Dimension of a Physical Quantity

Q 11 :

Match the List I with List II

List I List II

(A) Gravitational constant (I) [ $L T^{- 2}$ ]

(B) Gravitational potential energy (II) [ $L^{2} T^{- 2}$ ]

(C) Gravitational potential (III) [ $M L^{2} T^{- 2}$ ]

(D) Acceleration due to gravity (IV) [ $M^{- 1} L^{3} T^{- 2}$ ]

Choose the correct answer from the options given below: [2025]

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

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(1)

(A) $G = \frac{F r^{2}}{m^{2}}$

$[G] = \frac{[M L T^{- 2}] [L^{2}]}{[M^{2}]} = [M^{- 1} L^{3} T^{- 2}] \Rightarrow A \to IV$

(B) P.E. $U = m g h = [M L T^{- 2} L] = [M L^{2} T^{- 2}] \Rightarrow B \to III$

(C) Gravitational Potential, $V = \frac{G M}{r}$

$V = \frac{[M^{- 1} L^{3} T^{- 2}] [M]}{[L]} = [M^{0} L^{2} T^{- 2}] = [L^{2} T^{- 2}] \Rightarrow C \to II$

(D) Acceleration due to gravity $= [g] = [L T^{- 2}] \Rightarrow D \to I$

Q 12 :

Match the List I with List II

List I List II

(A) Boltzmann constant (I) $M L^{2} T^{- 1}$

(B) Coefficient of viscosity (II) $M L T^{- 3} K^{- 1}$

(C) Planck's constant (III) $M L^{2} T^{- 2} K^{- 1}$

(D) Thermal conductivity (IV) $M L^{- 1} T^{- 1}$

Choose the correct answer from the options given below: [2025]

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

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(1)

(A) $[k] = \frac{P V}{N T} = \frac{M L^{2} T^{- 2}}{K} = M L^{2} T^{- 2} K^{- 1}$

(B) $[η] = \frac{F}{6 π r v} = \frac{M L T^{- 2}}{L^{2} T^{- 1}} = M L^{- 1} T^{- 1}$

(C) $[h] = \frac{E}{f} = \frac{M L^{2} T^{- 2}}{T^{- 1}} = M L^{2} T^{- 1}$

(D) $\frac{d Q}{d t} = k \frac{A d T}{d x}, k = \frac{(M L^{2} T^{- 3}) L}{L^{2} \cdot K} = M L T^{- 3} K^{- 1}$

Q 13 :

In an electromagnetic system, the quantity representing the ratio of electric flux and magnetic flux has dimension of $M^{P} L^{Q} T^{R} A^{S}$ , where value of 'Q' and 'R' are [2025]

(3, –5)
(–2, 2)
(–2, 1)
(1, –1)

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(4)

$\frac{ϕ_{E}}{ϕ_{M}} = \frac{E A}{B A} = \frac{E}{B}$

$\frac{E}{B} = c \Rightarrow [\frac{E}{B}] = [c] = [L T^{- 1}]$

Q 14 :

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: [2025]

–1, 0
–1, 1
1, –1
0, –1

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(4)

Electric dipole moment $(\vec{P}) = q \times 2 l$

Magnetic dipole moment $= (\vec{M}) = I A$

$[\frac{P}{M}] = [\frac{L T A}{L^{2} A}] = L^{- 1} T = M^{0} L^{- 1} T^{1} A^{0}$

After comparing values of P and Q are 0, –1.

Q 15 :

The dimension of $\sqrt{\frac{μ_{0}}{ε_{0}}}$ is equal to that of : ( $μ_{0}$ = Vacuum permeability and $ε_{0}$ = Vacuum permittivity) [2025]

Voltage
Capacitance
Inductance
Resistance

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(4)

$[μ_{0}] = [M L T^{- 2} A^{- 2}]$

$[ε_{0}] = [M^{- 1} L^{- 3} T^{4} A^{2}]$

$[\sqrt{\frac{μ_{0}}{ε_{0}}}] = [M L^{2} T^{- 3} A^{- 2}] ≃ [R]$

Q 16 :

Match the List-I with List-II.

List-I List-II

(A) Mass density (I) $[M L^{2} T^{- 3}]$

(B) Impulse (II) $[M L T^{- 1}]$

(C) Power (III) $[M L^{2} T^{0}]$

(D) Moment of inertia (IV) $[M L^{- 3} T^{0}]$

Choose the correct answer from the options given below : [2025]

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

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(3)

(A) Mass density $= \frac{M}{V} = [M^{1} L^{- 3}]$ ... (IV)

(B) Impulse $= M \times u = [M^{1} L^{1} T^{- 1}]$ ... (II)

(C) Power $= F \cdot v = [M^{1} L^{2} T^{- 3}]$ ... (I)

(D) Moment of inertia $= M r^{2} = [M^{1} L^{2}]$ ... (III)

Q 17 :

In a measurement, it is asked to find modulus of elasticity per unit torque applied on the system. The measured quantity has dimension of $[M^{a} L^{b} T^{c}]$ . If b = –3, the value of c is __________. [2025]

(0)

[Modulus of elasticity] $= M L^{- 1} T^{- 2}$

[Torque] $= M L^{2} T^{- 2}$

[Modulus of elasticity per unit torque] $= \frac{M L^{- 1} T^{- 2}}{M L^{2} T^{- 2}} = L^{- 3}$ .

Q 18 :

The frequency (v) of an oscillating liquid drop may depend upon radius (r) of the drop, density ( $ρ$ ) of liquid and the surface tension (s) of the liquid as : $v = r^{a} ρ^{b} s^{c}$ . The values of a, b and c respectively are [2023]

$(- \frac{3}{2}, \frac{1}{2}, \frac{1}{2})$
$(- \frac{3}{2}, - \frac{1}{2}, \frac{1}{2})$
$(\frac{3}{2}, \frac{1}{2}, - \frac{1}{2})$
$(\frac{3}{2}, - \frac{1}{2}, \frac{1}{2})$

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(2)

$[T^{- 1}] = {[L^{1}]}^{a} {[M^{1} L^{- 3}]}^{b}$ ${[\frac{{MLT}^{- 2}}{L}]}^{c}$

$\Rightarrow T^{- 1} = M^{b + c} \cdot L^{a - 3 b} \cdot T^{- 2 c}$

$c = \frac{1}{2}, b = - \frac{1}{2}, a - 3 b = 0$

$a + \frac{3}{2} = 0 \Rightarrow a = - \frac{3}{2}$

Q 19 :

Match the following Column – I with Column – II

Column –I Column – II

A. Surface tension I. $kg m^{- 1} s^{- 1}$

B. Pressure II. $kg {ms}^{- 1}$

C. Viscosity III. $kg m^{- 1} s^{- 2}$

D. Impulse IV. $kg s^{- 2}$

Choose the correct answer from the options given below: [2023]

A – III; B – IV; C – I; D – II
A – IV; B – III; C – II; D – I
A – IV; B – III; C – I; D – II
A – II; B – I; C – III; D – IV

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(3)

Surface tension – N/m $\to kg s^{- 2}$

Pressure – $N / m^{2} \to kg m^{- 1} s^{- 2}$

Viscosity – $kg m^{- 1} s^{- 1}$

Impulse – $kg {ms}^{- 1}$

A –IV, B –III, C – I, D – II

Q 20 :

Match List I with List II

List I List II

A. Young's Modulus (Y) I. [ $M L^{- 1} T^{- 1}$ ]

B. Co-efficient of Viscosity ( $η$ ) II. [ $M L^{2} T^{- 1}$ ]

C. Planck's Constant (h) III. [ $M L^{- 1} T^{- 2}$ ]

D. Work Function ( $ϕ$ ) IV. [ $M L^{2} T^{- 2}$ ]

Choose the correct answer from the options given below: [2023]

A – I, B – III, C – IV, D – II
A – I, B – II, C – III, D – IV
A – II, B – III, C – IV, D – I
A – III, B – I, C – II, D – IV

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(4)

$Y = \frac{Stress}{Strain} = \frac{F / A}{△ l / l} = \frac{[M L T^{- 2}]}{[L^{2}]} = [M L^{- 1} T^{- 2}]$

$F = 6 π η r v \Rightarrow η = \frac{F}{6 π r v}$

$[η] = \frac{[M L T^{- 2}]}{[L] [L T^{- 1}]} = [M L^{- 1} T^{- 1}]$

$E = h v \Rightarrow h = \frac{E}{v} = \frac{[M L^{2} T^{- 2}]}{[T^{- 1}]} = [M L^{2} T^{- 1}]$

Work function has same dimension as that of energy, so

$[ϕ] = [M L^{2} T^{- 2}]$

Q 21 :

Match List I with List II:

List I List II

A. Pressure gradient I. [ $M^{0} L^{2} T^{- 2}$ ]

B. Energy density II. [ $M^{1} L^{- 1} T^{- 2}$ ]

C. Electric field III. [ $M^{1} L^{- 2} T^{- 2}$ ]

D. Latent heat IV. [ $M^{1} L^{1} T^{- 3} A^{- 1}$ ]

Choose the correct answer from the options given below: [2023]

A – III, B – II, C – I, D – IV
A – II, B – III, C – IV, D – I
A – III, B – II, C – IV, D – I
A – II, B – III, C – I, D – IV

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(3)

Pressure gradient = $\frac{d p}{d x} = \frac{[M L^{- 1} T^{- 2}]}{[L]} = [M^{1} L^{- 2} T^{- 2}]$

Energy density = $\frac{energy}{volume} = \frac{[M L^{2} T^{- 2}]}{[L^{3}]} = [M^{1} L^{- 1} T^{- 2}]$

Electric field = $\frac{Force}{charge} = \frac{[M L T^{- 2}]}{[A \cdot T]} = [M^{1} L^{1} T^{- 3} A^{- 1}]$

Latent heat = $\frac{heat}{mass} = \frac{[M L^{2} T^{- 2}]}{[M]} = [M^{0} L^{2} T^{- 2}]$

Q 22 :

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 : ${(x - A t)}^{2} + {(y - \frac{t}{B})}^{2} = a^{2}$

The dimensions of t is given as [ $T^{- 1}$ ]. [2023]

$A = [L^{- 1} T], B = [L^{- 1} T]$
$A = [L T], B = [L^{- 1} T^{- 1}]$
$A = [L^{- 1} T^{- 1}], B = [L^{- 1}]$
$A = [L^{- 1} T^{- 1}], B = [L T]$

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(2)

${(x - A t)}^{2} + {(y - \frac{t}{B})}^{2} = a^{2}$

$[A t] = A \times \frac{1}{T} = L$

$∴ [A] = T^{1} L^{1}$

$\frac{t}{B}$ is in meters

$∴ \frac{1}{T [B]} = L \Rightarrow [B] = T^{- 1} L^{- 1}$

Q 23 :

Match List I with List II.

List I List II

A. Torque I. $kg m^{- 1} s^{- 2}$

B. Energy density II. $kg {ms}^{- 1}$

C. Pressure gradient III. $kg m^{- 2} s^{- 2}$

D. Impulse IV. $kg m^{2} s^{- 2}$

Choose the correct answer from the options given below: [2023]

A – IV, B – III, C – I, D – II
A – I, B – IV, C – III, D – II
A – IV, B – I, C – II, D – III
A – IV, B – I, C – III, D – II

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(4)

Torque $\to kg m^{2} s^{- 2} (IV)$

Energy density $\to kg m^{- 1} s^{- 2} (I)$

Pressure gradient $\to kg m^{- 2} s^{- 2} (III)$

Impulse $\to kg {ms}^{- 1} (II)$

Q 24 :

Match List I with List II.

List I List II

A. Angular momentum I. [ $M L^{2} T^{- 2}$ ]

B. Torque II. [ $M L^{- 2} T^{- 2}$ ]

C. Stress III. [ $M L^{2} T^{- 1}$ ]

D. Pressure gradient IV. [ $M L^{- 1} T^{- 2}$ ]

Choose the correct answer from the options given below: [2023]

A – I, B – IV, C – III, D – II
A – III, B – I, C – IV, D – II
A – II, B – III, C – IV, D – I
A – IV, B – II, C – I, D – III

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(2)

$\bar{L} = \vec{r} \times \vec{p} \Rightarrow [L] = [M^{0} L^{1} T^{0}] [M^{1} L^{1} T^{- 1}] = [M^{1} L^{2} T^{- 1}]$

$\bar{τ} = \vec{r} \times \vec{F} \Rightarrow [τ] = [L^{1}] [M L T^{- 2}] = [M L^{2} T^{- 2}]$

Stress $\equiv$ Pressure = $\frac{F}{A} \Rightarrow [Stress] = [M L^{- 1} T^{- 2}]$

Pressure Gradient = $\frac{d P}{d x}$

$\Rightarrow$ [Pressure Gradient] = $[M L^{- 1} T^{- 2}]$

Q 25 :

$(P + \frac{a}{V^{2}}) (V - b) = R T$ 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 [2023]

bulk modulus
modulus of rigidity
compressibility
energy density

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(3)

[b] = [V]

$[\frac{a}{b^{2}}] = [P]$

$∴ [\frac{b^{2}}{a}] = \frac{1}{[P]} = \frac{1}{[B]} = [K]$

Q 26 :

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: [2023]

$[h^{\frac{1}{2}} c^{- \frac{1}{2}} G^{1}]$
$[h^{1} c^{1} G^{- 1}]$
$[h^{- \frac{1}{2}} c^{\frac{1}{2}} G^{\frac{1}{2}}]$
$[h^{\frac{1}{2}} c^{\frac{1}{2}} G^{- \frac{1}{2}}]$

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(4)

Dimensional formale of mass is $H^{x} C^{y} G^{z}$

$M^{1} = {(M L^{2} T^{- 1})}^{x} (L T^{- 1}) {(M^{- 1} L^{3} T^{- 2})}^{z}$

$M^{1} L^{0} T^{0} = M^{x - z} L^{2 x + y + 3 z} T^{- x - y - 2 z}$

on comparing both side

$x - z = 1$

$2 x + y + 3 z = 0$

$- x - y - 2 z = 0$

On solving above equations we get

$x = \frac{1}{2}, y = \frac{1}{2}, z = \frac{- 1}{2}$

Q 27 :

If R, $X_{L}$ and $X_{C}$ represent resistance, inductive reactance and capacitive reactance. Then which of the following is dimensionless: [2023]

$R X_{L} X_{C}$
$\frac{R}{\sqrt{X_{L} X_{C}}}$
$\frac{R}{X_{L} X_{C}}$
$R \frac{X_{L}}{X_{C}}$

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(2)

All three have same dimension therefore $\frac{R}{\sqrt{X_{L} X_{C}}}$ is dimensionless.

Q 28 :

Match List I with List II

List I List II

A. Planck's constant (h) I, [ $M^{1} L^{2} T^{- 2}$ ]

B. Stopping potential ( $V_{S}$ ) II. [ $M^{1} L^{1} T^{- 1}$ ]

C. Work function ( $ϕ$ ) III. [ $M^{1} L^{2} T^{- 1}$ ]

D. Momentum (p) IV. [ $M^{1} L^{2} T^{- 3} A^{- 1}$ ]

Choose the correct answer from the options given below: [2023]

A – III, B – IV, C – II, D – I
A – III, B – IV, C – I, D – II
A – I, B – III, C – IV, D – I
A – III, B – I, C – II, D – IV

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(2)

(A) Planck's constant

hv = E

$h = \frac{E}{v} = \frac{M^{1} L^{2} T^{- 2}}{T^{- 2}} = M^{1} L^{2} T^{- 1}$

(B) E = qv

$V = \frac{E}{q} = \frac{M^{1} L^{2} T^{- 2}}{A^{1} T^{1}} = M^{1} L^{2} T^{- 3} A^{- 1}$

(C) $ϕ$ (work function) = energy = $M^{1} L^{2} T^{- 2}$

(D) Momentum (p) = F.t = $M^{1} L^{1} T^{- 2} T^{1} = M^{1} L^{1} T^{- 1}$

Q 29 :

Dimension of $\frac{1}{μ_{0} ε_{0}}$ should be equal to [2023]

$\frac{L}{T}$
$\frac{L^{2}}{T^{2}}$
$\frac{T}{L}$
$\frac{T^{2}}{L^{2}}$

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(2)

$\frac{1}{μ_{0} ε_{0}} = c^{2} \Rightarrow [\frac{1}{μ_{0} ε_{0}}] = [c^{2}] = [L^{2} T^{- 2}]$

Q 30 :

Match List I with List II

List I List II

A. Torque I. $M L^{- 2} T^{- 2}$

B. Stress II. $M L^{2} T^{- 2}$

C, Pressure gradient III. $M L^{- 1} T^{- 1}$

D. Coefficient of viscosity IV. $M L^{- 1} T^{- 2}$

Choose the correct answer from the options given below: [2023]

A – II, B – I, C – IV, D – III
A – II, B – IV, C – I, D – III
A – III, B – IV, C – I, D – II
A – IV, B – II, C – III, D – I

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(2)

A. Torque $\Rightarrow \vec{τ} = \vec{r} \times \vec{F}$

$[τ] = [L] [M L T^{- 2}] \Rightarrow M L^{2} T^{- 2}$

B. Stress = $\frac{F}{A} \Rightarrow \frac{M L T^{- 2}}{L^{2}}$

[Stress] = $M L^{- 1} T^{- 2}$

C. Pressure gradient = $\frac{△ P}{△ X}$

$\Rightarrow \frac{[F / A]}{[L]} = \frac{M L T^{- 2}}{L^{3}} \Rightarrow M L^{- 2} T^{- 2}$

D. Coefficient of viscosity $\Rightarrow F = 6 π η r v$

$M L T^{- 2} = [η] L^{2} T^{- 1}$

$[η] = M L^{- 1} T^{- 1}$

Q 31 :

If force (F), velocity (V) and time (T) are considered as fundamental physical quantity, then dimensional formula of density will be [2023]

$F V^{- 2} T^{2}$
$F V^{- 4} T^{2}$
$F V^{4} T^{- 6}$
$F^{2} V^{- 2} T^{6}$

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(2)

$[M L^{- 3}] = {[M L T^{- 2}]}^{a} {[L T^{- 1}]}^{b}$ ${[T]}^{c} = [M^{a} L^{a + b} T^{- 2 a + b + c}]$

a = 1

a + b = –3 $\Rightarrow$ b = –4

Also, – 2a – b + c = 0

c = 2

Q 32 :

Match List-I with List-II

List I List II

A. Spring constant I. ( $T^{- 1}$ )

B. Angular speed II. ( $M T^{- 2}$ )

C. Angular momentum III. ( $M L^{2}$ )

D. Moment of Inertia IV. ( $M L^{2} T^{- 1}$ )

Choose the correct answer from the options given below: [2023]

A – II, B – I, C – IV, D – III
A – IV, B – I, C – III, D – II
A – II, B – III, C – I, D – IV
A – I, B – III, C – II, D – IV

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(1)

Spring Constant

$[K] = \frac{[F]}{[x]} = \frac{M L T^{- 2}}{L} = M T^{- 2}$

$[ω] = \frac{[θ]}{[t]} = \frac{1}{T} = T^{- 1}$

Q 33 :

In the equation $[X + \frac{a}{Y^{2}}] [Y - b] = R T$ , X is pressure, Y is volume, R is universal gas constant and T is temperature. The physical quantity a equivalent to the ratio $\frac{a}{b}$ is [2023]

Energy
Impulse
Pressure gradient
Coefficient of viscosity

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(1)

X and $\frac{a}{Y^{2}}$ have same dimensions

Y and b have same dimensions

$∴ [a] = [M L^{5} T^{- 2}]$

$[b] = [L^{3}]$

$\frac{[a]}{[b]} = [M L^{2} T^{- 2}]$ has dimensions of energy.

Q 34 :

The speed of a wave produced in water is given by $v = λ^{a} g^{b} ρ^{c}$ . When $λ$ , g and $ρ$ are wavelength of wave, acceleration due to gravity and density of water respectively. The values of a, b and c respectively, are [2023]

$\frac{1}{2}, \frac{1}{2}, 0$
1, 1, 0
1, –1, 0
$\frac{1}{2}, 0, \frac{1}{2}$

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(1)

$v = λ^{a} g^{b} ρ^{c}$

using dimension formula

$\Rightarrow [M^{0} L^{1} T^{- 1}] = {[L^{1}]}^{a} {[L^{1} T^{- 2}]}^{b} {[M^{1} L^{- 3}]}^{c}$

$\Rightarrow [M^{0} L^{1} T^{- 1}] = [M^{c} L^{a + b - 3 c} T^{- 2 b}]$

$∴ c = 0, a + b - 3 c = 1, - 2 b = - 1 \Rightarrow b = \frac{1}{2}$

Now, a + b –3c = 1

$\Rightarrow a + \frac{1}{2} - 0 = 1$

$\Rightarrow a = \frac{1}{2}$

$∴ a = \frac{1}{2}, b = \frac{1}{2}, c = 0$

Q 35 :

Consider a modified Bernoulli equation.

$(P + \frac{A}{B t^{2}}) + ρ g (h + B t) + \frac{1}{2} ρ v^{2} = constant .$

If t has the dimension of time then the dimensions of A and B are , respectively. [2026]

$[M L^{0} T^{- 1}] a n d [M^{0} L T^{- 1}]$
$[M L^{0} T^{- 2}] a n d [M^{0} L T^{- 2}]$
$[M L^{0} T^{- 1}] a n d [M^{0} L T]$
$[M L^{0} T^{- 2}] a n d [M^{0} L T^{- 1}]$

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(1)

$\Rightarrow [P] = [\frac{A}{B t^{2}}] . . . . . (1)$

$\Rightarrow [h] = [B t] . . . . . (2)$

$\Rightarrow [B] = [\frac{h}{t}] = [\frac{L}{T}] = [L T^{- 1}]$

$Putting B in equation (1)$

$[M L^{- 1} T^{- 2}] = [\frac{A}{L T^{- 1} \times T^{2}}]$

$[A] = [M L^{0} T^{- 1}]$

Q 36 :

Match the LIST-I with LIST-II [2026]

List - I List - II

A. Magnetic induction I. ${MLT}^{- 2} A^{- 2}$

B. Magnetic flux II. ${ML}^{2} T^{- 2} A^{- 2}$

C. Magnetic permeability III. ${ML}^{0} T^{- 2} A^{- 1}$

D. Self inductance IV. ${ML}^{2} T^{- 2} A^{- 1}$

Choose the correct answer from the options given below:

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

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(4)

$Magnetic induction$

$F = qvB$

$[B] = [\frac{F}{q v}]$

$[B] = [M T^{- 2} A^{- 1}]$

$Magnetic Flux (ϕ)$

$ϕ = (B) \cdot (Area)$

$[ϕ] = [M L^{2} T^{- 2} A^{- 1}]$

$Magnetic Permeability$

$[μ] = [M L T^{- 2} A^{- 2}]$

$Self inductance$

$Using U = \frac{1}{2} L I^{2}$

$[Self inductance] = [M L^{2} T^{- 2} A^{- 1}]$

$A - III, B - IV, C - I, D - II$

Q 37 :

Match the LIST-I with LIST-II: [2026]

List - I List - II

A. Spring constant I. ${ML}^{2} T^{- 2} K^{- 1}$

B. Thermal conductivity II. ${ML}^{0} T^{- 2}$

C. Boltzmann constant III. ${ML}^{2} T^{- 3} A^{- 2}$

D. Inductive reactance IV. ${MLT}^{- 3} K^{- 1}$

Choose the correct answer from the options given below:

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

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(1)

$(A) F = K x$

$[M L T^{- 2}] = [K] [L]$

$[K] = M L^{0} T^{- 2}$

$(B) Thermal conductivity$

$\frac{d Q}{d t} = \frac{k A}{ℓ} Δ T$

$M L^{2} T^{- 3} = \frac{[k] L^{2} K}{L}$

$[k] = M L T^{- 3} K^{- 1}$

$(C) Boltzmann constant$

$[K] = M L^{2} T^{- 2} K^{- 1}$

$(D) Inductive reactance$

$\frac{[V]}{[I]} = \frac{M L^{2} T^{- 3} A^{- 1}}{A}$

$= M L^{2} T^{- 3} A^{- 2}$

Q 38 :

Match List – I with List – II [2026]

List – I

List – II

A. Coefficient of viscosity

I. $[M L ⁻ ¹ T ⁻ ²]$

B. Surface tension

II. $[M L ² T ⁻ ²]$

C. Pressure

III. $[M L ⁰ T ⁻ ²]$

D. Surface energy

IV. $[M L ⁻ ¹ T ⁻ ¹]$

Choose the correct answer from the options given below :

List – I	List – II
A. Coefficient of viscosity	I. $[M L ⁻ ¹ T ⁻ ²]$
B. Surface tension	II. $[M L ² T ⁻ ²]$
C. Pressure	III. $[M L ⁰ T ⁻ ²]$
D. Surface energy	IV. $[M L ⁻ ¹ T ⁻ ¹]$

A–I, B–II, C–IV, D–III
A–I, B–III, C–II, D–IV
A–IV, B–III, C–I, D–II
A–IV, B–I, C–II, D–III

1

2

3

4

(3)

$(A) η = \frac{F d r}{A d v} = \frac{[M L T^{- 2}] [L]}{[L^{2}] [L T^{- 1}]} = [M L^{- 1} T^{- 1}]$

$(B) S = \frac{F}{L} = \frac{[M L T^{- 2}]}{[L]} = [M T^{- 2}]$

$(C) P = \frac{F}{A} = \frac{[M L T^{- 2}]}{[L^{2}]} = [M L^{- 1} T^{- 2}]$

$(D) E = S \times A = [M T^{- 2}] [L^{2}] = [M L^{2} T^{- 2}]$

Q 39 :

The force of interaction between two atoms is given by $F = α β \exp (- \frac{x^{2}}{α k T})$ ; where $' x'$ is the distance, $' k'$ is the Boltzmann constant and T is temperature and $α$ and $β$ are two constants. The dimensions of $β$ is:

$M^{0} L^{2} T^{- 4}$
$M^{2} {LT}^{- 4}$
${MLT}^{- 2}$
$M^{2} L^{2} T^{- 2}$

1

2

3

4

(2)

Force of interaction between two atoms, $F = α β e^{(\frac{- x^{2}}{α k T})}$

Since exponential terms are dimensionless

$∴$ $α β e^{(\frac{- x^{2}}{α k T})}$

$[F] = [α] [β]$

$M L T^{- 2} = M^{- 1} T^{2} [β]$

$\Rightarrow$ $[β] = M^{2} L T^{- 4}$

Q 40 :

Match the List-I with List-II and select the correct answer using the codes given below the list.

List-I List-II

A. Boltzmann constant I. $[M L^{2} T^{- 1}]$

B. Coefficient of viscosity II. $[M L^{- 1} T^{- 1}]$

C. Planck constant III. $[M L T^{- 3} K^{- 1}]$

D. Thermal conductivity IV. $[M L^{2} T^{- 2} K^{- 1}]$

Choose the correct answer from the options given below:

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

1

2

3

4

(3)

conceptual

Topic Question Set

Q 13 :

In an electromagnetic system, the quantity representing the ratio of electric flux and magnetic flux has dimension of $M^{P} L^{Q} T^{R} A^{S}$ , where value of 'Q' and 'R' are [2025]

Q 14 :

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: [2025]

Q 15 :

The dimension of $\sqrt{\frac{μ_{0}}{ε_{0}}}$ is equal to that of : ( $μ_{0}$ = Vacuum permeability and $ε_{0}$ = Vacuum permittivity) [2025]

Q 16 :

Match the List-I with List-II.

List-I List-II

(A) Mass density (I) $[M L^{2} T^{- 3}]$

(B) Impulse (II) $[M L T^{- 1}]$

(C) Power (III) $[M L^{2} T^{0}]$

(D) Moment of inertia (IV) $[M L^{- 3} T^{0}]$

Choose the correct answer from the options given below : [2025]

Q 17 :

In a measurement, it is asked to find modulus of elasticity per unit torque applied on the system. The measured quantity has dimension of $[M^{a} L^{b} T^{c}]$ . If b = –3, the value of c is __________. [2025]

Q 18 :

The frequency (v) of an oscillating liquid drop may depend upon radius (r) of the drop, density ( $ρ$ ) of liquid and the surface tension (s) of the liquid as : $v = r^{a} ρ^{b} s^{c}$ . The values of a, b and c respectively are [2023]

Q 19 :

Match the following Column – I with Column – II

Column –I Column – II

A. Surface tension I. $kg m^{- 1} s^{- 1}$

B. Pressure II. $kg {ms}^{- 1}$

C. Viscosity III. $kg m^{- 1} s^{- 2}$

D. Impulse IV. $kg s^{- 2}$

Choose the correct answer from the options given below: [2023]

Q 23 :

Match List I with List II.

List I List II

A. Torque I. $kg m^{- 1} s^{- 2}$

B. Energy density II. $kg {ms}^{- 1}$

C. Pressure gradient III. $kg m^{- 2} s^{- 2}$

D. Impulse IV. $kg m^{2} s^{- 2}$

Choose the correct answer from the options given below: [2023]

Q 24 :

Match List I with List II.

List I List II

A. Angular momentum I. [ $M L^{2} T^{- 2}$ ]

B. Torque II. [ $M L^{- 2} T^{- 2}$ ]

C. Stress III. [ $M L^{2} T^{- 1}$ ]

D. Pressure gradient IV. [ $M L^{- 1} T^{- 2}$ ]

Choose the correct answer from the options given below: [2023]

Q 25 :

$(P + \frac{a}{V^{2}}) (V - b) = R T$ 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 [2023]

Q 26 :

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: [2023]

Q 27 :

If R, $X_{L}$ and $X_{C}$ represent resistance, inductive reactance and capacitive reactance. Then which of the following is dimensionless: [2023]

Q 29 :

Dimension of $\frac{1}{μ_{0} ε_{0}}$ should be equal to [2023]

Q 30 :

Match List I with List II

List I List II

A. Torque I. $M L^{- 2} T^{- 2}$

B. Stress II. $M L^{2} T^{- 2}$

C, Pressure gradient III. $M L^{- 1} T^{- 1}$

D. Coefficient of viscosity IV. $M L^{- 1} T^{- 2}$

Choose the correct answer from the options given below: [2023]

Q 31 :

If force (F), velocity (V) and time (T) are considered as fundamental physical quantity, then dimensional formula of density will be [2023]

Q 32 :

Match List-I with List-II

List I List II

A. Spring constant I. ( $T^{- 1}$ )

B. Angular speed II. ( $M T^{- 2}$ )

C. Angular momentum III. ( $M L^{2}$ )

D. Moment of Inertia IV. ( $M L^{2} T^{- 1}$ )

Choose the correct answer from the options given below: [2023]

Q 33 :

In the equation $[X + \frac{a}{Y^{2}}] [Y - b] = R T$ , X is pressure, Y is volume, R is universal gas constant and T is temperature. The physical quantity a equivalent to the ratio $\frac{a}{b}$ is [2023]

Q 34 :

The speed of a wave produced in water is given by $v = λ^{a} g^{b} ρ^{c}$ . When $λ$ , g and $ρ$ are wavelength of wave, acceleration due to gravity and density of water respectively. The values of a, b and c respectively, are [2023]

Q 35 :

Consider a modified Bernoulli equation.

$(P + \frac{A}{B t^{2}}) + ρ g (h + B t) + \frac{1}{2} ρ v^{2} = constant .$

If t has the dimension of time then the dimensions of A and B are , respectively. [2026]

Q 39 :

The force of interaction between two atoms is given by $F = α β \exp (- \frac{x^{2}}{α k T})$ ; where $' x'$ is the distance, $' k'$ is the Boltzmann constant and T is temperature and $α$ and $β$ are two constants. The dimensions of $β$ is:

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	List I		List II
(A)	Gravitational constant	(I)	[ $L T^{- 2}$ ]
(B)	Gravitational potential energy	(II)	[ $L^{2} T^{- 2}$ ]
(C)	Gravitational potential	(III)	[ $M L^{2} T^{- 2}$ ]
(D)	Acceleration due to gravity	(IV)	[ $M^{- 1} L^{3} T^{- 2}$ ]

	List I		List II
(A)	Boltzmann constant	(I)	$M L^{2} T^{- 1}$
(B)	Coefficient of viscosity	(II)	$M L T^{- 3} K^{- 1}$
(C)	Planck's constant	(III)	$M L^{2} T^{- 2} K^{- 1}$
(D)	Thermal conductivity	(IV)	$M L^{- 1} T^{- 1}$

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

	Column –I		Column – II
A.	Surface tension	I.	$kg m^{- 1} s^{- 1}$
B.	Pressure	II.	$kg {ms}^{- 1}$
C.	Viscosity	III.	$kg m^{- 1} s^{- 2}$
D.	Impulse	IV.	$kg s^{- 2}$

	List I		List II
A.	Young's Modulus (Y)	I.	[ $M L^{- 1} T^{- 1}$ ]
B.	Co-efficient of Viscosity ( $η$ )	II.	[ $M L^{2} T^{- 1}$ ]
C.	Planck's Constant (h)	III.	[ $M L^{- 1} T^{- 2}$ ]
D.	Work Function ( $ϕ$ )	IV.	[ $M L^{2} T^{- 2}$ ]

	List I		List II
A.	Torque	I.	$kg m^{- 1} s^{- 2}$
B.	Energy density	II.	$kg {ms}^{- 1}$
C.	Pressure gradient	III.	$kg m^{- 2} s^{- 2}$
D.	Impulse	IV.	$kg m^{2} s^{- 2}$

	List I		List II
A.	Angular momentum	I.	[ $M L^{2} T^{- 2}$ ]
B.	Torque	II.	[ $M L^{- 2} T^{- 2}$ ]
C.	Stress	III.	[ $M L^{2} T^{- 1}$ ]
D.	Pressure gradient	IV.	[ $M L^{- 1} T^{- 2}$ ]

	List - I		List - II
A.	Magnetic induction	I.	${MLT}^{- 2} A^{- 2}$
B.	Magnetic flux	II.	${ML}^{2} T^{- 2} A^{- 2}$
C.	Magnetic permeability	III.	${ML}^{0} T^{- 2} A^{- 1}$
D.	Self inductance	IV.	${ML}^{2} T^{- 2} A^{- 1}$

	List-I		List-II
A.	Boltzmann constant	I.	$[M L^{2} T^{- 1}]$
B.	Coefficient of viscosity	II.	$[M L^{- 1} T^{- 1}]$
C.	Planck constant	III.	$[M L T^{- 3} K^{- 1}]$
D.	Thermal conductivity	IV.	$[M L^{2} T^{- 2} K^{- 1}]$

Topic Question Set

Q 12 : Match the List I with List II List I List II (A) Boltzmann constant (I) ML2T–1 (B) Coefficient of viscosity (II) MLT–3K–1 (C) Planck's constant (III) ML2T–2K–1 (D) Thermal conductivity (IV) ML–1T–1 Choose the correct answer from the options given below: [2025]

Q 13 : In an electromagnetic system, the quantity representing the ratio of electric flux and magnetic flux has dimension of MPLQTRAS, where value of 'Q' and 'R' are [2025]

Q 14 : In an electromagnetic system, a quantity defined as the ratio of electric dipole moment and magnetic dipole moment has dimension of [MPLQTRAS]. The value of P and Q are: [2025]

Q 15 : The dimension of μ0ε0 is equal to that of : (μ0 = Vacuum permeability and ε0 = Vacuum permittivity) [2025]

Q 16 : Match the List-I with List-II. List-I List-II (A) Mass density (I) [ML2T-3] (B) Impulse (II) [MLT-1] (C) Power (III) [ML2T0] (D) Moment of inertia (IV) [ML-3T0] Choose the correct answer from the options given below : [2025]

Q 17 : In a measurement, it is asked to find modulus of elasticity per unit torque applied on the system. The measured quantity has dimension of [MaLbTc]. If b = –3, the value of c is __________. [2025]

Q 18 : The frequency (v) of an oscillating liquid drop may depend upon radius (r) of the drop, density (ρ) of liquid and the surface tension (s) of the liquid as : v=raρbsc. The values of a, b and c respectively are [2023]

Q 19 : Match the following Column – I with Column – II Column –I Column – II A. Surface tension I. kg m–1s–1 B. Pressure II. kg ms–1 C. Viscosity III. kg m–1s–2 D. Impulse IV. kg s–2 Choose the correct answer from the options given below: [2023]

Q 20 : Match List I with List II List I List II A. Young's Modulus (Y) I. [ML–1T–1] B. Co-efficient of Viscosity (η) 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: [2023]

Q 21 : Match List I with List II: List I List II A. Pressure gradient I. [M0L2T–2] B. Energy density II. [M1L–1T–2] C. Electric field III. [M1L–2T–2] D. Latent heat IV. [M1L1T–3A–1] Choose the correct answer from the options given below: [2023]

Q 22 : The equation of a circle is given by x2+y2=a2, 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+(y–tB)2=a2 The dimensions of t is given as [T–1]. [2023]

Q 23 : Match List I with List II. List I List II A. Torque I. kg m–1s–2 B. Energy density II. kg ms–1 C. Pressure gradient III. kg m–2s–2 D. Impulse IV. kg m2s–2 Choose the correct answer from the options given below: [2023]

Q 24 : Match List I with List II. List I List II A. Angular momentum I. [ML2T–2] B. Torque II. [ML–2T–2] C. Stress III. [ML2T–1] D. Pressure gradient IV. [ML–1T–2] Choose the correct answer from the options given below: [2023]

Q 25 : (P+aV2)(V–b)=RT 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 b2a, will be [2023]

Q 26 : 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: [2023]

Q 27 : If R, XL and XC represent resistance, inductive reactance and capacitive reactance. Then which of the following is dimensionless: [2023]

Q 28 : Match List I with List II List I List II A. Planck's constant (h) I, [M1L2T–2] B. Stopping potential (VS) II. [M1L1T–1] C. Work function (ϕ) III. [M1L2T–1] D. Momentum (p) IV. [M1L2T–3A–1] Choose the correct answer from the options given below: [2023]

Q 29 : Dimension of 1μ0ε0 should be equal to [2023]

Q 30 : Match List I with List II 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: [2023]

Q 31 : If force (F), velocity (V) and time (T) are considered as fundamental physical quantity, then dimensional formula of density will be [2023]

Q 32 : Match List-I with List-II List I List II A. Spring constant I. (T–1) B. Angular speed II. (MT–2) C. Angular momentum III. (ML2) D. Moment of Inertia IV. (ML2T–1) Choose the correct answer from the options given below: [2023]

Q 33 : In the equation [X+aY2][Y–b]=RT, X is pressure, Y is volume, R is universal gas constant and T is temperature. The physical quantity a equivalent to the ratio ab is [2023]

Q 34 : The speed of a wave produced in water is given by v=λagbρc. When λ, g and ρ are wavelength of wave, acceleration due to gravity and density of water respectively. The values of a, b and c respectively, are [2023]

Q 35 : Consider a modified Bernoulli equation. (P+ABt2)+ρg(h+Bt)+12ρv2=constant. If t has the dimension of time then the dimensions of A and B are ________, ________ respectively. [2026]

Q 36 : Match the LIST-I with LIST-II [2026] List - I List - II A. Magnetic induction I. MLT-2A-2 B. Magnetic flux II. ML2T-2A-2 C. Magnetic permeability III. ML0T-2A-1 D. Self inductance IV. ML2T-2A-1 Choose the correct answer from the options given below:

Q 37 : Match the LIST-I with LIST-II: [2026] List - I List - II A. Spring constant I. ML2T-2K-1 B. Thermal conductivity II. ML0T-2 C. Boltzmann constant III. ML2T-3A-2 D. Inductive reactance IV. MLT-3K-1 Choose the correct answer from the options given below:

Q 38 : Match List – I with List – II [2026] List – I List – II A. Coefficient of viscosity I. [ML⁻¹T⁻²] B. Surface tension II. [ML²T⁻²] C. Pressure III. [ML⁰T⁻²] D. Surface energy IV. [ML⁻¹T⁻¹] Choose the correct answer from the options given below :

Q 39 : The force of interaction between two atoms is given by F=αβexp(-x2αkT); where 'x' is the distance, 'k' is the Boltzmann constant and T is temperature and α and β are two constants. The dimensions of β is:

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Q 13 :

In an electromagnetic system, the quantity representing the ratio of electric flux and magnetic flux has dimension of $M^{P} L^{Q} T^{R} A^{S}$ , where value of 'Q' and 'R' are [2025]

Q 14 :

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: [2025]

Q 15 :

The dimension of $\sqrt{\frac{μ_{0}}{ε_{0}}}$ is equal to that of : ( $μ_{0}$ = Vacuum permeability and $ε_{0}$ = Vacuum permittivity) [2025]

Q 16 :

Match the List-I with List-II.

List-I List-II

(A) Mass density (I) $[M L^{2} T^{- 3}]$

(B) Impulse (II) $[M L T^{- 1}]$

(C) Power (III) $[M L^{2} T^{0}]$

(D) Moment of inertia (IV) $[M L^{- 3} T^{0}]$

Choose the correct answer from the options given below : [2025]

Q 17 :

In a measurement, it is asked to find modulus of elasticity per unit torque applied on the system. The measured quantity has dimension of $[M^{a} L^{b} T^{c}]$ . If b = –3, the value of c is __________. [2025]

Q 18 :

The frequency (v) of an oscillating liquid drop may depend upon radius (r) of the drop, density ( $ρ$ ) of liquid and the surface tension (s) of the liquid as : $v = r^{a} ρ^{b} s^{c}$ . The values of a, b and c respectively are [2023]

Q 19 :

Match the following Column – I with Column – II

Column –I Column – II

A. Surface tension I. $kg m^{- 1} s^{- 1}$

B. Pressure II. $kg {ms}^{- 1}$

C. Viscosity III. $kg m^{- 1} s^{- 2}$

D. Impulse IV. $kg s^{- 2}$

Choose the correct answer from the options given below: [2023]

Q 23 :

Match List I with List II.

List I List II

A. Torque I. $kg m^{- 1} s^{- 2}$

B. Energy density II. $kg {ms}^{- 1}$

C. Pressure gradient III. $kg m^{- 2} s^{- 2}$

D. Impulse IV. $kg m^{2} s^{- 2}$

Choose the correct answer from the options given below: [2023]

Q 24 :

Match List I with List II.

List I List II

A. Angular momentum I. [ $M L^{2} T^{- 2}$ ]

B. Torque II. [ $M L^{- 2} T^{- 2}$ ]

C. Stress III. [ $M L^{2} T^{- 1}$ ]

D. Pressure gradient IV. [ $M L^{- 1} T^{- 2}$ ]

Choose the correct answer from the options given below: [2023]

Q 25 :

$(P + \frac{a}{V^{2}}) (V - b) = R T$ 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 [2023]

Q 26 :

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: [2023]

Q 27 :

If R, $X_{L}$ and $X_{C}$ represent resistance, inductive reactance and capacitive reactance. Then which of the following is dimensionless: [2023]

Q 29 :

Dimension of $\frac{1}{μ_{0} ε_{0}}$ should be equal to [2023]

Q 30 :

Match List I with List II

List I List II

A. Torque I. $M L^{- 2} T^{- 2}$

B. Stress II. $M L^{2} T^{- 2}$

C, Pressure gradient III. $M L^{- 1} T^{- 1}$

D. Coefficient of viscosity IV. $M L^{- 1} T^{- 2}$

Choose the correct answer from the options given below: [2023]

Q 31 :

If force (F), velocity (V) and time (T) are considered as fundamental physical quantity, then dimensional formula of density will be [2023]

Q 32 :

Match List-I with List-II

List I List II

A. Spring constant I. ( $T^{- 1}$ )

B. Angular speed II. ( $M T^{- 2}$ )

C. Angular momentum III. ( $M L^{2}$ )

D. Moment of Inertia IV. ( $M L^{2} T^{- 1}$ )

Choose the correct answer from the options given below: [2023]

Q 33 :

In the equation $[X + \frac{a}{Y^{2}}] [Y - b] = R T$ , X is pressure, Y is volume, R is universal gas constant and T is temperature. The physical quantity a equivalent to the ratio $\frac{a}{b}$ is [2023]

Q 34 :

The speed of a wave produced in water is given by $v = λ^{a} g^{b} ρ^{c}$ . When $λ$ , g and $ρ$ are wavelength of wave, acceleration due to gravity and density of water respectively. The values of a, b and c respectively, are [2023]

Q 35 :

Consider a modified Bernoulli equation.

$(P + \frac{A}{B t^{2}}) + ρ g (h + B t) + \frac{1}{2} ρ v^{2} = constant .$

If t has the dimension of time then the dimensions of A and B are , respectively. [2026]

Q 39 :

The force of interaction between two atoms is given by $F = α β \exp (- \frac{x^{2}}{α k T})$ ; where $' x'$ is the distance, $' k'$ is the Boltzmann constant and T is temperature and $α$ and $β$ are two constants. The dimensions of $β$ is: