everyone who has stood in a corner of a mirrored room knows the answer there is three images, not four.

Concept: Two plane mirrors inclined at an angle form a system in which images are produced by repeated reflection, each mirror imaging both the object and the images already formed by the other. All of these images lie on a circle centred on the line where the mirror planes meet, with radius equal to the distance of the object from that line, and they are distributed around the circle by successive reflection in the two mirror planes. If the angle between the mirrors is theta, the quantity 360 divided by theta counts how many sectors of theta the full turn contains. For an even value of that quantity the two chains of images, one built by starting at the first mirror and one by starting at the second, terminate at the same point behind the mirrors, so the last image is shared and the number of distinct images is one less than the number of sectors. For an odd value the two chains terminate at different points unless the object is placed symmetrically on the bisector, which is why the odd case has to be stated with a condition attached and the even case does not. The same reasoning explains the familiar special cases: mirrors at 90 degrees give three images, at 60 degrees five, at 45 degrees seven and at 30 degrees eleven, the count rising as the angle narrows. The kaleidoscope is the everyday application, its three mirrors set at 60 degrees so that the pattern is repeated in a closed ring.

State PCS 2022 Q41 — Science & Technology PYQ with Answer

State PCS 2022 Q41 — Science & Technology PYQ with Answer