NCERT Exemplar Solutions Class 11 Physics Units and Measurement — Multiple Choice Questions Q1–18

NCERT Exemplar Problems, Class 11 Physics — Units and Measurement, MCQ I and MCQ II. All 18 questions solved.

The book numbers these questions 2.1 to 2.18 (the exemplar counts this as its second unit, while the rationalised textbook calls it Chapter 1); here they are Questions 1 to 18. Questions 1 to 12 are MCQ I (exactly one option is correct) and Questions 13 to 18 are MCQ II (one or more options may be correct). Three ideas carry the whole page:

• Significant figures. In multiplication and division the result keeps the fewest significant figures of the data; in addition and subtraction it keeps the fewest decimal places.
• Propagation of error. For a product or quotient the relative errors add, each multiplied by its power: if $X = a^p b^q$, then $\dfrac{\Delta X}{X} = p\dfrac{\Delta a}{a} + q\dfrac{\Delta b}{b}$.
• Dimensions. Only quantities with the same dimensions can be added, subtracted or equated, and the argument of a sine, exponential or power series must be dimensionless.

Key insight. Most errors on this page come from mixing up the two rounding rules: Question 2 is an addition, so it is decimal places that count, while Question 3 is a division, so it is significant figures.

Choose the correct answer from the given options. (MCQ I: one correct. MCQ II: one or more correct.)

Question 1

The number of significant figures in $0.06900$ is

Solution. The zeros before the first non-zero digit ($0.0$) only locate the decimal point, so they are not significant. Writing the number as $6.900\times10^{-2}$ makes this clear. The trailing zeros after the decimal point are significant, because someone deliberately recorded them as measured digits. That leaves $6, 9, 0, 0$: four significant figures.

Question 2

The sum of the numbers $436.32$, $227.2$ and $0.301$ in appropriate significant figures is

Solution. The raw sum is $436.32 + 227.2 + 0.301 = 663.821$. In addition the rule is about decimal places, not the count of significant figures: the result can be no more precise than the least precise term. $227.2$ is known only to the first decimal place, so the sum is rounded to one decimal place: $663.8$.

Printed key error. The key prints (b) $664$. That would round away the first decimal place, which $227.2$ does determine. Nothing in the data justifies dropping to the units place, and $664$ throws away a digit that is genuinely known. The textbook’s rule for addition gives $663.8$.

Question 3

The mass and volume of a body are $4.237$ g and $2.5\ \text{cm}^3$, respectively. The density of the material of the body in correct significant figures is

Solution. $\rho = \dfrac{4.237}{2.5} = 1.6948\ \text{g cm}^{-3}$. This is a division, so the result keeps as many significant figures as the least precise factor. The volume $2.5$ has two, so the density is rounded to two significant figures: $1.7\ \text{g cm}^{-3}$.

Question 4

The numbers $2.745$ and $2.735$ on rounding off to 3 significant figures will give

Solution. In both numbers the digit being dropped is exactly $5$. The textbook rule for this case is to make the preceding digit even: leave it if it is already even, raise it by one if it is odd. In $2.745$ the preceding digit is $4$ (even), so it stays: $2.74$. In $2.735$ the preceding digit is $3$ (odd), so it becomes $4$: $2.74$. The rule exists so that, over many roundings, as many go up as go down and no bias builds up.

Question 5

The length and breadth of a rectangular sheet are $16.2$ cm and $10.1$ cm, respectively. The area of the sheet in appropriate significant figures and error is

Solution. Each length is given to $0.1$ cm, so take $\Delta l = \Delta b = 0.1$ cm. The area is $A = 16.2\times10.1 = 163.62\ \text{cm}^2$. For a product the relative errors add:

$$\frac{\Delta A}{A} = \frac{0.1}{16.2} + \frac{0.1}{10.1}, \qquad \Delta A = 0.1\times10.1 + 0.1\times16.2 = 2.63\ \text{cm}^2.$$

The data have three significant figures, so $A$ is quoted as $164$, and an error of $2.63$ on a number known only to the units place is quoted as $3$. The error should never be stated more finely than the value itself.

Question 6

Which of the following pairs of physical quantities does not have same dimensional formula?

Solution. Work and torque are both force times distance, $[\text{ML}^2\text{T}^{-2}]$. Angular momentum ($mvr$) and Planck’s constant ($E/\nu$) are both $[\text{ML}^2\text{T}^{-1}]$. Impulse ($F\,\Delta t$) equals the change in momentum, so both are $[\text{MLT}^{-1}]$. Tension is a force, $[\text{MLT}^{-2}]$, but surface tension is force per unit length, $[\text{MT}^{-2}]$. The similar names hide different dimensions.

