MPSC 2022 Prelims Q41: A cat is sitting in between two plane mirrors. If the angle

MPSC 2022 Prelims Paper-I · Question 41 of 100

State Services Preliminary Examination 2022, held 21 August 2022. Transcribed from the commission's own booklet (संच A); answer as per the official MPSC final answer key.

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 ?

(1)0
(2)1
(3)11
(4)12

Answer

(3)

Why

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 others are wrong

(1)0 — 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.
(2)1 — 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.
(4)12 — 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.

This is a formula question with a mechanism behind it, and MPSC's science section is fond of exactly that combination because a candidate who has memorised only half the rule will fall into a distractor that has been placed for the purpose. The two halves are the division and the treatment of the remainder: divide the full turn by the angle, then subtract one when the result is even. Option (4) is the answer of a candidate who did the division and stopped, which is the most common way this question is lost. The mechanism is worth learning as well as the rule, because the Commission also asks the qualitative versions of the same idea — what happens as the angle between the mirrors is reduced, why a kaleidoscope uses 60 degrees, why two parallel mirrors give an unlimited series of images — and none of those can be answered from the formula alone. Two parallel mirrors are the limiting case worth carrying away: the angle is zero, the division is not defined, and the images repeat without end, growing fainter with every reflection because each reflection loses a little light. Questions of this family are quick marks under time pressure provided the rule is remembered whole, and quick losses when it is remembered in part.

Key facts

For two plane mirrors inclined at an angle theta, let n be 360 divided by theta. If n is an even whole number the number of images is n minus one, whatever the position of the object between the mirrors.
If n is an odd whole number the number of images is n minus one when the object lies on the bisector of the angle between the mirrors, and n when the object lies anywhere else between them.
For an angle of 30 degrees, n is 12, which is even, so the number of images formed is 11 — the case asked in this question.
The standard values follow from the same rule: mirrors at 90 degrees give three images, at 60 degrees five images, and at 45 degrees seven images, the number rising as the angle between the mirrors is narrowed.
Two parallel plane mirrors are the limiting case in which the angle is zero and the formula does not apply; the reflections repeat without end, each image fainter than the last because every reflection absorbs a little of the light.

Divide the full turn by the angle, then take the cat itself off the count

angle between the mirrors = 30 degrees→The cat and all its images lie on one circle centred on the line where the two mirrors meet, evenly spaced by this angle. Each mirror keeps forming images of the images made by the other, so what is being counted is a ring of positions and not one image per mirror

n = 360 / 30 = 12→The number of evenly spaced positions round the full turn — and the reason the series ends rather than running on for ever: once the ring is filled there is nowhere left to put a new image

n is EVEN, so images = n - 1 = 11→ONE of the twelve positions is occupied by the cat, which is a real animal and not a reflection. Subtract once for the object — not once per mirror. The two series, one running clockwise from each mirror, meet on the far side, where the last image of each is the same image and is counted once

the condition on the rule→n EVEN gives n - 1 wherever the cat sits between the mirrors. n ODD gives n - 1 only when the object stands on the bisector of the angle, and n anywhere else. Here n = 12 is even, so the question is answerable without being told where the cat is — at 40 degrees, where n = 9, it would not be

12 is the count of positions, not of images, and option (4) prints exactly that number for a candidate who stops one line early. The same rule gives the values worth carrying: 3 images at 90 degrees, 5 at 60, 7 at 45, the count rising as the mirrors close on each other. Two parallel mirrors are the limiting case where the angle is zero, the division is impossible and the reflections repeat without end, each one fainter than the last.

Study next

Image formation by a single plane mirror — the image is virtual, erect, of the same size and as far behind the mirror as the object is in front of it, with lateral inversionMultiple reflection between inclined mirrors, the circle on which the images lie, and why the even case loses one image while the odd case depends on the position of the objectThe kaleidoscope and the periscope as applications of plane mirrors, and why the kaleidoscope uses mirrors at 60 degreesReflection at spherical mirrors — the mirror formula, magnification, and the nature of images formed by concave and convex mirrorsLaws of reflection and regular against diffuse reflection, which decide whether an image is formed at all

Common traps

Dividing 360 by the angle and stopping there, which gives 12 instead of 11 and is the single commonest error on this question
Applying the even-case rule to an odd value of n without asking where the object is placed, since the odd case gives a different count on and off the bisector
Assuming that each mirror simply contributes one image, so that two mirrors give two images, which ignores the successive reflection that the arrangement is designed to produce
Trying to use the formula for two parallel mirrors, where the angle is zero, the division is undefined and the series of images has no end

MPSC sets optics from the school syllabus in two forms. The first is a direct calculation like this one, where an angle is supplied and a number of images is wanted, and the angles chosen are almost always 30, 45, 60 or 90 degrees so that the division comes out whole. The second is conceptual — which mirror gives an erect and diminished image, why the image in a plane mirror is called virtual, what use is made of a convex mirror on a vehicle, how a kaleidoscope produces its pattern — and these reward understanding of the mechanism rather than the formula. Preparing the pair of mirrors properly means being able to state the rule in both its cases and to reproduce the standard values for 90, 60, 45 and 30 degrees from memory, because that turns the calculation into a recall and saves the time that the arithmetic would otherwise cost in the examination hall.

Related PYQs

No directly related past PYQ was found.

Practice

practice — not a real PYQ
An object is placed between two plane mirrors inclined at an angle of 60 degrees to each other. How many images of the object will be formed ?
- (a)3
- (b)5
- (c)6
- (d)7
Answer(b) 5 — dividing the full turn of 360 degrees by 60 gives 6, which is an even number, so the number of images is one less than that, namely five, and this holds wherever the object is placed between the mirrors. The answer 6 is what a candidate gets by performing the division and forgetting the subtraction, which is the standard error in this family of questions.
practice — not a real PYQ
Two plane mirrors are placed parallel and facing each other, with an object between them. Which of the following statements about the images is correct ?
- (a)Exactly two images are formed, one in each mirror
- (b)No image is formed because the reflections cancel
- (c)An unending series of images is formed, each fainter than the one before
- (d)Exactly four images are formed
Answer(c) An unending series of images is formed, each fainter than the one before — with the mirrors parallel the angle between them is zero, so the rule that divides 360 by the angle cannot be applied, and light is reflected back and forth without ever completing a turn. Each successive image is dimmer because a small fraction of the light is absorbed at every reflection, which is why the series appears to fade away rather than to stop.