MPSC 2023 Prelims Paper-I · Question 37 of 100

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.

• (1)8 — 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.
• (3)10 — 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.
• (4)11 — 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.

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.

The general science section of an MPSC paper always carries a few short calculations, and they are chosen so that the arithmetic is trivial and the knowledge required is a single rule. This item is typical: the division is 360 by 36, which anyone can do, and the entire question turns on whether the candidate knows what to do with the quotient. That makes it a high-value question for a prepared candidate and a coin toss for an unprepared one, since all four options are plausible small integers clustered around the right answer and none can be eliminated by inspection. The lesson for preparation is to learn the small stock of optics rules with their conditions attached rather than as bare formulae, because the conditions are where the commission places its difficulty — here the phrase 'placed symmetrically' is doing no work, but the identical phrase in a question about mirrors at 72 degrees would decide the answer. The lesson for the examination hall is to compute the intermediate quantity, write it down, and only then apply the rule; most of the wrong answers to a question of this shape come from a slip before the rule is reached rather than from ignorance of the rule itself.

• Two plane mirrors inclined at an angle produce multiple images by repeated reflection, and the object together with all its images lies on a circle centred on the line where the mirrors meet, with the positions spaced at intervals equal to the angle between them.
• Writing n for 360 divided by the angle in degrees, the number of images is n minus one when n is an even whole number, and this holds wherever the object is placed between the mirrors.
• When n is an odd whole number the position matters: an object on the bisector of the angle, that is, placed symmetrically, gives n minus one images, while an object placed off the bisector gives n images.
• For mirrors at 36 degrees, n is 10 and nine images are formed; at 90 degrees three images are formed, at 60 degrees five, and between two parallel mirrors the series of images is unlimited.
• The image formed by a plane mirror is always virtual, erect, of the same size as the object, laterally inverted, and situated as far behind the mirror as the object is in front of it.

• Stopping at the quotient of 360 divided by the angle and reporting it as the number of images, when one of those positions is occupied by the object itself
• Getting the direction of the correction wrong and adding one instead of subtracting, which is what happens when the formula is learnt without the underlying picture
• Ignoring the condition attached to the rule; the position of the object matters only when the quotient is odd, and a stem mentioning symmetry may be relevant or may not
• Making the arithmetic slip before the rule is applied, since most wrong answers to this kind of question come from a misread angle rather than from a misremembered rule

Optics supplies MPSC with both definitional and numerical questions, and the numerical ones are deliberately light on arithmetic. The recurring types are the number of images formed by inclined mirrors, the nature and position of an image in a plane or spherical mirror, simple applications of the mirror or lens formula, the conditions for total internal reflection, and the identification of a defect of vision with the lens that corrects it. In every case the calculation is short and the difficulty is placed in a condition or a definition. Expect the option set for a counting question to consist of consecutive integers around the correct value, which removes any possibility of elimination by rough estimate, and expect the stem to include a phrase describing the object's position, which will be decisive in some geometries and irrelevant in others.

No directly related past PYQ was found.

• practice — not a real PYQ

  Two plane mirrors are placed at right angles to one another. How many images of an object placed between them will be formed ?

  • (a)Two
  • (b)Three
  • (c)Four
  • (d)An unlimited number

  Answer(b) Three — dividing 360 by 90 gives four, which is an even whole number, so the number of images is one less than four, and the placement of the object makes no difference. The four equally spaced positions on the circle consist of the object and its three images, which is the familiar arrangement seen when two mirrors are set into a corner. An unlimited series of images requires the mirrors to be parallel, so that the angle between them is zero.

• practice — not a real PYQ

  An object is placed between two plane mirrors inclined at an angle of 72 degrees, but not on the bisector of that angle. How many images will be formed ?

  • (a)Four
  • (b)Five
  • (c)Six
  • (d)Seven

  Answer(b) Five — dividing 360 by 72 gives five, which is an odd whole number, and for an odd quotient the answer depends on where the object stands. Placed off the bisector it gives five images, while placed symmetrically on the bisector it would give one fewer, that is four. This is the case in which the phrase describing the object's position is decisive, and it is the reason the condition must be learnt along with the rule.