MPSC 2021 Prelims Paper-I · Question 52 of 100

Correct — option (1), 8.9. Relative density, also called specific gravity, is defined as the ratio of the density of a substance to the density of water, water being taken at 4 °C where its density is at a maximum of 1000 kg m⁻³, equivalently 1 g cm⁻³. Both numbers needed are printed in the stem, so the working is a single division: 8.9 × 10³ kg m⁻³ divided by 1000 kg m⁻³, which is 8.9 × 10³ divided by 1 × 10³, which is 8.9. That is the whole calculation, and the entire question is what happens to the units while you do it. Because the numerator and the denominator are both densities, the kilograms cancel against kilograms and the cubic metres against cubic metres, and nothing is left. Relative density is therefore a pure number — it is dimensionless, it carries no unit, and it has the same value whatever consistent system of units the two densities were measured in. Work the same problem in grams per cubic centimetre and it comes out identically: copper at 8.9 g cm⁻³ divided by water at 1 g cm⁻³ is again 8.9. That invariance is the practical point of defining the quantity at all, and it is why the relative density of a substance is numerically equal to its density expressed in grams per cubic centimetre, a shortcut worth carrying. So the answer is the bare number 8.9, and any option that attaches a unit to it, or that shifts its power of ten, has misunderstood what a ratio is rather than made an arithmetic slip. It is worth sanity-checking the result against what it means physically: a relative density of 8.9 says that a given volume of copper is 8.9 times as heavy as the same volume of water, which is why copper sinks, and it sits sensibly between aluminium at about 2.7 and lead, mercury and gold higher up the scale.

• (2)8.9 kgm⁻³ — This option carries the right number and attaches a unit to it, and that is the error the question was written to expose. Relative density is a ratio of two densities, so the unit of the numerator cancels against the identical unit of the denominator and the result is a pure number with no dimensions at all. Writing 8.9 kg m⁻³ does not merely add a redundant label; it states something physically absurd, because 8.9 kilograms per cubic metre is roughly the density of a gas rather than of a metal — copper's actual density, given in the stem, is a thousand times larger. The habit that prevents this is to carry the units through the arithmetic rather than to attach them at the end: divide kg m⁻³ by kg m⁻³ and watch them disappear. The same discipline settles a whole family of dimensionless quantities that MPSC asks about, including the coefficient of friction, refractive index, strain and the mechanical advantage of a machine, none of which takes a unit.
• (3)8.9 × 10⁶ — This is what a candidate gets by multiplying the two densities instead of dividing them: 8.9 × 10³ multiplied by 1 × 10³ is 8.9 × 10⁶. The error is one of operation rather than of arithmetic, and it usually comes from reading 'relative density' as an unfamiliar phrase and reaching for whichever operation the numbers seem to invite. The definition is a quotient, and the order of the quotient matters too — the density of the substance goes on top and the density of water underneath, so that substances denser than water score above 1 and substances lighter than water score below it. A second check catches this option instantly. Relative density is a comparison, and a comparison of a metal with water should produce a number of ordinary size: no everyday solid is millions of times denser than water, and even the densest metals sit around twenty. An answer in the millions is not a slightly wrong value but a value of the wrong order entirely.
• (4)8.9 × 10⁻⁶ — This option is the previous one with the sign of the exponent reversed, and no correct handling of the two densities in the stem produces it. It is the answer of a candidate who has decided that a power of ten must appear somewhere and is guessing at its direction — the sort of error that follows from manipulating exponents mechanically rather than dividing 8.9 × 10³ by 1 × 10³ and seeing the powers cancel. The physical check disposes of it even faster than the arithmetic. A relative density of 8.9 × 10⁻⁶ would mean that copper is about a hundred thousand times less dense than water, which would make it lighter than any gas and would put a copper coin somewhere above the atmosphere rather than at the bottom of a bucket. Getting into the habit of asking whether an answer is physically possible, before checking whether it is arithmetically right, disposes of two of the three wrong options here.

Density is mass per unit volume, measured in kilograms per cubic metre in SI units and often quoted in grams per cubic centimetre in the laboratory, where 1 g cm⁻³ equals 1000 kg m⁻³. Relative density, or specific gravity, is the density of a substance divided by the density of water at 4 °C, the temperature at which water is densest. Because it is a ratio of two like quantities it has no dimensions and no unit, and because water's density is 1 g cm⁻³ the relative density of any substance is numerically the same as its density in grams per cubic centimetre — copper at 8.9, aluminium at about 2.7, mercury at about 13.6, gold at about 19.3, ice at about 0.92. That last figure is why ice floats, and why only about a tenth of an iceberg stands above the water. The quantity matters because it decides floating and sinking: a body placed in a fluid is buoyed up by a force equal to the weight of the fluid it displaces, which is Archimedes' principle, so a body of relative density greater than 1 sinks in water and one of relative density less than 1 floats with the fraction of its volume submerged equal to its relative density. Relative density is measured directly with a hydrometer, a weighted float that sinks to a depth set by the density of the liquid; the lactometer used to check milk and the instruments used to test the electrolyte of a lead-acid battery are hydrometers built for particular ranges.

MPSC's science section includes a small number of numerical items in each paper, and they are almost always one-step calculations that test whether a definition is held precisely rather than whether a candidate can compute. This is a clear example: the arithmetic is a single division that most candidates could do without writing anything down, and the marks turn entirely on knowing that the result of dividing one density by another is a bare number. The option set is built accordingly — the correct value appears twice, once naked and once wearing a unit, and the other two rows shift its power of ten in either direction. That design is a useful warning about how such questions are constructed. When the same digits appear in more than one option, the question is testing something other than arithmetic, and the something is usually a unit, a power of ten or the direction of a ratio. Note also how the exponents are printed. The stem and the options use typeset superscripts — kgm⁻³, 10³, 10⁶, 10⁻⁶ — and the card reproduces them as printed. A candidate reading quickly can miss a minus sign in a superscript, which is exactly the difference between the third and the fourth option here.

• Relative density, or specific gravity, is the density of a substance divided by the density of water at 4 °C; because it is a ratio of two like quantities it is dimensionless and carries no unit.
• The density of water at 4 °C is 1000 kg m⁻³, equivalently 1 g cm⁻³, which is why the relative density of a substance is numerically equal to its density expressed in grams per cubic centimetre.
• For copper: 8.9 × 10³ kg m⁻³ divided by 1000 kg m⁻³ gives 8.9, and the same division in grams per cubic centimetre — 8.9 divided by 1 — gives the same value, which is the point of defining the quantity as a ratio.
• A body with relative density greater than 1 sinks in water and one with relative density less than 1 floats, with the submerged fraction of its volume equal to its relative density; ice at about 0.92 is the standard example.
• Other dimensionless quantities examined in the same syllabus include the coefficient of friction, refractive index, strain and mechanical advantage; relative density is measured directly with a hydrometer, of which the lactometer is a familiar form.

• Attaching a unit to a dimensionless quantity, which is the specific error the second option in this set was printed to catch
• Multiplying two densities instead of dividing them, or inverting the ratio so that the density of water goes on top
• Misreading the sign of an exponent printed as a superscript, which is the only difference between two of the wrong options here
• Failing to sanity-check an answer against physical reality — a metal cannot be millions of times denser, or a hundred thousand times lighter, than water
• Confusing density with relative density in a stem that supplies both, and reporting the substance's density where a ratio was asked for

Numerical science items in MPSC papers are short by design and test a definition rather than a technique. Relative density arrives in three shapes. The first is this one — two densities given, the ratio asked for, and the marks turning on the absence of a unit. The second is a flotation item: a block of stated relative density is placed in water and the fraction submerged, or the volume above the surface, is asked for, which follows directly from the law of flotation. The third is qualitative, asking which of several substances will float, or what a hydrometer reading indicates, or why ice floats on water. All three are answerable from one definition and one principle, so the preparation is to hold the definition of relative density exactly, to memorise a handful of standard values, and to be able to state Archimedes' principle in a single sentence.

No directly related past PYQ was found.

• practice — not a real PYQ

  Which of the following physical quantities is dimensionless ?

  • (a)Density
  • (b)Relative density
  • (c)Pressure
  • (d)Momentum

  Answer(b) Relative density — it is the ratio of the density of a substance to the density of water, so the units of the numerator cancel against identical units in the denominator and the result is a pure number. Density is mass per unit volume and carries kg m⁻³, pressure is force per unit area and carries the pascal, and momentum is mass times velocity and carries kg m s⁻¹. Other dimensionless quantities in the same syllabus include the refractive index, the coefficient of friction, strain and mechanical advantage.

• practice — not a real PYQ

  A block of ice of relative density 0.92 floats in water. What fraction of its volume remains above the surface of the water ?

  • (a)About 8 per cent
  • (b)About 20 per cent
  • (c)About 46 per cent
  • (d)About 92 per cent

  Answer(a) About 8 per cent — by the law of flotation, a floating body displaces its own weight of fluid, so the submerged fraction of its volume equals its relative density, which here is 0.92 or 92 per cent. The remaining 8 per cent stands above the surface, which is the standard explanation for why only a small part of an iceberg is visible. The figure of 92 per cent in the last option is the submerged fraction rather than the exposed one, and choosing it is the commonest way this calculation is reversed.