Question 7

Measure of two quantities along with the precision of respective measuring instrument is $A = 2.5\ \text{m s}^{-1} \pm 0.5\ \text{m s}^{-1}$, $B = 0.10\ \text{s} \pm 0.01\ \text{s}$. The value of $AB$ will be

Solution. $AB = 2.5\times0.10 = 0.25$ m. For a product the relative errors add:

$$\frac{\Delta(AB)}{AB} = \frac{0.5}{2.5} + \frac{0.01}{0.10} = 0.2 + 0.1 = 0.3,$$

so $\Delta(AB) = 0.3\times0.25 = 0.075 \approx 0.08$ m.

Question 8

You measure two quantities as $A = 1.0\ \text{m} \pm 0.2\ \text{m}$, $B = 2.0\ \text{m} \pm 0.2\ \text{m}$. We should report correct value for $\sqrt{AB}$ as:

Solution. $\sqrt{AB} = \sqrt{2.0} = 1.414$ m, which is reported to two significant figures as $1.4$ m because the data have two. A square root halves the relative error:

$$\frac{\Delta R}{R} = \frac12\left(\frac{0.2}{1.0} + \frac{0.2}{2.0}\right) = \frac12(0.2+0.1) = 0.15,$$

so $\Delta R = 0.15\times1.414 = 0.21 \approx 0.2$ m. Option (b) has the right fraction but quotes both value and error more finely than the data allow.

Question 9

Which of the following measurements is most precise?

Solution. Precision is the fineness of the smallest division to which a reading is recorded. Each value is written to two decimal places of its own unit, so the uncertainties are $0.01$ mm, $0.01$ cm, $0.01$ m and $0.01$ km. The smallest of these is $0.01$ mm, so $5.00$ mm is the most precise.

Question 10

The mean length of an object is $5$ cm. Which of the following measurements is most accurate?

Solution. Accuracy is closeness to the true (here, mean) value, not the number of digits written. The deviations from $5$ cm are $0.1$, $0.195$, $0.25$ and $0.4$ cm. The smallest is for $4.9$ cm. Option (b) looks impressive because of its three decimal places, but that is precision, and it is further from the mean.

Question 11

Young’s modulus of steel is $1.9\times10^{11}\ \text{N/m}^2$. When expressed in CGS units of dynes/cm$^2$, it will be equal to ($1\ \text{N} = 10^5$ dyne, $1\ \text{m}^2 = 10^4\ \text{cm}^2$)

Solution. Replace each unit by its CGS equivalent:

$$1.9\times10^{11}\ \frac{\text{N}}{\text{m}^2} = 1.9\times10^{11}\times\frac{10^5\ \text{dyne}}{10^4\ \text{cm}^2} = 1.9\times10^{12}\ \text{dyne/cm}^2.$$

Question 12

If momentum ($P$), area ($A$) and time ($T$) are taken to be fundamental quantities, then energy has the dimensional formula

Solution. Write $[E] = P^aA^bT^c$ and compare powers of M, L and T. With $[P] = \text{MLT}^{-1}$ and $[A] = \text{L}^2$,

$$\text{ML}^2\text{T}^{-2} = \text{M}^a\,\text{L}^{a+2b}\,\text{T}^{-a+c}.$$

M gives $a = 1$; L gives $1 + 2b = 2$, so $b = \tfrac12$; T gives $-1 + c = -2$, so $c = -1$. A quick check: momentum times speed is energy, and $A^{1/2}T^{-1}$ is a length per unit time.

Question 13

On the basis of dimensions, decide which of the following relations for the displacement of a particle undergoing simple harmonic motion is not correct:

Solution. Two checks apply: the argument of a sine must be dimensionless, and both sides must have the dimension of length (with $a$ an amplitude, a length).

• (a) $t/T$ is time over time: dimensionless, and $y$ has the dimension of $a$. Correct.
• (b) $vt$ (speed times time) is a length, so it cannot be the argument of a sine. Not correct.
• (c) $t/a$ is time over length, not dimensionless; and $a/T$ is a speed, not a length. Not correct on both counts.
• (d) The arguments are dimensionless and the factor $a\sqrt2$ is a length. Correct.

Question 14

If $P$, $Q$, $R$ are physical quantities, having different dimensions, which of the following combinations can never be a meaningful quantity?

Solution. Addition and subtraction need equal dimensions. Since $P$, $Q$ and $R$ all have different dimensions, $P – Q$ and $R + Q$ are meaningless whatever else is done to them, so (a) and (e) can never be meaningful. In (b), $PQ$ could have the same dimensions as $R$ (for instance, force times time and momentum), so $PQ – R$ can make sense. In (d), $PR$ could match $Q^2$. Products and quotients such as (c) are always allowed.

Question 15

Photon is quantum of radiation with energy $E = h\nu$ where $\nu$ is frequency and $h$ is Planck’s constant. The dimensions of $h$ are the same as that of

Solution. $[h] = [E]/[\nu] = \text{ML}^2\text{T}^{-2}/\text{T}^{-1} = \text{ML}^2\text{T}^{-1}$. Angular momentum $mvr$ has exactly these dimensions, and angular impulse (torque times time) equals a change in angular momentum, so it does too. Linear impulse and linear momentum are $\text{MLT}^{-1}$, one power of length short.

Question 16

If Planck’s constant ($h$) and speed of light in vacuum ($c$) are taken as two fundamental quantities, which one of the following can, in addition, be taken to express length, mass and time in terms of the three chosen fundamental quantities?

Solution. Three base quantities can replace M, L and T only if their dimensional formulas are independent. $h$ is $\text{ML}^2\text{T}^{-1}$ and $c$ is $\text{LT}^{-1}$; together they still cannot separate mass from the rest, so the third must bring in mass in a new way.

• $m_e$ or $m_p$ (a mass, $\text{M}$) does this: for example $h/(m c)$ is a length and $h/(mc^2)$ a time.
• $G$ ($\text{M}^{-1}\text{L}^3\text{T}^{-2}$) also does: $\sqrt{hc/G}$ is a mass, $\sqrt{hG/c^3}$ a length and $\sqrt{hG/c^5}$ a time (these are worked out in Question 40).
• $e$ does not help. Electric charge brings in a new dimension of its own, and the combination $e^2/(4\pi\varepsilon_0 hc)$ (the fine-structure constant) is a pure number, so $e$ with $h$ and $c$ fixes no new scale of mass, length or time.

Question 17

Which of the following ratios express pressure?

Solution. Pressure is force per unit area, $\text{ML}^{-1}\text{T}^{-2}$. Energy per volume is $\text{ML}^2\text{T}^{-2}/\text{L}^3 = \text{ML}^{-1}\text{T}^{-2}$ as well, which is why pressure can be read as an energy density. Energy per area is $\text{MT}^{-2}$ (the dimension of surface tension) and force per volume is $\text{ML}^{-2}\text{T}^{-2}$.

Question 18

Which of the following are not a unit of time?

Solution. The second and the year measure time. The parsec is the distance at which one astronomical unit subtends one second of arc, and the light year is the distance light travels in a year; both are units of length despite the words “sec” and “year” in their names.

Common mistakes

• Rounding a sum to significant figures in Question 2. For addition count decimal places: $663.8$, not $664$.
• Treating trailing zeros as non-significant in Question 1. After a decimal point they are measured digits and do count.
• Quoting the error more finely than the value in Questions 5 and 8. If the value is known to the units place, so is its error.
• Forgetting the power when propagating error in Question 8. A square root halves the relative error.
• Confusing precision with accuracy in Questions 9 and 10. Precision is the size of the smallest division; accuracy is closeness to the true value.
• Reading “sec” or “year” as a unit of time in Question 18. Parsec and light year are lengths.

Practise next

• How many significant figures are there in $0.003040$?

Show answer

$4$. The leading zeros only place the decimal point; $3, 0, 4, 0$ are significant, including the trailing zero after the decimal point.

• A quantity is given by $X = \dfrac{a^3b^2}{\sqrt c}$. The percentage errors in $a$, $b$ and $c$ are $1\%$, $2\%$ and $4\%$. Find the percentage error in $X$.

Show answer

$9\%$. Each relative error is multiplied by its power and the results add: $3(1) + 2(2) + \tfrac12(4) = 3 + 4 + 2 = 9\%$.

• Exemplar Units and Measurement: Very Short and Short Answer Questions — the same ideas applied to orders of magnitude, the vernier, systems of units and precision.
• Exemplar Units and Measurement: Long Answer Questions — dimensional analysis, error propagation and astronomical estimates worked in full.
• NCERT textbook Chapter 1 — Units and Measurement for the theory of SI units, significant figures, errors and dimensions.