Units and Measurements

Q1 FREE PREVIEW

PYQ

Given below are two statements :
Statement I: In a vernier callipers, one vernier scale division is always smaller than one main scale division.
Statement II: The vernier constant is given by one main scale division multiplied by the number of vernier scale divisions.
In the light of the above statements, choose the correct answer from the options given below.

a

Statement I is false but Statement II is true

b

Statement I is true but Statement II is false

c

Both Statement I and Statement II are false

d

Both Statement I and Statement II are true

✓ Correct answer: b)

Statement I is true but Statement II is false

Explanation

A vernier scale is designed such that its divisions are slightly smaller than the main scale divisions. This difference allows precise measurements by aligning a particular vernier division with the main scale.
The vernier constant (least count) is actually given by:
Least Count = Value of One Main Scale Division-Value of One Vernier Scale Division
Thus, the correct option is: 2.

Q2 FREE PREVIEW

PYQ

Find the dimensions of $\frac{B}{\mu_0}$ (Shift I Memory Based)

a

[AL]

b

$\left[{\mathrm{AL}}^{-1}\right]$

c

[MAL]

d

$\left[{\mathrm{MALT}}^{-1}\right]$

✓ Correct answer: b)

$\left[{\mathrm{AL}}^{-1}\right]$

Explanation

Magnetic field B :
The dimensional formula for B is derived from the Lorentz force law:

$F=q v B \quad \Rightarrow \quad B=\frac{F}{q v}$

1. Force (F) has the dimensional formula:

$[F]=\left[\mathrm{MLT}^{-2}\right]$

2. Charge (q) has the dimensional formula:

$[q]=[\mathrm{AT}]$

3. Velocity (v) has the dimensional formula:

$[v]=\left[\mathrm{LT}^{-1}\right]$

Thus, the dimensional formula for B is:

$[B]=\frac{\left[\mathrm{MLT}^{-2}\right]}{[\mathrm{AT}]\left[\mathrm{LT}^{-1}\right]}=\frac{[\mathrm{M}]\left[\mathrm{T}^{-2}\right]}{[\mathrm{A}]}=[\mathrm{M}]\left[\mathrm{A}^{-1}\right]\left[\mathrm{T}^{-2}\right]$

Magnetic permeability ( $\mu_0$ ):
The dimensional formula of $\mu_0$ is:

$\left[\mu_0\right]=[\mathrm{M}][\mathrm{L}]\left[\mathrm{T}^{-2}\right]\left[\mathrm{A}^{-2}\right]$

Dimensions of $\frac{B}{\mu_0}$ :
Using the formula:

$\frac{B}{\mu_0}=\frac{[\mathrm{M}]\left[\mathrm{A}^{-1}\right]\left[\mathrm{T}^{-2}\right]}{[\mathrm{M}][\mathrm{L}]\left[\mathrm{T}^{-2}\right]\left[\mathrm{A}^{-2}\right]}$

Cancel the terms:

$\frac{B}{\mu_0}=[\mathrm{A}]\left[\mathrm{L}^{-1}\right]$

Q3 FREE PREVIEW

PYQ

The displacement of a particle as a function of time is given by:$x(t)=A\sin ⁡(t)+B{\cos ⁡}^{2}(t)+C{t}^{2}+D$.

Find the dimension of $\frac{ABC}{D}$

(Shift I Memory Based)

a

$\left[{\mathrm{L}}^{2}\right]$

b

$\left[{\mathrm{L}}^{2}{\mathrm{T}}^{−2}\right]$

c

$\left[{\mathrm{LT}}^{−2}\right]$

d

$\left[{\mathrm{L}}^{3}\mathrm{T}\right]$

✓ Correct answer: b)

$\left[{\mathrm{L}}^{2}{\mathrm{T}}^{−2}\right]$

Explanation

The displacement $x(t)$ has the dimension $[L]$. Each term in the equation must also have the same dimension $[L]$:

$A\sin ⁡(t)$: $\sin ⁡(t)$ is dimensionless, so $A$ has $[L]$.
$B{\cos ⁡}^{2}(t)$: ${\cos ⁡}^{2}(t)$ is dimensionless, so $B$ has $[L]$.
$C{t}^{2}$: ${t}^{2}$ has dimensions $[{T}^{2}]$, so $C$ must have $[L{T}^{−2}]$.
$D$: As part of the displacement, $D$ has $[L]$.

Thus, the dimensions of $\frac{ABC}{D}$ are:

$\frac{[L]⋅[L]⋅[L{T}^{−2}]}{[L]}=[{L}^{2}{T}^{−2}]$

Therefore, the correct answer is $(b)$.

Q4 FREE PREVIEW

PYQ

What is the dimensional formula of $a{b}^{-1}$ in the equation $\left(P+\frac{a}{{V}^{2}}\right)(V-b)=RT$, where letters have their usual meaning.

[JEE Main 2024, 5 Apr (Shift 2)]

a

$\left[{M}^{0}{L}^{3}{T}^{-2}\right]$

b

$\left[M{L}^{2}{T}^{-2}\right]$

c

$\left[{M}^{-1}{L}^{5}{T}^{3}\right]$

d

$\left[{M}^{6}{L}^{7}{T}^{4}\right]$

✓ Correct answer: b)

$\left[M{L}^{2}{T}^{-2}\right]$

Explanation

$$\begin{aligned}&& {[P]=\left[\frac{a}{v^2}\right]} \\& \Rightarrow[a]=\left[P V^2\right] \\& {[a]=\left[M L^{-1} T^{-2} L^6\right]=\left[M L^5 T^{-2}\right]}$$
&\text { Now, }[v]=[b]
&$$\begin{aligned}& {[b]=\left[L^3\right]} \\& \therefore\left[a b^{-1}\right]=\left[M L^5 T^{-2} L^{-3}\right]=\left[M L^2 T^{-2}\right]\end{aligned}$$
\end{aligned}

Q5

PYQ

Find the correct dimensional formula for the capacitance in terms of M, L, T and C, where they stand for units of mass, length, time, and charge.

(Shift II Memory Based)

a

$[{M}^{−1}{L}^{−2}{T}^{4}{C}^{2}]$

b

$[{M}^{−1}{L}^{−2}{T}^{2}{C}^{2}]$

c

$[{M}^{−1}{L}^{−1}{T}^{3}C]$

d

$[{M}^{0}{L}^{1}{T}^{−1}{C}^{−1}]$

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Q6

PYQ

Match List - I with List - II.

List - I	List - II
(A) Permeability of free space	(I) $\left[{\mathrm{ML}}^{2}{\mathrm{T}}^{-2}\right]$
(B) Magnetic field	(II) $\left[{\mathrm{MT}}^{-2}{\mathrm{A}}^{-1}\right]$
(C) Magnetic moment	(III) $\left[{\mathrm{MLT}}^{-2}{\mathrm{A}}^{-2}\right]$
(D) Torsional constant	(IV) $\left[{\mathrm{L}}^{2}\mathrm{A}\right]$

Choose the correct answer from the options given below :

[JEE Main 2025, 23 Jan (Shift 2)]

a

(A)-(IV), (B)-(III), (C)-(I), (D)-(II)

b

(A)-(III), (B)-(II), (C)-(IV), (D)-(I)

c

(A)-(II), (B)-(I), (C)-(III), (D)-(IV)

d

(A)-(I), (B)-(IV), (C)-(II), (D)-(III)

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Q7

PYQ

If $\epsilon_{ o }$ is the permittivity of free space and $E$ is the electric field, then $\epsilon_{ o } E ^2$ has the dimensions :

[JEE Main 2024, 8 Apr (Shift 2)]

a

$\left[ M L ^{-1} T ^{-2}\right]$

b

$\left[ M L ^2 T ^{-2}\right]$

c

$\left[ M ^{\circ} L ^{-2} T A \right]$

d

$\left[ M ^{-1} L ^{-3} T ^4 A ^2\right]$

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Q8

PYQ

The velocity of a particle moving along a straight line varies with time as:

$v=A{t}^{2}+\frac{Bt}{C+t}$

where A, B, and C are constants. Determine the dimensional formula of ABC.

(Shift - I Memory Based)

a

${L}^{2}{T}^{−2}$

b

${L}^{2}{T}^{−1}$

c

${L}^{2}{T}^{−3}$

d

${L}^{2}{T}^{−4}$

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Q9

PYQ

Time periods of oscillation of the same simple pendulum measured using four different measuring clocks were recorded as 4.62s , 4.632s , 4.6s and 4.64s. The arithmetic mean of these readings in correct significant figure is :

a

4.623s

b

4.62s

c

4.6s

d

5s

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Q10

PYQ

The measurements for a wire are given as:

Mass, $m=(0.60\pm 0.003)\text{ }\text{g}$,
Radius, $r=(0.50\pm 0.01)\text{ }\text{cm}$,
Length, $l=(10.00\pm 0.05)\text{ }\text{cm}$.

Find the maximum percentage error in the measurement of the density of the wire.

(Shift II Memory Based)

a

2%

b

3%

c

4%

d

5%

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Q11

PYQ

The expression given below shows the variation of velocity (v) with time (t), $v=A{t}^{2}+\frac{Bt}{C+t}$. The dimension of ABC is :

[JEE Main 2025, 29 Jan (Shift 1)]

a

$\left[{M}^{0}{L}^{2}{T}^{-2}\right]$

b

$\left[{M}^{0}{L}^{1}{T}^{-3}\right]$

c

$\left[{M}^{0}{L}^{2}{T}^{-3}\right]$

d

$\left[{M}^{0}{L}^{1}{T}^{-2}\right]$

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Q12

PYQ

10 divisions on the main scale of a Vernier calliper coincide with 11 divisions on the Vernier scale. If each division on the main scale is of 5 units, the least count of the instrument is :

[JEE Main 2024, 1 Feb (Shift 1)]

a

$\frac{1}{2}$

b

$\frac{5}{11}$

c

$\frac{50}{11}$

d

$\frac{10}{11}$

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Q13

PYQ

Which of the following pairs of physical quantities have the same dimensions?(Shift - I Memory Based)

a

Angular momentum and Planck's constant

b

Torque and moment of inertia

c

Impulse and surface tension

d

Momentum and work done

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Q14

PYQ

A physical quantity $Q$ is given by:$Q=\frac{a{b}^{4}}{cd}$.If the percentage errors in $a,b,c$ and $d$ are 2%, 1%, 2%, and 1% respectively, determine the percentage error in $Q$.

(Shift - II Memory Based)

a

2%

b

5%

c

9%

d

6%

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Q15

PYQ

A temperature difference can generate e.m.f. in some materials. Let $S$ be the e.m.f. produced per unit temperature difference between the ends of a wire, $\sigma$ the electrical conductivity and $\kappa$ the thermal conductivity of the material of the wire. Taking $M, L, T, I$ and $K$ as dimensions of mass, length, time, current and temperature, respectively, the dimensional formula of the quantity $Z=\frac{S^2 \sigma}{\kappa}$ is:

[JEE Advanced 2025]

a

$\left[{M}^{0}{L}^{0}{T}^{0}{I}^{0}{K}^{0}\right]$

b

$\left[{M}^{0}{L}^{0}{T}^{0}{I}^{0}{K}^{-1}\right]$

c

$\left[{M}^{1}{L}^{2}{T}^{-2}{I}^{-1}{K}^{-1}\right]$

d

$\left[{M}^{1}{L}^{2}{T}^{-4}{I}^{-1}{K}^{-1}\right]$

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Q16

PYQ

Match List - I with List - II.

List - I List - II
(A) Young's Modulus (I) $M{L}^{-1}{T}^{-1}$
(B) Torque (II) $M{L}^{-1}{T}^{-2}$
(C) Coefficient of Viscosity (III) ${M}^{-1}{L}^{3}{T}^{-2}$
(D) Gravitational Constant (IV) $M{L}^{2}{T}^{-2}$

Choose the correct answer from the options given below :

[JEE Main 2025, 29 Jan (Shift 2)]

a

(A)-(II), (B)-(IV), (C)-(I), (D)-(III)

b

(A)-(II), (B)-(I), (C)-(IV), (D)-(III)

c

(A)-(IV), (B)-(II), (C)-(III), (D)-(I)

d

(A)-(I), (B)-(III), (C)-(II), (D)-(IV)

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Q17

PYQ

The pair of physical quantities not having same dimensions is :

[JEE Main 2025, 29 Jan (Shift 1)]

a

Pressure and Young's modulus

b

Surface tension and impulse

c

Angular momentum and Planck's constant

d

Torque and energy

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Q18

PYQ

Find the dimensions of $\frac{B}{\mu_0}$ (Shift I Memory Based)

a

[AL]

b

$\left[{\mathrm{AL}}^{-1}\right]$

c

[MAL]

d

$\left[{\mathrm{MALT}}^{-1}\right]$

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Q19

PYQ

The maximum percentage error in the measurment of density of a wire is
[Given, mass of wire $=(0.60\pm 0.003)\mathrm{g}$
radius of wire $=(0.50\pm 0.01)\mathrm{cm}$
length of wire $=(10.00\pm 0.05)\mathrm{cm}]$

[JEE Main 2025, 22 Jan (Shift 2)]

a

5

b

8

c

4

d

7

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Q20

PYQ

Match List - I with List - II.

List - I List - II
(A) Young's Modulus (I) $M{L}^{-1}{T}^{-1}$
(B) Torque (II) $M{L}^{-1}{T}^{-2}$
(C) Coefficient of Viscosity (III) ${M}^{-1}{L}^{3}{T}^{-2}$
(D) Gravitational Constant (IV) $M{L}^{2}{T}^{-2}$

Choose the correct answer from the options given below :

[JEE Main 2025, 29 Jan (Shift 2)]

a

(A)-(II), (B)-(IV), (C)-(I), (D)-(III)

b

(A)-(II), (B)-(I), (C)-(IV), (D)-(III)

c

(A)-(IV), (B)-(II), (C)-(III), (D)-(I)

d

(A)-(I), (B)-(III), (C)-(II), (D)-(IV)

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Q21

PYQ

The energy in a system varies with position and time as E(x, t) = ${x}^{3}$ ${e}^{–\beta t}$, where β = 0.3 $se{c}^{–1}$. Given that the P% error in x = 1.2 % and that the % error in t = 1.6%. Find the maximum % error in E at t = 5 sec.

(Shift - II Memory Based)

a

1%

b

2%

c

3%

d

6%

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Q22

PYQ

If 50 Vernier divisions are equal to 49 main scale divisions of a travelling microscope and one smallest reading of main scale is 0.5 mm, the Vernier constant of travelling microscope is:

a

0.1 mm

b

0.1 cm

c

0.01 cm

d

0.01 mm

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Q23

PYQ

For an experimental expression $y=\frac{32.3\times 1125}{27.4}$, where all the digits are significant. Then to report the value of y we should write

a

y = 1326.19

b

y = 1326.2

c

y = 1326.186

d

y = 1330

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Q24

PYQ

If $\epsilon_{ o }$ is the permittivity of free space and $E$ is the electric field, then $\epsilon_{ o } E ^2$ has the dimensions :

[JEE Main 2024, 8 Apr (Shift 2)]

a

$\left[ M L ^{-1} T ^{-2}\right]$

b

$\left[ M L ^2 T ^{-2}\right]$

c

$\left[ M ^{\circ} L ^{-2} T A \right]$

d

$\left[ M ^{-1} L ^{-3} T ^4 A ^2\right]$

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Q25

PYQ

Find the dimensions of $\frac{B}{\mu_0}$ (Shift I Memory Based)

a

[AL]

b

$\left[{\mathrm{AL}}^{-1}\right]$

c

[MAL]

d

$\left[{\mathrm{MALT}}^{-1}\right]$

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Q26

PYQ

If $B$ is magnetic field and ${\mu }_{0}$ is permeability of free space, then the dimensions of $\left(B/{\mu }_{0}\right)$ is

[JEE Main 2025, 22 Jan (Shift 1)]

a

${\mathrm{ML}}^{2}{\mathrm{T}}^{-2}{\mathrm{A}}^{-1}$

b

${\mathrm{MT}}^{-2}{\mathrm{A}}^{-1}$

c

${\mathrm{LT}}^{-2}{\mathrm{A}}^{-1}$

d

${\mathrm{L}}^{-1}\mathrm{A}$

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Q27

PYQ

Electric flux $ϕ$ is related to linear charge density $\lambda$and surface charge density $\sigma$ as $ϕ=\alpha \lambda +\beta \sigma$, where $\alpha$ and $\beta$ are constants. The dimensions of $(\beta \mathrm{/}\alpha )$ are:

(Shift I Memory Based)

a

Displacement

b

Area

c

Electric field

d

Velocity

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Q28

PYQ

The position of a particle moving on x -axis is given by $\mathrm{x}(\mathrm{t})=\mathrm{Asint}+{\mathrm{Bcos}}^{2}\mathrm{t}+{\mathrm{Ct}}^{2}+\mathrm{D}$, where $t$ is time. The dimension of $\frac{ABC}{D}$ is

[JEE Main 2025, 23 Jan (Shift 1)]

a

${{}^{\text{ }}\mathrm{L}}^{3}{\mathrm{T}}^{-2}$

b

${L}^{2}$

c

${\mathrm{L}}^{2}{\mathrm{T}}^{-2}$

d

$L$

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Q29

PYQ

The position of a particle moving on x -axis is given by $\mathrm{x}(\mathrm{t})=\mathrm{Asint}+{\mathrm{Bcos}}^{2}\mathrm{t}+{\mathrm{Ct}}^{2}+\mathrm{D}$, where $t$ is time. The dimension of $\frac{ABC}{D}$ is

[JEE Main 2025, 23 Jan (Shift 1)]

a

${{}^{\text{ }}\mathrm{L}}^{3}{\mathrm{T}}^{-2}$

b

${L}^{2}$

c

${\mathrm{L}}^{2}{\mathrm{T}}^{-2}$

d

$L$

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Q30

PYQ

While measuring diameter of a wire using a screw gauge the following readings were noted Main scale reading is 1 mm and circular scale reading is equal to 42 division Pitch of screw gauge is 1 mm and it has 100 divisions on circular scale The diameter of wire is x/50 mm The value of x is

a

142

b

71

c

42

d

21

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Q31

PYQ

A physical quantity $Q$ is found to depend on quantities $a,b,c$ by the relation $Q=\frac{{a}^{4}{b}^{3}}{{c}^{2}}$. The percentage error in $a,b$ and $c$ are $3\%,4\%$ and $5\%$respectively. Then, the percentage error in $Q$ is :

[JEE Main 2024, 29 Jan (Shift 2)]

a

34%

b

43%

c

14%

d

66%

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Q32

PYQ

Consider two physical quantities A and B related to each other as $E=\frac{B−{x}^{2}}{At}$
= where E, x and t have dimensions of energy, length and time respectively. The dimension of AB is

a

${L}^{–2}{M}^{1}{T}^{0}$

b

${L}^{2}{M}^{–1}{T}^{1}$

c

${L}^{–2}{M}^{–1}{T}^{1}$

d

${L}^{0}{M}^{–1}{T}^{1}$

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Q33

PYQ

In a vernier calipers, (N+1) divisions of vernier scale coincide with N divisions of main scale. If 1 MSD represents 0.1 mm, the vernier constant (in cm) is :

a

$\frac{1}{10\mathrm{N}}$

b

$\frac{1}{100(\mathrm{N}+1)}$

c

100N

d

10(N+1)

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Q34

PYQ

Match List - I with List - II.

List - I	List - II
(A) Permeability of free space	(I) $\left[{\mathrm{ML}}^{2}{\mathrm{T}}^{-2}\right]$
(B) Magnetic field	(II) $\left[{\mathrm{MT}}^{-2}{\mathrm{A}}^{-1}\right]$
(C) Magnetic moment	(III) $\left[{\mathrm{MLT}}^{-2}{\mathrm{A}}^{-2}\right]$
(D) Torsional constant	(IV) $\left[{\mathrm{L}}^{2}\mathrm{A}\right]$

Choose the correct answer from the options given below :

[JEE Main 2025, 23 Jan (Shift 2)]

a

(A)-(IV), (B)-(III), (C)-(I), (D)-(II)

b

(A)-(III), (B)-(II), (C)-(IV), (D)-(I)

c

(A)-(II), (B)-(I), (C)-(III), (D)-(IV)

d

(A)-(I), (B)-(IV), (C)-(II), (D)-(III)

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Q35

PYQ

The pitch of an error free screw gauge is $1\mathrm{mm}$ and there are 100 divisions on the circular scale. While measuring the diameter of a thick wire, the pitch scale reads $1\mathrm{mm}$ and ${63}^{\text{rd }}$ division on the circular scale coincides with the reference line. The diameter of the wire is:

[Re-NEET 2024]

a

$1.63\mathrm{cm}$

b

$0.163\mathrm{cm}$

c

$0.163\mathrm{m}$

d

$1.63\mathrm{m}$

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Q36

PYQ

The energy of a system is given as $\mathrm{E}(\mathrm{t})={\alpha }^{3}{e}^{-\beta t}$, where $t$ is the time and $\beta =0.3{\mathrm{s}}^{-1}$. The errors in the measurement of $\alpha$ and $t$ are$1.2\%$ and $1.6\%$, respectively. At $t=5\mathrm{s}$, maximum percentage error in the energy is :

[JEE Main 2025, 23 Jan (Shift 2)]

a

$6\%$

b

$11.6\%$

c

$8.4\%$

d

$4\%$

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Q37

PYQ

In an experiment to measure focal length (f) of convex lens, the least counts of the measuring scales for the position of object (u) and for the position of image (v) are $∆$u and $∆$v, respectively. The error in the measurement of the focal length of the convex lens will be:

a

$\frac{∆u}{u}+\frac{∆v}{v}$

b

${f}^{2}\left[\frac{∆u}{{u}^{2}}+\frac{∆v}{{v}^{2}}\right]$

c

$2f\left[\frac{∆u}{u}+\frac{∆v}{v}\right]$

d

$f\left[\frac{∆u}{u}+\frac{∆v}{n}\right]$

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Q38

PYQ

Which one of the following is the correct dimensional formula for the capacitance (F) in terms of M, L, T and $C$ stand for unit of mass, length, time and charge,

[JEE Main 2025, 22 Jan (Shift 2)]

a

$[\mathrm{F}]=\left[{\mathrm{C}}^{2}{\mathrm{M}}^{-1}{\mathrm{L}}^{-2}{\mathrm{T}}^{2}\right]$

b

$[\mathrm{F}]=\left[{\mathrm{CM}}^{-2}{\mathrm{L}}^{-2}{\mathrm{T}}^{-2}\right]$

c

$[\mathrm{F}]=\left[{\mathrm{CM}}^{-1}{\mathrm{L}}^{-2}{\mathrm{T}}^{2}\right]$

d

$[\mathrm{F}]=\left[{\mathrm{C}}^{2}{\mathrm{M}}^{-2}{\mathrm{L}}^{2}{\mathrm{T}}^{2}\right]$

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Q39

PYQ

A physical quantity $Q$ is given by:$Q=\frac{a{b}^{4}}{cd}$.If the percentage errors in $a,b,c$ and $d$ are 2%, 1%, 2%, and 1% respectively, determine the percentage error in $Q$.

a

2%

b

5%

c

9%

d

6%

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Q40

PYQ

If force $ (\mathrm{F}) $, velocity (V) and time $ (\mathrm{T}) $ are taken as fundamental units, then the dimensions of mass are

a

$ \left[\mathrm{F} \mathrm{V} \mathrm{T}^{-1}\right] $

b

$ \left[\mathrm{F} \mathrm{V} \mathrm{T}^{-2}\right] $

c

$ \left[\mathrm{F} \mathrm{V}^{-1} \mathrm{~T}^{-1}\right] $

d

$ \left[\mathrm{F} \mathrm{V}^{-1} \mathrm{~T}\right] $

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Q41

PYQ

If force (F), length (L) and time (T) are taken as the fundamental quantities, then what will be the dimension of density?

[JEE Main 2021, 27 Aug (Shift 2)]

a

$\left[F{L}^{-3}{T}^{2}\right]$

b

$\left[F{L}^{-5}{T}^{2}\right]$

c

$\left[F{L}^{-4}{T}^{2}\right]$

d

$\left[F{L}^{-3}{T}^{3}\right]$

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