Units and Measurements
166 JEE Physics previous year questions on Units and Measurements — options free on every question; 17 include the answer & explanation free, the rest unlock with PYQ Pass.
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)]
\(6\%\)
\(E(t)={\alpha }^{3}{e}^{-\beta t}\\ dE(t)=3{\alpha }^{2}{e}^{-\beta t}d\alpha -\beta {\alpha }^{3}{e}^{-\beta t}dt\\ \frac{dE(t)}{E(t)}=\frac{3{\alpha }^{2}{e}^{-\beta t}d\alpha }{E(t)}+\frac{\beta {\alpha }^{3}{e}^{-\beta t}dt}{E(t)}\\ \frac{\Delta E}{E}=\frac{3\Delta \alpha }{\alpha }+\beta \Delta t\\ \frac{\Delta t}{t}\times 100=1.6\\ \frac{\Delta t}{5}=0.016\Rightarrow \Delta t=0.08\)
\(\frac{\Delta E}{E}=3\times 0.012+0.3\times 0.08\\ \frac{\Delta E}{E}=0.036+0.024\Rightarrow \frac{\Delta E}{E}=0.06\\ \%E=6\%\)
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)]
\({\mathrm{L}}^{-1}\mathrm{A}\)
Magnetic field in long solenoid
\(\left[\frac{B}{{\mu }_{0}}\right]=[ni]={L}^{-1}{A}^{1}\). (n=number of turns per unit length)
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)
9%
We are given the equation:
\(Q=\frac{a{b}^{4}}{cd}\)
To determine the percentage error \(Q\), we use the error propagation rule:
\(\frac{\Delta Q}{Q}=\frac{\Delta a}{a}+4\frac{\Delta b}{b}+\frac{\Delta c}{c}+\frac{\Delta d}{d}\)
Step 1: Substituting Given Errors- Percentage error in a = 2%
- Percentage error in b = 1%
- Percentage error in c = 2%
- Percentage error in d = 1%
Now, substituting these values into the equation:
\(\text{Percentage error in }Q=2+4(1)+2+1\) \(=2+4+2+1=9\mathrm{\%}\)
Final Answer:\(C\ 9\mathrm{\%}\)C
The potential energy of a particle changes with distance \(x\) from a fixed origin as \(V=\frac{A\sqrt{x}}{x+B}\), where \(A\) and \(B\) are constant with appropriate dimensions. The dimensions of \(AB\) are_____ .
[JEE Main 2026, 6 Apr (Shift 1)]
\(\left[{M}^{1}{L}^{7/2}{T}^{-2}\right]\)
(d) \(B\) must have the same dimensions as \(x:[B]=[L]\)
The formula for potential energy is \(V=\frac{A\sqrt{x}}{x+B}\).
Rearranging for \(A\) :
\(A=\frac{V(x+B)}{\sqrt{x}}\\\)
\([A]=\frac{\left[{M}^{1}{L}^{2}{T}^{-2}\right][L]}{\left[{L}^{1/2}\right]}=\frac{\left[{M}^{1}{L}^{3}{T}^{-2}\right]}{\left[{L}^{1/2}\right]}\\\)
\(=\left[{M}^{1}{L}^{5/2}{T}^{-2}\right]\)
Dimensions for AB ;
\([AB]=\left[{M}^{1}{L}^{5/2}{T}^{-2}\right]\times \left[{L}^{1}\right]\\\)
\([AB]=\left[{M}^{1}{L}^{7/2}{T}^{-2}\right]\)
The dimension of \(\sqrt{\frac{{\mu }_{0}}{{ϵ}_{0}}}\) is equal to that of :
( \({\mu }_{0}=\) Vacuum permeability and \({ϵ}_{0}=\)Vacuum permittivity)
[JEE Main 2025, 7 Apr (Shift 2)]
Resistance
$$$$$$\begin{aligned}& L=\frac{\mu_0 N A}{\ell} \\& C=\frac{A \epsilon_0}{d} \\& \frac{L}{C} \propto \frac{\mu_0}{\epsilon_0} \\& \sqrt{\frac{\mu_0}{\epsilon_0}} \propto \sqrt{\frac{L}{C}} \\& \frac{L}{C}=\frac{\tau R}{(\tau / R)}=R^2 \\& \sqrt{\frac{\mu_0}{\epsilon_0}}=R\end{aligned}$$$$$$
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)]
\({\mathrm{L}}^{2}{\mathrm{T}}^{-2}\)
\(x(t)=A\sin (t)+{Bcos}^{2}(t)+C{t}^{2}+D\)
Here, A, B, C, and D are constants, and t is time.
\([A]=L,[B]=L,[C]=\frac{L}{{T}^{2}},[D]=L\\ \left[\frac{ABC}{D}\right]=\left[\frac{L\times L\times L{T}^{-2}}{L}\right]=\left[{L}^{2}{T}^{-2}\right]\)
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)-(III), (B)-(II), (C)-(IV), (D)-(I)
\(\tau =C\theta \\ c=\left[M{L}^{2}{T}^{-2}\right]\)
\(M=NIA\\ M=\left[{L}^{2}A\right]\)
\(\text{ }F=qVB\\ \left[ML{T}^{-2}\right]=[AT]\left[L{T}^{-1}\right]B\\ B=\left[M{T}^{-2}{A}^{-1}\right]\)
\(B=\frac{{\mu }_{0}i}{2r}\\ {\mu }_{0}=\frac{\left[M{T}^{-2}{A}^{-1}\right][L]}{[A]}\)
\({\mu }_{0}=\left[ML{T}^{-2}{A}^{-2}\right]\)
Dimensions of universal gravitational constant (G) in terms of Planck's constant (h), distance (L), mass (M) and time (T) are
[02 April, 2026 (Shift-2)]
\(\left[h{T}^{-1}L{M}^{-2}\right]\)
Energy of a photon: \(E=\frac{hc}{\lambda }\)
Gravitational potential energy: \(E=\frac{G{M}_{1}{M}_{2}}{r}\)
\(\left[M{L}^{2}{T}^{-2}\right]=[h]\left[{T}^{-1}\right]=[G]\left[{M}^{2}{L}^{-1}\right]\) \([G]=\left[{M}^{-2}L{T}^{-1}h\right]\)
The time period of a simple harmonic oscillator is \(T=2\pi \sqrt{\frac{k}{m}}.\) Measured value of mass (m) of the object is 10 g with an accuracy of 10 mg and time for 50 oscillations of the spring is found to be 60 s using a watch of 2s resolution. Percentage error in determination of spring constant (k) is _____ %.
[JEE Main 2026, 28 Jan (Shift 2)]
6.76
\(\frac{\Delta K}{K}=\frac{2\Delta T}{T}+\frac{\Delta m}{m}\)
\(T=\frac{60}{50}=1.2\sec\)
\(\Delta T=\frac{2}{50}\)
\(∴\text{ }\frac{\Delta K}{K}=\frac{2\times 2}{50\times 1.2}+\frac{10\times {10}^{−3}}{10}=0.0676\)
\(∴\) %Error = 6.76%
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)-(II), (B)-(IV), (C)-(I), (D)-(III)
\(\text{ (A) }[Y]=\frac{F}{A\left(\frac{\Delta ℓ}{ℓ}\right)}=M{L}^{-1}{T}^{-2}\left(\frac{\Delta ℓ}{ℓ}\right.=\text{ Dimensionless })\)
(B) Torque
\((\vec{\tau })=\vec{r}\times \vec{F}=L\times ML{T}^{-2}=M{L}^{2}{T}^{-2}\)
(C) Coefficient of viscosity \(\Rightarrow \frac{F}{A\cdot \frac{dv}{dt}}\)
\(\eta \to Pa\text{ sec. }\\ [\eta ]=\frac{ML{T}^{-2}}{{L}^{2}}\times T=M{L}^{-1}{T}^{-1}\)
\(\text{ (D) Gravitational constant ( }G\text{ ) }\\ [G]=\frac{F\cdot {r}^{2}}{{m}_{1}{m}_{2}}=\frac{ML{T}^{-2}\times {L}^{2}}{{M}^{2}}={M}^{-1}{L}^{3}{T}^{-2}\)
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)]
\(\frac{5}{11}\)
$$\begin{aligned}& \text { Given: } 10 \mathrm{MSD}=11 \mathrm{VSD} \\& \Rightarrow 1 \mathrm{VSD}=\frac{10}{11} \mathrm{MSD} \\& \text { As, } \mathrm{LC}=1 \mathrm{MSD}-1 \mathrm{VSD} \\& =1 \mathrm{MSD}-\frac{10}{11} \mathrm{MSD}=\frac{1 \mathrm{MSD}}{11} \\& =\frac{5}{11} \text { units }\end{aligned}$$
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:
0.01 mm
$$\begin{aligned}& \text { (d) } 50 \mathrm{~V}+\mathrm{S}=49 \mathrm{~S}+\mathrm{S} \\& \mathrm{~S}=50(\mathrm{~S}-\mathrm{V}) \\& 0.5=50(\mathrm{~S}-\mathrm{V})\end{aligned}$$
The Vernier constant of travelling microscope is \(S-V=\frac{0.5}{50}=\frac{1}{100}=0.01mm\)
If \(G\) be the gravitational constant and u be the energy density then which of the following quantity have the dimensions as that of the \(\sqrt{uG}\) :
Force per unit mass
Dimension of energy density:
\([u]=\frac{M{L}^{2}{T}^{-2}}{{L}^{3}}\) \([u]=M{L}^{-1}{T}^{-2}\)Dimension of gravitational constant:
\([G]={M}^{-1}{L}^{3}{T}^{-2}\)Multiply the dimensions:
\([\mathrm{uG}]={M}^{0}{L}^{2}{T}^{-4}\)Take square root:
\([\sqrt{\mathrm{uG}}]=L{T}^{-2}\)Dimension of force per unit mass:
\([\frac{\text{Force}}{\text{Mass}}]=ML{T}^{-2}M\) \([\frac{\text{Force}}{\text{Mass}}]=L{T}^{-2}\)Hence, the required quantity is force per unit mass.
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)]
\(\left[ M L ^{-1} T ^{-2}\right]\)
$$\begin{aligned}& \text { } E=\frac{Q}{4 \pi \varepsilon_0 R^2} \Rightarrow \varepsilon_0=\frac{Q}{4 \pi R^2 E} \\& \text { Now, } \varepsilon_0 E^2=\frac{Q}{4 \pi R^2 E} \cdot E^2=\frac{Q}{4 \pi R^2} \cdot E \\& {\left[\varepsilon_0 E^2\right]=\left[\frac{Q E}{R^2}\right]=\frac{[Q][E]}{\left[R^2\right]}=\frac{[Q]}{\left[R^2\right][Q][R]}[W]} \\& =\frac{[W]}{\left[R^3\right]}=\frac{M L^2 T^{-2}}{L^3}=M L^{-1} T^{-2}\end{aligned}$$
Which one of the following is not a measurable quantity?
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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)
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In an electromagnetic system, a quantity defined as the ratio of electric dipole moment and magnetic dipole moment has dimension of \([{M}^{P}{L}^{Q}{T}^{R}{A}^{S}]\). The value of P and Q are :
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Find the dimensions of \(\frac{B}{\mu_0}\) (Shift I Memory Based)
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The pair of physical quantities not having same dimensions is :
[JEE Main 2025, 29 Jan (Shift 1)]
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The diameter of a wire measured by a screw gauge of least count 0.001 cm is 0.08 cm . The length measured by a scale of least count 0.1 cm is 150 cm . When a weight of 100 N is applied to the wire, the extension in length is 0.5 cm , measured by a micrometer of least count 0.001 cm . The error in the measured Young's modulus is \(\alpha \times {10}^{9}N/{m}^{2}\). The value of \(\alpha\) is ____________ (Ignore the contribution of the load to Young's modulus error calculation)
[JEE Main 2026, 2 Apr (Shift 1)]
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The dimensional formula of \(\frac{1}{2}{\epsilon }_{0}{E}^{2}\left({\epsilon }_{0}=\right.\) permittivity of vacuum and E = electric field) is \({M}^{a}{L}^{b}{T}^{c}.\) The value of 2 a - b + c =_________
[JEE Main 2026, 2 Apr (Shift 1)]
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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)]
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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)
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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
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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)
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For the determination of refractive index of glass slab, a travelling microscope is used whose main scale contains 300 equal divisions equals to 15 cm. The vernier scale attached to the microscope has 25 divisions equals to 24 divisions of main scale. The least count (LC) of the travelling microscope is (in cm) :
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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)
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A new unit \((\alpha )\) of length is chosen such that it is equal to the speed of light in vacuum. What is the distance between Venus and Earth in terms of \(\alpha\) units if light takes 6 min 40 s to cover this distance?
[JEE Main 2026, 8 Apr (Shift 2)]
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Which of the following pairs of physical quantities have the same dimensions?(Shift - I Memory Based)
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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)]
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Consider two physical quantities \(A\) and \(B\) related to each other as \(E=\frac{B-x^2}{A t}\) where \(E, x\) and \(t\) have dimensions of energy, length and time respectively. The dimension of \(A B\) is:
[JEE Main 2024, 31 Jan (Shift 2)]
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Consider two physical quantities \(A\) and \(B\) related to each other as \(E=\frac{B-x^2}{A t}\) where \(E, x\) and \(t\) have dimensions of energy, length and time respectively. The dimension of \(A B\) is:
[JEE Main 2024, 31 Jan (Shift 2)]
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The maximum percentage error in the measurement 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)]
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In finding out refractive index of glass slab the following observations were made through travelling microscope 50 vernier scale division \(=49\) MSD; 20 divisions on main scale in each \(cm\)
For mark on paper
\(MSR =8.45 cm , VC =26\)
For mark on paper seen through slab
\(MSR =7.12 cm , VC =41\)
For powder particle on the top surface of the glass slab
\(\text { MSR }=4.05 cm , VC =1\)
(MSR \(=\) Main Scale Reading, \(VC =\) Vernier Coincidence \()\)
Refractive index of the glass slab is :
[JEE Main 2024, 06 Apr (Shift 2)]
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In finding out refractive index of glass slab the following observations were made through travelling microscope 50 vernier scale division \(=49\) MSD; 20 divisions on main scale in each \(cm\)
For mark on paper
\(MSR =8.45 cm , VC =26\)
For mark on paper seen through slab
\(MSR =7.12 cm , VC =41\)
For powder particle on the top surface of the glass slab
\(\text { MSR }=4.05 cm , VC =1\)
(MSR \(=\) Main Scale Reading, \(VC =\) Vernier Coincidence \()\)
Refractive index of the glass slab is :
[JEE Main 2024, 06 Apr (Shift 2)]
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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)]
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The time period of a simple harmonic oscillator is \(T=2\pi \sqrt{\frac{k}{m}}.\) Measured value of mass (m) of the object is 10 g with an accuracy of 10 mg and time for 50 oscillations of the spring is found to be 60 s using a watch of 2s resolution. Percentage error in determination of spring constant (k) is _____ %.
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Which of the following pairs of physical quantities have the same dimensions?(Shift - I Memory Based)
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To compare EMF of two cells using potentiometer the balancing lengths obtained are 200 cm and 150 cm. The least count of scale is 1 cm. The percentage error in the ratio of EMFs is ______
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A vernier callipers has 20 divisions on the vernier scale, which coincides with \({19}^{th}\) division on the main scale. The least count of the instrument is \(0.1mm\) . One main scale division is equal to____ mm.
[JEE Main 2024, 5 Apr (Shift 2)]
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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)
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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)
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In a screw gauge, the zero of the circular scale lies 3 divisions above the horizontal pitch line when their metallic studs are brought in contact. Using this instrument thickness of a sheet is measured. If pitch scale reading is 1 mm and the circular scale reading is 51 then the correct thickness of the sheet is_____ mm. [Assume least count is 0.01 mm]
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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.
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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:
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Four persons measure the length of a rod as 20.00 cm, 19.75 cm, 17.01 cm and 18.25 cm. The relative error in the measurement of average length of the rod is:
[JEE Main 2026, 23 Jan (Shift 1)]
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Identify the physical quantity that cannot be measured using spherometer :
[JEE Main 2024, 27 Jan (Shift 1)]
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Identify the physical quantity that cannot be measured using spherometer :
[JEE Main 2024, 27 Jan (Shift 1)]
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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:
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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)
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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
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In an electromagnetic system, the quantity representing the ratio of electric flux and magnetic flux has dimension of \({\mathrm{M}}^{\mathrm{P}}{\mathrm{L}}^{\mathrm{Q}}{\mathrm{T}}^{\mathrm{R}}{\mathrm{A}}^{\mathrm{S}}\), where value of 'Q' and 'R' are
[JEE Main 2025, 4 Apr (Shift 1)]
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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)]
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Match List - I with List - II.
|
| List - I (Number) | List - II (Significant figure) | |
| (A) | 1001 | (I) | 3 |
| (B) | 010.1 | (II) | 4 |
| (C) | 100.100 | (III) | 5 |
| (D) | 0.0010010 | (IV) | 6 |
Choose the correct answer from the options given below :
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Match List - I with List - II.
|
| List - I (Number) | List - II (Significant figure) | |
| (A) | 1001 | (I) | 3 |
| (B) | 010.1 | (II) | 4 |
| (C) | 100.100 | (III) | 5 |
| (D) | 0.0010010 | (IV) | 6 |
Choose the correct answer from the options given below :
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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)
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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
[JEE Main 2025, 24 Jan (Shift 1)]
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In a screw gauge the zero of main scale reference line coincides with the fifth division of the circular scale when two studs are in contact. There are 100 divisions in circular scale and pitch of screw gauge is 0.1 mm . When diameter of a sphere is measured, the reading of main scale is 5 mum and \({50}^{\text{th }}\) division of circular scale coincides with the reference line of main scale. The diameter of sphere is___________ mm.
[02 April, 2026 (Shift-2)]
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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)
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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.
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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)]
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If \(\epsilon_{ o }\) is the permittivity of free space and \(E\) is the electric field, then \(\epsilon_{ o } E ^2\) has the dimensions :
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In a vernier callipers, 50 vernier scale divisions are equal to 48 main scale divisions. If one main scale division = 0.05 mm, then the least count of the vernier callipers is ____ mm.
[JEE Main 2026, 24 Jan (Shift 2)]
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In a vernier callipers, 50 vernier scale divisions are equal to 48 main scale divisions. If one main scale division = 0.05 mm, then the least count of the vernier callipers is ____ mm.
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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)]
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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]
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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)]
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In a vernier calliper, when both jaws touch each other, zero of the vernier scale shifts towards left and its \(4^{\text {th }}\) division coincides exactly with a certain division on main scale. If 50 vernier scale divisions equal to 49 main scale divisions and zero error in the instrument is \(0.04 mm\) then how many main scale divisions are there in \(1 cm\) ?
[JEE Main 2024, 06 Apr (Shift 2)]
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In a vernier calliper, when both jaws touch each other, zero of the vernier scale shifts towards left and its \(4^{\text {th }}\) division coincides exactly with a certain division on main scale. If 50 vernier scale divisions equal to 49 main scale divisions and zero error in the instrument is \(0.04 mm\) then how many main scale divisions are there in \(1 cm\) ?
[JEE Main 2024, 06 Apr (Shift 2)]
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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.
[JEE Main 2025, 22 Jan (Shift 1)]
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Which one of the following is not a measurable quantity?
[JEE Main 2026, 28 Jan (Shift 2)]
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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 :
[JEE Main 2024, 5 Apr (Shift 1)]
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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
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The percentage error in the calculated volume of a sphere, if there is 2% error in its diameter measurement, is ______.
[JEE Main 2026, 6 Apr (Shift 2)]
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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)
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The radius (r), length \((l)\) and resistance (R) of a metal wire was measured in the laboratory as:
r = 0.35 ± 0.05 cm, R = 100 ± 10 ohm, l = 15 ± 0.2 cm
The percentage error in resistivity of the material of the wire is :
[JEE Main 2024, 01 Feb (Shift 1)]
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The radius (r), length \((l)\) and resistance (R) of a metal wire was measured in the laboratory as:
r = 0.35 ± 0.05 cm, R = 100 ± 10 ohm, l = 15 ± 0.2 cm
The percentage error in resistivity of the material of the wire is :
[JEE Main 2024, 01 Feb (Shift 1)]
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The equation for real gas is given by \(\left(\mathrm{P}+\frac{\mathrm{a}}{{\mathrm{V}}^{2}}\right)(\mathrm{V}-\mathrm{b})=\mathrm{RT}\), where P, V, T and R are the pressure, volume, temperature and gas constant, respectively. The dimension of \({\mathrm{ab}}^{-2}\) is equivalent to that of :
[JEE Main 2025, 2 Apr (Shift 1)]
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In a screw gauge, the zero of the circular scale lies 3 divisions above the horizontal pitch line when their metallic studs are brought in contact. Using this instrument thickness of a sheet is measured. If pitch scale reading is 1 mm and the circular scale reading is 51 then the correct thickness of the sheet is_____ mm. [Assume least count is 0.01 mm]
[JEE Main 2026, 23 Jan (Shift 1)]
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Match List - I with List - II.
| List - I | List - II | ||
| A. | Meter (L) | I. | \(\sqrt{\frac{hc}{G}}\) |
| B. | Second (S) | II. | \(\sqrt{\frac{Gh}{{c}^{5}}}\) |
| C. | Kilogram (M) | III. | \(\sqrt{\frac{{K}^{2}{L}^{2}{c}^{3}}{Gh}}\) |
| D. | Kelvin (K) | IV. | \(\sqrt{\frac{Gh}{{c}^{3}}}\) |
where h (Planck’s constant), G (gravitational constant) and c (speed of light in vacuum) as fundamental units.
Choose the correct answer from the options given below:
[JEE Main 2026, 5 Apr (Shift 2)]
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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
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In an experiment, a set of reading are obtained as follows - 1.24 mm, 1.25 mm, 1.23 mm, 1.21 mm. The expected least count of the instrument used in recording these readings is _____ mm.
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The density \(\rho\) of a uniform cylinder is determined by measuring its mass \(m\), length \(l\) and diameter \(d\). The measured values of \(m,l\) and \(d\) are \(97.42\pm 0.02g,8.35\pm 0.05mm\) and \(20.20\pm 0.02mm\), respectively. Calculated percentage fractional error in \(\rho\) is _____ .
[JEE Main 2026, 6 Apr (Shift 1)]
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Given a charge q, current I and permeability of vacuum \({\mu }_{0}\). Which of the following quantity has the dimension of momentum?
[JEE Main 2025, 2 Apr (Shift 2)]
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Length, breadth and thickness of a strip having a uniform cross section are measured to be 10.5 cm , 0.05 mm , and \(6.0 \mu \mathrm{~m}\), respectively. Which of the following option(s) give(s) the volume of the strip in \(\mathrm{cm}^3\) with correct significant figures:
[JEE Advanced 2025]
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A person measures mass of 3 different particles as 435.42 g, 226.3 g and 0.125 g. According to the rules for arithmetic operations with significant figures, the additions of the masses of 3 particles will be.
[JEE Main 2025, 3 Apr (Shift 1)]
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The unit of \(\sqrt{\frac{2\mathrm{I}}{{ϵ}_{0}c}}\)is :
( I = intensity of an electromagnetic wave, c : speed of light)
[JEE Main 2025, 7 Apr (Shift 2)]
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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.
[JEE Main 2025, 22 Jan (Shift 1)]
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Four persons measure the length of a rod as 20.00 cm, 19.75 cm, 17.01 cm and 18.25 cm. The relative error in the measurement of average length of the rod is:
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L, C and R represent physical quantities inductance, capacitance and resistance respectively. The dimensional formula \(M{L}^{2}{T}^{-4}{A}^{-2}\) corresponds to:
[JEE Main 2026, 5 Apr (Shift 1)]
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A quantity Q is formulated as \({\mathrm{X}}^{-2}{\mathrm{Y}}^{+\frac{3}{2}}{\mathrm{Z}}^{-\frac{2}{5}}.\)\(\mathrm{X},\mathrm{Y}\) and \(Z\) are independent parameters which have fractional errors of \(0.1,0.2\) and \(0.5\), respectively in measurement. The maximum fractional error of Q is.
[JEE Main 2025, 8 Apr (Shift 1)]
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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 :
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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)
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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)]
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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)]
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If mass is written as \(m=k c ^{ P } G^{-1 / 2} h^{1 / 2}\) then the value of \(P\) will be : (Constants have their usual meaning with \(k\) a dimensionless constant)
[JEE Main 2024, 30 Jan (Shift 2)]
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If mass is written as \(m=k c ^{ P } G^{-1 / 2} h^{1 / 2}\) then the value of \(P\) will be : (Constants have their usual meaning with \(k\) a dimensionless constant)
[JEE Main 2024, 30 Jan (Shift 2)]
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In a Vernier calipers, when both jaws touch each other, zero of the Vernier scale is shifted to the right of zero of the main scale and 7th Vernier division coincides with a main scale reading. If the value of 1 main scale division is 1 mm and there are 10 Vernier scale divisions, then the Vernier caliper has:
[JEE Main 2026, 5 Apr (Shift 1)]
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Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : In Vernier calliper if positive zero error exists, then while taking measurements, the reading taken will be more than the actual reading.
Reason (R) : The zero error in Vernier Calliper might have happened due to manufacturing defect or due to rough handling.
In the light of the above statements, choose the correct answer from the options given below :
[JEE Main 2024, 27 Jan (Shift 2)]
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Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : In Vernier calliper if positive zero error exists, then while taking measurements, the reading taken will be more than the actual reading.
Reason (R) : The zero error in Vernier Calliper might have happened due to manufacturing defect or due to rough handling.
In the light of the above statements, choose the correct answer from the options given below :
[JEE Main 2024, 27 Jan (Shift 2)]
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In an experiment, a set of reading are obtained as follows - 1.24 mm, 1.25 mm, 1.23 mm, 1.21 mm. The expected least count of the instrument used in recording these readings is _____ mm.
[JEE Main 2026, 28 Jan (Shift 2)]
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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)
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The equation for real gas is given by \(\left(\mathrm{P}+\frac{\mathrm{a}}{{\mathrm{V}}^{2}}\right)(\mathrm{V}-\mathrm{b})=\mathrm{RT}\), where P, V, T and R are the pressure, volume, temperature and gas constant, respectively. The dimension of \({\mathrm{ab}}^{-2}\) is equivalent to that of :
[JEE Main 2025, 2 Apr (Shift 1)]
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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:
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The maximum percentage error in the measurement 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)]
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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)
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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.
[JEE Main 2025, 22 Jan (Shift 1)]
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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)]
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A vernier callipers has 20 divisions on the vernier scale, which coincides with \({19}^{th}\) division on the main scale. The least count of the instrument is \(0.1mm\) . One main scale division is equal to____ mm.
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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)]
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In an experiment using Ohm's law, voltmeter reads 10 V and ammeter reads 5 A. Least counts are 500 mV and 200 mA respectively. The estimated error in resistance measurement is:
[JEE Main 2026, 5 Apr (Shift 2)]
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The pair of physical quantities not having same dimensions is :
[JEE Main 2025, 29 Jan (Shift 1)]
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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)]
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Find the dimensions of \(\frac{B}{\mu_0}\) (Shift I Memory Based)
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In a screw gauge when the circular scale is given five complete rotations it moves linearly by 2.5 mm.If the circular scale has 100 divisions, the least count of screw gauge is ____ mm.
[04 April, 2026 (Shift-I)]
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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)]
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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]
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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
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A force is represented by \(F=a{x}^{2}+b{t}^{\frac{1}{2}}\) , where x = distance and t = time. The dimensions of \({b}^{2}/a\) are:
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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)]
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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.
[JEE Main 2025, 22 Jan (Shift 1)]
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Consider the equation \(H=\frac{{x}^{p}{ϵ}^{q}{E}^{r}}{{t}^{s}}\) Where H= magnetizing field intensity; E= electric field, \(\in =\) permittivity, x= distance, t= time. The values of p, q, r and s respectively are:
[JEE Main 2026, 8 Apr (Shift 2)]
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The pair of physical quantities not having same dimensions is :
[JEE Main 2025, 29 Jan (Shift 1)]
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To compare EMF of two cells using potentiometer the balancing lengths obtained are 200 cm and 150 cm. The least count of scale is 1 cm. The percentage error in the ratio of EMFs is ______
[JEE Main 2026, 23 Jan (Shift 2)]
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Dimensional formula for thermal conductivity is (here K denotes the temperature):
[JEE Main 2020, 4 Sep (Shift 1)]
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If velocity \( [V] \), time \( [T] \) and force \( [F] \) are chosen as the base quantities, the dimensions of the mass will be
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The pitch of the screw gauge is \(1\mathrm{mm}\) and there are 100 divisions on the circular scale. When nothing is put in between the jaws, the zero of the circular scale lies 8 divisions below the reference line. When a wire is placed between the jaws, the first linear scale division is clearly visible while \({72}^{\text{nd }}\)division on circular scale coincides with the reference line. The radius of the wire is:
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The pitch of the screw gauge is \(1\mathrm{mm}\) and there are 100 divisions on the circular scale. When nothing is put in between the jaws, the zero of the circular scale lies 8 divisions below the reference line. When a wire is placed between the jaws, the first linear scale division is clearly visible while \({72}^{\text{nd }}\)division on circular scale coincides with the reference line. The radius of the wire is:
[JEE Main 2021, 25 Feb (Shift 1)]
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If E and H represents the intensity of electric field and magnetising field respectively, then the unit of E / H will be:
[JEE Main 2021, 27 Aug (Shift 1)]
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A physical quantity \( A \) is related to four observable \( a, b, c \) and \( d \) as \( A=\frac{a^{2} b^{3}}{c \sqrt{d}} \). The percentage errors of measurement in \( a, b, c \) and \( d \) are \( 1 \%, 3 \%, 2 \% \) and \( 2 \% \) respectively. What is the percentage error in the quantity \( A \) ?
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Which of the following equations is dimensionally incorrect?
Where \(t =\) time, \(h =\) height, \(s =\) surface tension, \(\theta =\) angle, \(\rho =\) density, \(a,r =\) radius, \(g =\) acceleration due to gravity, \(V=\) volume, \(p =\) pressure, \(W =\) work done, \(\Gamma =\) torque, \(\in =\) permittivity, \(E =\) electric field, \(J =\) current density, \(L =\) length.
[JEE Main 2021, 31 Aug (Shift 1)]
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If force (\(\mathrm{F}\)), length (\(\mathrm{L}\)) and time (\(\mathrm{T}\)) are assumed to be fundamental units, then the dimensional formula of the mass will be
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Amount of solar energy received on the earths surface per unit area per unit time is defined a solar constant. Dimension of solar constant is
[JEE Main 2020, 3 Sep (Shift 2)]
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In the equation \(\left[X+\frac{a}{{Y}^{2}}\right][Y-b]=RT,X\) is pressure, Y is volume, R is universal gas constant and T is temperature. The physical quantity equivalent to the ratio \(\frac{a}{b}\) is :
[JEE Main 2023, 13 Apr (Shift 2)]
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\(\left(P+\frac{a}{V^2}\right)(V-b)=R T\) represents the equation of state of some gases. Where \(P\) is the pressure, \(V\) is the volume, \(T\) is the temperature and \(a, b, R\) are the constants. The physical quantity, which has dimensional formula as that of \(\frac{b^2}{a}\), will be:
[JEE Main 2023, 1 Feb (Shift 1)]
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Consider the following statements about vernier calipers:
Statement I: In a vernier calipers, one vernier scale division is 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.
(Shift I Memory Based)
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Match the physical quantities in Column-I with their corresponding dimensions in Column-II, where symbols have their usual meanings.
| Column-I (Physical Quantity) | Column-II (Dimensions) |
|---|---|
| (A) Young’s modulus | (i) \([A{L}^{2}]\) |
| (B) Magnetic moment | (ii) \([M{L}^{2}{T}^{−2}{A}^{−1}]\) |
| (C) Magnetic flux | (iii) \([A{L}^{−1}]\) |
| (D) Magnetic intensity | (iv) \([M{L}^{−1}{T}^{−2}]\) |
(Shift - II Memory Based)
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The dimension of stopping potential \( V_{0} \) in photoelectric effect in units of Plancks constant \( h \), speed of light \( c \) and gravitational constant \( G \) and ampere \( A \) is
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A cylindrical wire of mass \((0.4\pm 0.01)g\) has length \((8\pm 0.04)\mathrm{cm}\) and radius \((6\pm 0.03)\mathrm{mm}\). The maximum error in its density will be:
[JEE Main 2023, 8 Apr (Shift 1)]
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A 2 meter long scale with least count of \(0.2\mathrm{cm}\) is used to measure the locations of objects on an optical bench. While measuring the focal length of a convex lens, the object pin and the convex lens are placed at \(80\mathrm{cm}\) mark and \(1\mathrm{m}\) mark respectively. The image of the object pin on the other side of lens coincides with image pin that is kept at \(180\mathrm{cm}\) mark. The % error in the estimation of focal length is:
[JEE Main 2023, 6 Apr (Shift 2)]
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The frequency \(\nu\) of an oscillating liquid drop may depend upon radius (r) of the drop, density (\(\rho\)) of liquid and the surface tension (s) of the liquid as: \(\nu\)\(={r}^{a}{\rho }^{b}{s}^{c}\). The values of a, b and c respectively are; [Dimensions of surface tension are MT–2]
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Which of the following is not a dimensionless quantity?
[JEE Main 2021, 27 Aug (Shift 1)]
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Match List-I with List-II
(Torque = Force × Length) \(\left(\text{Stress = }\frac{\text{Force}}{\text{Area}}\right)\)
| List-I | List-II | ||
| A. | Torque | I. | ML–2T–2 |
| B. | Stress | II. | ML2T–2 |
| C. | Pressure gradient | III. | ML–1T–1 |
| D. | Coefficient of viscosity | IV. | ML–1T–2 |
Choose the correct answer from the options given below:
[JEE Main 2023, 8 Apr (Shift 2)]
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If velocity \( [V] \), time \( [T] \) and force \( [F] \) are chosen as the base quantities, the dimensions of the mass will be
[JEE Main 2021, 31 Aug (Shift 2)]
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The frequency \( (v) \) of an oscillating liquid drop may depend upon radius ( \( r \) ) of the drop, density ( \( \rho \) ) of liquid and the surface tension \( (S) \) of the liquid as: \( v=r^{a} \rho^{b} S^{c} \) The values of \( a, b \) and \( c \) respectively are
[JEE Main 2023, 24 Jan (Shift 2)]
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The time period of a simple pendulum is given by \( T=2 \pi \sqrt{\frac{l}{g}} \). The measured value of the length of pendulum is \( 10 \mathrm{~cm} \) known to a \( 1 \mathrm{~mm} \) accuracy. The time for 200 oscillations of the pendulum is found to be 100 second using a clock of 1 s resolution. The percentage accuracy in the determination of \( g \) using this pendulum is \( x \) . The value of \( x \) to be nearest integer is:-
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If \(E\) and \(G\) respectively denote energy and gravitational constant, then \(\frac{E}{G}\)has the dimensions of:
[NEET 2021]
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A physical quantity \( A \) is related to four observable \( a, b, c \) and \( d \) as \( A=\frac{a^{2} b^{3}}{c \sqrt{d}} \). The percentage errors of measurement in \( a, b, c \) and \( d \) are \( 1 \%, 3 \%, 2 \% \) and \( 2 \% \) respectively. What is the maximum percentage error in the quantity \( A \) ?
[JEE Main 2023, 10 Apr (Shift 1)]
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A body of mass \((5\pm 0.5)\mathrm{kg}\) is moving with a velocity of \((20\pm 0.4)\mathrm{m}/\mathrm{s}\). Its kinetic energy will be
[JEE Main 2023, 13 Apr (Shift 1)]
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The frequency \(\nu\) of an oscillating liquid drop may depend upon radius (r) of the drop, density (\(\rho\)) of liquid and the surface tension (s) of the liquid as: \(\nu\)\(={r}^{a}{\rho }^{b}{s}^{c}\). The values of a, b and c respectively are; [Dimensions of surface tension are MT–2]
[JEE Main 2023, 24 Jan (Shift 2)]
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If \(R,{X}_{L}\) and \({X}_{C}\) represent resistance, inductive reactance and capacitive reactance. Then which of the following is dimensionless:
[JEE Main 2023, 31 Jan (Shift 1)]
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The equation of a circle is given by \({x}^{2}+{y}^{2}={a}^{2}\), where a is the radius. If the equation is modified to change the origin other than (0,0), then find out the correct dimensions of \(A\) and \(B\) in a new equation: \((x-At{)}^{2}+{\left(y-\frac{t}{B}\right)}^{2}={a}^{2}\). The dimensions of t is given as \(\left[{T}^{-1}\right]\).
[JEE Main 2023, 29 Jan (Shift 2)]
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In a typical combustion engine the work done by a gas molecule is given by \(W={\alpha }^{2}\beta {e}^{-\frac{\beta {x}^{2}}{kT}}\), where x is the displacement, k is the Boltzmann constant and T is the temperature. If \(\alpha\) and \(\beta\) are constants, dimension of \(\alpha\) will be :
[JEE Main 2021, 26 Feb (Shift 1)]
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Match List I with List II :
| List – I (Physical Quantity) | List – II (Dimensional Formula) |
| A. Pressure gradient | I. \(\left[{M}^{0}{L}^{2}{T}^{-2}\right]\) |
| B. Energy density | II. \(\left[{M}^{1}{L}^{-1}{T}^{-2}\right]\) |
| C. Electric Field | III. \(\left[{M}^{1}{L}^{-2}{T}^{-2}\right]\) |
| D. Latent heat | IV. \(\left[{M}^{1}{L}^{1}{T}^{-3}{A}^{-1}\right]\) |
Choose the correct answer from the options given below :
[JEE Main 2023, 29 Jan (Shift 1)]
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Two resistances are given as \({R}_{1}=(10\pm 0.5)\Omega\) and \({R}_{2}=(15\pm 0.5)\Omega\). The percentage error in the measurement of equivalent resistance when they are connected in parallel is
[JEE Main 2023, 6 Apr (Shift 1)]
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One main scale division of a vernier calipers is ' a ' cm and \({n}^{\text{th }}\) division of the vernier scale coincide with \((n-1{)}^{\text{th }}\) division of the main scale. The least count of the calipers (in mm ) is:
[JEE Main 2021, 16 Mar (Shift 1)]
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If \({\mu }_{0}\) and \({\epsilon }_{0}\) are the permeability and permittivity of free space,
respectively, then the Dimensions of \( 1 /\left(\mu_{0} \varepsilon_{0}\right) \) is:
[JEE Main 2023, 8 Apr (Shift 1)]
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If the velocity of light c, universal gravitational constant G and planck's constant h are chosen as fundamental quantities. The dimensions of mass in the new system is:
[JEE Main 2023, 1 Feb (Shift 2)]
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Using screw gauge of pitch \( 0.1 \mathrm{~cm} \) and 50 divisions on its circular scale, the thickness of an object is measured. It should correctly be recorded as:
[JEE Main 2020, 3 Sep (Shift 1)]
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Using screw gauge of pitch \( 0.1 \mathrm{~cm} \) and 50 divisions on its circular scale, the thickness of an object is measured. It should correctly be recorded as:
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The period of oscillation of a simple pendulum is \(T=2\pi \sqrt{\frac{L}{g}}\). Measured value of ' L ' is \(1.0\mathrm{m}\) from meter scale having a minimum division of \(1\mathrm{mm}\) and time of one-complete oscillation is \(1.95\mathrm{s}\) measured from stopwatch of \(0.01\mathrm{s}\) resolution. The percentage error in the determination of ' g ' will be:
[JEE Main 2021, 24 Feb (Shift 2)]
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Match List-I with List-II
| List-I | List-II | ||
| A. | Angular momentum | I. | [ML2 T–2] |
| B. | Torque | II. | [ML–2 T–2] |
| C. | Stress | III. | [ML2 T–1] |
| D. | Pressure gradient | IV. | [ML–1 T–2] |
Choose the correct answer from the options given below:
[JEE Main 2023, 31 Jan (Shift 2)]
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Match List-I with List-II.
| List-I | List-II | ||
| (A) | Young’s modulus (Y) | (I) | [ML–1T–1] |
| (B) | Coefficient of viscosity (h) | (II) | [ML2T–1] |
| (C) | Planck’s constant (h) | (III) | [ML–1T–2] |
| (D) | Work function \((ϕ)\) | (IV) | [ML2T–2] |
Choose the correct answer from the options given below:
[JEE Main 2023, 25 Jan (Shift 2)]
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