For an experimental expression y=(32.3x1125/27.4) , where all the digits are significant. Then to report the value of y…

Year	2026	2025	2024	Total
Questions	14	22	14	50

Q jee_main_2025_02_april_morning Dimensional Analysis

Match List-I with List-II.

array|l|l| List-I & List-II (A) Coefficient of viscosity & (I) [ML⁰T⁻³] (B) Intensity of wave & (II) [ML⁻²T⁻²] (C) Pressure gradient & (III) [M⁻¹LT²] (D) Compressibility & (IV) [ML⁻¹T⁻¹] array

Choose the correct answer from the options given below:

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

Solution

Related Formula

Definitions of physical quantities:

Viscosity: $F = -η A (dv)/(dx)$ $F = -\eta A \frac{dv}{dx}$
Intensity: $I = PowerArea$ $I = \frac{\text{Power}}{\text{Area}}$
Pressure gradient: $(dP)/(dx)$ $\frac{dP}{dx}$
Compressibility: $K = (1)/(B) = StrainStress$

Core Logic

Let's calculate each dimensional formula:

Coefficient of viscosity ( $η$ $\eta$):

[η] = ([F])/([A][(dv)/(dx)]) = M L T⁻²L² ( L T⁻¹L) = M L⁻¹T⁻¹

Matches (IV).

Intensity of wave ( $I$ $I$):

[I] = [Power][Area] = M L²T⁻³L² = M T⁻³ = M L⁰T⁻³

Matches (I).

Pressure gradient ( $(dP)/(dx)$ $\frac{dP}{dx}$):

[(dP)/(dx)] = [Pressure][Length] = M L⁻¹T⁻²L = M L⁻²T⁻²

Matches (II).

Compressibility ( $K$ $K$):

Compressibility is the inverse of Bulk Modulus:

[K] = 1[Pressure] = 1M L⁻¹T⁻² = M⁻¹LT²

Matches (III).

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

Step 1: Final Conclusion

The correct option is (2).

Pattern Recognition

Target basic matching terms first: Compressibility is the reciprocal of pressure, giving $[M⁻¹LT²]$ $[M^{-1}LT^2]$. Wave intensity has units of power per unit area, giving $[MT⁻³]$ $[MT^{-3}]$. This immediately isolates option (2).

Chapter Mix

Class 11 Physics: Units and Measurements Class 11 Physics: Mechanical Properties of Fluids Class 11 Physics: Mechanical Properties of Solids

Q2 jee_main_2025_02_april_morning Dimensional Analysis

The equation for real gas is given by

(P + (a)/(V²))(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:

A. Planck's constant
B. Compressibility
C. Strain
D. Energy density

Solution

Related Formula

By the principle of dimensional homogeneity, terms added or subtracted must have the same dimensions:

[P] = [(a)/(V²)] [a] = [P][V]²

$[V] = [b]$ $[V] = [b]$

Core Logic

Let's find the dimensional formula of the quantities:

Pressure $P$ $P$:

[P] = M L⁻¹T⁻²

Volume $V$ $V$:

[V] = L³

Substituting these to find $[a]$ $[a]$ and $[b]$ $[b]$:

[a] = (M L⁻¹T⁻²)(L⁶) = M L⁵T⁻²

[b] = L³ [b⁻²] = L⁻⁶

Now, compute the dimensions of $ab⁻²$ $ab^{-2}$:

[ab⁻²] = (M L⁵T⁻²)(L⁻⁶) = M L⁻¹T⁻²

This matches the dimensions of pressure.

Let's evaluate the options:

Planck's constant: $[h] = M L²T⁻¹$ $[h] = \text{M L}^2\text{T}^{-1}$
Compressibility: $[β] = M⁻¹LT²$
Strain: dimensionless
Energy density (energy per unit volume):

[(E)/(V)] = M L²T⁻²L³ = M L⁻¹T⁻²

Step 1: Final Conclusion

Therefore, the dimension of $ab⁻²$ $ab^{-2}$ is equivalent to that of Energy density.

Pattern Recognition

By writing the relation directly as $[ab⁻²] = ([a])/([b]²)$ $[ab^{-2}] = \frac{[a]}{[b]^2}$, and noting $[a] = [P][V]²$ $[a] = [P][V]^2$ and $[b] = [V]$ $[b] = [V]$, we get $[ab⁻²] = ([P][V]²)/([V]²) = [P]$ (Pressure). Since pressure and energy density have identical dimensions, the answer is immediately Energy density.

Chapter Mix

Class 11 Physics: Units and Measurements Class 11 Physics: Kinetic Theory of Gases

Q jee_main_2025_03_april_evening Dimensional Analysis and Constants

Match the LIST-I with LIST-II

LIST-I	LIST-II
A. Boltzmann constant	I. $ML²T⁻¹$ $ML^2T^{-1}$
B. Coefficient of viscosity	II. $MLT⁻³K⁻¹$ $MLT^{-3}K^{-1}$
C. Planck's constant	III. $ML²T⁻²K⁻¹$ $ML^2T^{-2}K^{-1}$
D. Thermal conductivity	IV. $ML⁻¹T⁻¹$ $ML^{-1}T^{-1}$

Choose the correct answer from the options given below :

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

Solution

Related Formula

Formulas to find dimensional formulas:

Boltzmann constant:

kB = EnergyTemperature

Coefficient of viscosity:

η = (F)/(A (dv)/(dx))

Planck's constant:

h = (E)/(ν)

Thermal conductivity:

(dQ)/(dt) = K A (dT)/(dx) \Rightarrow K = Heat flow \cdot thicknessArea \cdot Temperature difference

Core Logic

Evaluate each constant individually:

Step 1: Dimensions of Boltzmann constant ( $kB$ $k_B$)

[kB] = [ML²T⁻²][K] = [ML²T⁻²K⁻¹] \Rightarrow Matches III

Step 2: Dimensions of Coefficient of viscosity ( $η$ $\eta$)

[η] = [MLT⁻²][L²] [T⁻¹] = [ML⁻¹T⁻¹] \Rightarrow Matches IV

Step 3: Dimensions of Planck's constant ( $h$ $h$)

[h] = [ML²T⁻²][T⁻¹] = [ML²T⁻¹] \Rightarrow Matches I

Step 4: Dimensions of Thermal conductivity ( $K$ $K$)

[K] = [ML²T⁻³] [L][L²] [K] = [MLT⁻³K⁻¹] \Rightarrow Matches II

This sequence yields A-III, B-IV, C-I, D-II, matching Option (1).

Pattern Recognition

To solve matching sets efficiently, search for the most recognizable dimensions first. Planck's constant $h$ $h$ ( $ML²T⁻¹$ $ML^2T^{-1}$) and viscosity coefficient $η$ $\eta$ ( $ML⁻¹T⁻¹$ $ML^{-1}T^{-1}$) are highly unique and usually resolve the options instantly.

Chapter Mix

Class 11 Physics: Units and Measurements

Q25 jee_main_2025_03_april_evening Error Analysis

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

C = pq²r³\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 ________ \%.

Numerical Answer. Answer: 15 to 15

Solution

Related Formula

For a physical quantity defined by algebraic powers $C = (p^a q^b)/(r^c s^d)$ $C = \frac{p^a q^b}{r^c s^d}$, the maximum fractional error is calculated by summing absolute scaled fractional errors:

(Δ C)/(C) = a (Δ p)/(p) + b (Δ q)/(q) + c (Δ r)/(r) + d (Δ s)/(s)

Expressed as percentages:

% error in C = a(% error in p) + b(% error in q) + c(% error in r) + d(% error in s)

Core Logic

Given expression:

C = p¹ q² r⁻³ s-1/2

Max fractional error equation:

(Δ C)/(C) = 1 ((Δ p)/(p)) + 2 ((Δ q)/(q)) + 3 ((Δ r)/(r)) + (1)/(2) ((Δ s)/(s))

Step 1: Calculate the total percentage error

Substitute the individual percentage errors:

Error in $p = 1%$ $p = 1\%$
Error in $q = 2%$ $q = 2\%$
Error in $r = 3%$ $r = 3\%$
Error in $s = 2%$ $s = 2\%$

% error in C = 1(1%) + 2(2%) + 3(3%) + (1)/(2)(2%)

% error in C = 1% + 4% + 9% + 1% = 15%

The total percentage error in $C$ $C$ is $15%$ $15\%$.

Pattern Recognition

In error propagation, individual errors always combine constructively to produce the maximum possible uncertainty limit. Hence, negative powers (like division by $r³$ $r^3$ or $s1/2$ $s^{1/2}$) are integrated using positive coefficients during maximum absolute error summation.

Chapter Mix

Class 11 Physics: Units and Measurements

Q10 jee_main_2025_08_april_evening Error Analysis

A quantity

Q

is formulated as

X⁻²Y(3)/(2)Z-(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:

A. $0.1$ $0.1$
B. $0.8$ $0.8$
C. $0.7$ $0.7$
D. $0.6$ $0.6$

Solution

Related Formula

For a quantity $Q = X^a Y^b Z^c$ $Q = X^a Y^b Z^c$, the maximum fractional error is:

(Δ Q)/(Q) = |a| (Δ X)/(X) + |b| (Δ Y)/(Y) + |c| (Δ Z)/(Z)

where, $(Δ X)/(X), (Δ Y)/(Y), (Δ Z)/(Z)$ are fractional errors of individual variables

Core Logic

Given formula: $Q = X⁻² Y3/2 Z-2/5$ $Q = X^{-2} Y^{3/2} Z^{-2/5}$.

Identify the absolute exponents:

$|a| = |-2| = 2$ $|a| = |-2| = 2$
$|b| = |(3)/(2)| = (3)/(2)$
$|c| = |-(2)/(5)| = (2)/(5)$

Now write the error expression:

(Δ Q)/(Q) = 2 (Δ X)/(X) + (3)/(2) (Δ Y)/(Y) + (2)/(5) (Δ Z)/(Z)

Substitute the given values:

$(Δ X)/(X) = 0.1$ $\frac{\Delta X}{X} = 0.1$
$(Δ Y)/(Y) = 0.2$ $\frac{\Delta Y}{Y} = 0.2$
$(Δ Z)/(Z) = 0.5$ $\frac{\Delta Z}{Z} = 0.5$

Step 1: Compute Maximum Fractional Error

Calculate term by term:

(Δ Q)/(Q) = 2 (0.1) + (3)/(2) (0.2) + (2)/(5) (0.5)

(Δ Q)/(Q) = 0.2 + 0.3 + 0.2 = 0.7

Pattern Recognition

Sees: Exponential algebraic relation for errors. Trap: Exponents are negative, but maximum error is cumulative. Always take the absolute value of exponents when summing errors! ✓

Chapter Mix

Class 11 Physics: Units and Measurements

For an experimental expression y=(32.3×1125)/(27.4)$y=\frac{32.3\times1125}{27.4}$ , where all the digits are significant. Then to report the value of y we should write :-

Solution & Explanation

Related Formula

Core Logic

Step 1: Rounding Off

Pattern Recognition

Chapter Mix

Reference Study Guides

More Units and Measurements Previous-Year Questions — Page 4

Solution

Related Formula

Core Logic

Step 1: Final Conclusion

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 1: Final Conclusion

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 1: Dimensions of Boltzmann constant (kB$k_B$)

Step 2: Dimensions of Coefficient of viscosity (η$\eta$)

Step 3: Dimensions of Planck's constant (h$h$)

Step 4: Dimensions of Thermal conductivity (K$K$)

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 1: Calculate the total percentage error

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 1: Compute Maximum Fractional Error

Pattern Recognition

Chapter Mix

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For an experimental expression $y=(32.3\times1125)/(27.4)$ $y=\frac{32.3\times1125}{27.4}$ , where all the digits are significant. Then to report the value of y we should write :-

Step 1: Dimensions of Boltzmann constant ( $kB$ $k_B$)

Step 2: Dimensions of Coefficient of viscosity ( $η$ $\eta$)

Step 3: Dimensions of Planck's constant ( $h$ $h$)

Step 4: Dimensions of Thermal conductivity ( $K$ $K$)