Units and Measurements — JEE Main PYQs

Dimensions of universal gravitational constant ($G$) in terms of Planck's constant ($h$), distance ($L$), mass ($M$) and time ($T$) are ________.

Match the LIST-I with LIST-II

List-I	List-II
A. Planck's constant	I. $\mathrm{M\,L^2\,T^{-2}}$
B. Stopping potential	II. $\mathrm{T^{-1}}$
C. Work function	III. $\mathrm{M\,L^2\,T^{-1}}$
D. Threshold frequency	IV. $\mathrm{M\,L^2\,T^{-3}\,A^{-1}}$

Choose the correct answer from the options given below:

$L$, $C$ and $R$ represents physical quantities inductance, capacitance and resistance respectively. The dimensional formula $\mathrm{M\,L^2\,T^{-4}\,A^{-2}}$ corresponds to __________.

Match List - I with List - II.

List - I | List - II

A. Meter (L) — I. $\sqrt{\dfrac{hc}{G}}$

B. Second (S) — II. $\sqrt{\dfrac{Gh}{c^5}}$

C. Kilogram (M) — III. $\sqrt{\dfrac{K^2 L^2 c^3}{Gh}}$

D. Kelvin (K) — IV. $\sqrt{\dfrac{Gh}{c^3}}$

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 :

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 $97.42 \pm 0.02$ g, $8.35 \pm 0.05$ mm and $20.20 \pm 0.02$ mm, respectively. Calculated percentage fractional error in $\rho$ is ________.

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 ________.

The percentage error in the calculated volume of a sphere, if there is 2% error in its diameter measurement, is ________.

Match List - I with List - II.

List - I	List - II
A. Boltzmann constant	I. $[\text{M}^{-1}\text{L}^{3}\text{T}^{-2}]$
B. Stefan's constant	II. $[\text{M}\,\text{L}^{2}\text{T}^{-1}]$
C. Planck's constant	III. $[\text{M}\,\text{L}^{2}\text{T}^{-2}\text{K}^{-1}]$
D. Gravitational constant	IV. $[\text{M}\,\text{L}^{0}\text{T}^{-3}\text{K}^{-4}]$

Choose the correct answer from the options given below :

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?

Consider the equation $H = \dfrac{x^p \epsilon^q E^r}{t^s}$

Where $H$ = magnetic field; $E$ = electric field, $\epsilon$ = permittivity, $x$ = distance, $t$ = time.

The values of $p$, $q$, $r$ and $s$ respectively are :

The equation for real gas is given by $\left(P+\frac{a}{V^2}\right)(V-b)=RT$, where P,V,T and R are the pressure, volume, temperature and gas constant, respectively. The dimension of $ab^{-2}$ is equivalent to that of :

Match List - I with List - II.

List - I	List - II
(A) Coefficient of viscosity	(I) $[\mathrm{ML^0T^{-3}}]$
(B) Intensity of wave	(II) $[\mathrm{ML^{-2}T^{-2}}]$
(C) Pressure gradient	(III) $[\mathrm{M^{-1}LT^{2}}]$
(D) Compressibility	(IV) $[\mathrm{ML^{-1}T^{-1}}]$

Choose the correct answer from the options given below :

Given a charge $q$, current $I$ and permeability of vacuum $\mu_0$. Which of the following quantity has the dimension of momentum ?

If $\mu_0$ and $\epsilon_0$ are the permeability and permittivity of free space, respectively, then the dimension of $\left(\dfrac{1}{\mu_0\epsilon_0}\right)$ is :

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 addition of the masses of 3 particles will be.

Match the LIST-I with LIST-II

	LIST-I		LIST-II
A.	Gravitational constant	I.	$[\mathrm{LT^{-2}}]$
B.	Gravitational potential energy	II.	$[\mathrm{L^2T^{-2}}]$
C.	Gravitational potential	III.	$[\mathrm{ML^2T^{-2}}]$
D.	Acceleration due to gravity	IV.	$[\mathrm{M^{-1}L^3T^{-2}}]$

Choose the correct answer from the options given below:

Match the LIST-I with LIST-II

	LIST-I		LIST-II
A.	Boltzmann constant	I.	$\mathrm{ML^2T^{-1}}$
B.	Coefficient of viscosity	II.	$\mathrm{MLT^{-3}K^{-1}}$
C.	Planck's constant	III.	$\mathrm{ML^2T^{-2}K^{-1}}$
D.	Thermal conductivity	IV.	$\mathrm{ML^{-1}T^{-1}}$

Choose the correct answer from the options given below:

A physical quantity $C$ is related to four other quantities $p$, $q$, $r$ and $s$ as follows

$$C = \frac{pq^2}{r^3\sqrt{s}}.$$

The percentage errors in the measurement of $p$, $q$, $r$ and $s$ are 1%, 2%, 3% and 2%, respectively. The percentage error in the measurement of $C$ will be __________ %.

In an electromagnetic system, the quantity representing the ratio of electric flux and magnetic flux has dimension of $\mathrm{M^P L^Q T^R A^S}$, where value of 'Q' and 'R' are

In an electromagnetic system, a quantity defined as the ratio of electric dipole moment and magnetic dipole moment has dimension of $[\mathrm{M^P\, L^Q\, T^R\, A^S}]$. The value of P and Q are :

Given below are two statements :

Statement (I) : The dimensions of Planck's constant and angular momentum are same.

Statement (II) : In Bohr's model electron revolve around the nucleus only in those orbits for which angular momentum is integral multiple of Planck's constant.

In the light of the above statements, choose the most appropriate answer from the options given below :

If $\epsilon_0$ denotes the permittivity of free space and $\Phi_E$ is the flux of the electric field through the area bounded by the closed surface, then dimensions of $\left(\epsilon_0 \frac{d\phi_E}{dt}\right)$ are that of :

The dimension of $\sqrt{\dfrac{\mu_0}{\epsilon_0}}$ is equal to that of :

($\mu_0$ = Vacuum permeability and $\epsilon_0$ = Vacuum permittivity)

Match List - I with List - II.

List - I	List - II
(A) Mass density	(I) $[\mathrm{ML^2T^{-3}}]$
(B) Impulse	(II) $[\mathrm{MLT^{-1}}]$
(C) Power	(III) $[\mathrm{ML^2T^{0}}]$
(D) Moment of inertia	(IV) $[\mathrm{ML^{-3}T^{0}}]$

Choose the correct answer from the options given below :

A quantity Q is formulated as $X^{-2} Y^{+\frac{3}{2}} Z^{-\frac{2}{5}}$. X, 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

If B is magnetic field and $\mu_0$ is permeability of free space, then the dimensions of $(B/\mu_0)$ is

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

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 $= (10.00 \pm 0.05)\,\mathrm{cm}$]

Match List - I with List - II.

	List - I		List - II
(A)	Permeability of free space	(I)	$[\mathrm{M\,L^2\,T^{-2}}]$
(B)	Magnetic field	(II)	$[\mathrm{M\,T^{-2}\,A^{-1}}]$
(C)	Magnetic moment	(III)	$[\mathrm{M\,L\,T^{-2}\,A^{-2}}]$
(D)	Torsion constant	(IV)	$[\mathrm{L^2\,A}]$

Choose the correct answer from the options given below :

The energy of a system is given as $E(t) = \alpha^3 e^{-\beta t}$, 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 $t = 5$ s, maximum percentage error in the energy is :

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

The energy E and momentum p of a moving body of mass m are related by some equation. Given that c represents the speed of light, identify the correct equation

In a measurement, it is asked to find modulus of elasticity per unit torque applied on the system. The measured quantity has dimension of $[\mathrm{M^a\, L^b\, T^c}]$. If b = 3, the value of c is _____

[Figure: a bar magnet with poles S and N, total length $2l$, and a field point P located on the perpendicular bisector at distance $d$ from the centre of the magnet]

A bar magnet has total length $2l = 20$ units and the field point P is at a distance $d = 10$ units from the centre of the magnet. If the relative uncertainty of length measurement is 1%, then uncertainty of the magnetic field at point P is :

Match List - I with List - II.

List - I	List - II
(A) Angular Impulse	(I) $[\mathrm{M^0 L^2 T^{-2}}]$
(B) Latent Heat	(II) $[\mathrm{M L^2 T^{-3} A^{-1}}]$
(C) Electrical resistivity	(III) $[\mathrm{M L^2 T^{-1}}]$
(D) Electromotive force	(IV) $[\mathrm{M L^3 T^{-3} A^{-2}}]$

Choose the correct answer from the options given below :

The equation of stationary wave is :

$$y = 2a \sin\left(\frac{2\pi nt}{\lambda}\right) \cos\left(\frac{2\pi x}{\lambda}\right).$$

Which of the following is NOT correct :

Applying the principle of homogeneity of dimensions, determine which one is correct,

where $T$ is time period, $G$ is gravitational constant, $M$ is mass, $r$ is radius of orbit.

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 $\sqrt{uG}$ :

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

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

Match List I with List II

	LIST I		LIST II
A.	Torque	I.	$[M^1L^1T^{-2}A^{-2}]$
B.	Magnetic field	II.	$[L^2A^1]$
C.	Magnetic moment	III.	$[M^1T^{-2}A^{-1}]$
D.	Permeability of free space	IV.	$[M^1L^2T^{-2}]$

Choose the correct answer from the options given below:

To find the spring constant $(k)$ of a spring experimentally, a student commits 2% positive error in the measurement of time and 1% negative error in measurement of mass. The percentage error in determining value of $k$ is :

Given below are two statements :

Statement (I) : Dimensions of specific heat is $[\mathrm{L^2 T^{-2} K^{-1}}]$.

Statement (II) : Dimensions of gas constant is $[\mathrm{M^1 L^2 T^{-1} K^{-1}}]$.

In the light of the above statements, choose the most appropriate answer from the options given below.

In an expression $a \times 10^{b}$:

If $\epsilon_0$ is the permittivity of free space and $\mathrm{E}$ is the electric field, then $\epsilon_0 \mathrm{E}^2$ has the dimensions :

The dimensional formula of latent heat is :

The de-Broglie wavelength associated with a particle of mass $m$ and energy $E$ is $h/\sqrt{2mE}$. The dimensional formula for Planck's constant is :

The dimensional formula of angular impulse is :

The radius (r), length ($l$) and resistance (R) of a metal wire was measured in the laboratory as

$r = (0.35 \pm 0.05)$ cm

$R = (100 \pm 10)$ ohm

$l = (15 \pm 0.2)$ cm

The percentage error in resistivity of the material of the wire is :

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 :

Given below are two statements :

Statement (I) : Planck's constant and angular momentum have same dimensions.

Statement (II) : Linear momentum and moment of force have same dimensions.

In the light of the above statements, choose the correct answer from the options given below :

The resistance $R = \frac{V}{I}$ where $V = (200 \pm 5)$ V and $I = (20 \pm 0.2)$ A, the percentage error in the measurement of R is :

Match List - I with List - II.

	List - I		List - II
(A)	Coefficient of viscosity	(I)	$[\mathrm{M\,L^2\,T^{-2}}]$
(B)	Surface tension	(II)	$[\mathrm{M\,L^2\,T^{-1}}]$
(C)	Angular momentum	(III)	$[\mathrm{M\,L^{-1}\,T^{-1}}]$
(D)	Rotational kinetic energy	(IV)	$[\mathrm{M\,L^0\,T^{-2}}]$

Choose the correct answer from the options given below :

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)

A force is represented by $F = ax^2 + bt^{\frac{1}{2}}$

where $x$ = distance and $t$ = time. The dimensions of $b^2/a$ are:

If the percentage errors in measuring the length and the diameter of a wire are 0.1% each. The percentage error in measuring its resistance will be:

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

The measured value of the length of a simple pendulum is 20 cm with 2 mm accuracy. The time for 50 oscillations was measured to be 40 seconds with 1 second resolution. From these measurements, the accuracy in the measurement of acceleration due to gravity is N%. The value of N is: