Consider two physical quantities A and B related to each other as E = (B - x2/At) where E, x and t have dimensions of energy, length and time respectively. The dimension of AB is

By the Principle of Homogeneity, terms added or subtracted must have the same dimensions: [B] = [x2] Known dimensional formulas: Length x [L] Energy E [ML2T-2] Time t [T] Since x2 is subtracted from B: From the equation E = (B - x2/At): Identify sums/differences first to instantly isolate B. Once [B] is fixed, the entire numerator is just L2. Swap out variables to isolate [A]. Combining is just standard exponent addition. Class 11 Physics: Units and Measurements

Consider two physical quantities A and B related to each other as E = (B - x2/At) where E, x and t have dimensions of energy, length and time respectively. The dimension of AB is

By the Principle of Homogeneity, terms added or subtracted must have the same dimensions: [B] = [x2] Known dimensional formulas: Length x [L] Energy E [ML2T-2] Time t [T] Since x2 is subtracted from B: From the equation E = (B - x2/At): Identify sums/differences first to instantly isolate B. Once [B] is fixed, the entire numerator is just L2. Swap out variables to isolate [A]. Combining is just standard exponent addition. Class 11 Physics: Units and Measurements

Consider two physical quantities A and B related to each other as E = (B - x2/At) where E, x and t have dimensions of energy, length and time respectively. The dimension of AB is

By the Principle of Homogeneity, terms added or subtracted must have the same dimensions: [B] = [x2] Known dimensional formulas: Length x [L] Energy E [ML2T-2] Time t [T] Since x2 is subtracted from B: From the equation E = (B - x2/At): Identify sums/differences first to instantly isolate B. Once [B] is fixed, the entire numerator is just L2. Swap out variables to isolate [A]. Combining is just standard exponent addition. Class 11 Physics: Units and Measurements

Consider two physical quantities A and B related to each other as E = (B - x2/At) where E, x and t have dimensions of energy, length and time respectively. The dimension of AB is

By the Principle of Homogeneity, terms added or subtracted must have the same dimensions: [B] = [x2] Known dimensional formulas: Length x [L] Energy E [ML2T-2] Time t [T] Since x2 is subtracted from B: From the equation E = (B - x2/At): Identify sums/differences first to instantly isolate B. Once [B] is fixed, the entire numerator is just L2. Swap out variables to isolate [A]. Combining is just standard exponent addition. Class 11 Physics: Units and Measurements

Consider two physical quantities A and B related to each other as E = (B - x2/At) where E, x and t have dimensions of…

Year	2026	2025	2024	Total
Questions	14	22	14	50

Year

2026

2025

2024

Total

Questions

Solution & Explanation

Related Formula

By the Principle of Homogeneity, terms added or subtracted must have the same dimensions: $[B] = [x²]$ $[B] = [x^2]$

Core Logic

Known dimensional formulas: Length $x \to [L]$ $x \to [L]$ Energy $E \to [ML²T⁻²]$ $E \to [ML^2T^{-2}]$ Time $t \to [T]$ $t \to [T]$

Step 1: Dimension of B

Since $x²$ $x^2$ is subtracted from $B$ $B$:

[B] = [x²] = [L²]

Step 2: Dimension of A

From the equation $E = (B - x²)/(At)$ $E = \frac{B - x^2}{At}$:

[A] = ([B - x²])/([E][t])

[A] = [L²][ML²T⁻²][T] = [L²][ML²T⁻¹]

[A] = [M⁻¹T¹]

Step 3: Dimension of AB

[AB] = [A] \times [B]

[AB] = [M⁻¹T¹] \times [L²]

[AB] = [L² M⁻¹ T¹]

Pattern Recognition

Identify sums/differences first to instantly isolate $B$ $B$. Once $[B]$ $[B]$ is fixed, the entire numerator is just $L²$ $L^2$. Swap out variables to isolate $[A]$ $[A]$. Combining is just standard exponent addition.

Chapter Mix

Class 11 Physics: Units and Measurements

More Units and Measurements Previous-Year Questions — Page 8

Q jee_main_2025_29_jan_morning Dimensional Homogeneity

The expression given below shows the variation of velocity (v) with time (t),

v = At² + (Bt)/(C + t)

. The dimension of ABC is:

A. $[M⁰ ~L² T⁻³]$
B. $[M⁰ ~L¹ T⁻³]$
C. $[M⁰L¹T⁻²]$
D. $[M⁰ ~L² T⁻²]$

Solution

Related Formula

[v] = [At²] = [(Bt)/(C+t)]

Core Logic

By the principle of dimensional homogeneity:

$[C] = [t] = [T]$ $[C] = [t] = [T]$
$[At²] = [v] [A][T²] = [LT⁻¹] [A] = [LT⁻³]$
$[(Bt)/(T)] = [v] [B] = [LT⁻¹]$

Step 1: Calculate Dimensions of ABC

[ABC] = [LT⁻³] \cdot [LT⁻¹] \cdot [T] = [L² T⁻³]

Pattern Recognition

Denominator terms matched first give $C$ $C$, tracking linear velocity units sets the balance for $A$ $A$ and $B$ $B$

Chapter Mix

Class 11 Physics: Units and Measurements

Q42 jee_main_2024_01_february_morning Vernier Calliper

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:

A. $(1)/(2)$ $\frac{1}{2}$
B. $(10)/(11)$ $\frac{10}{11}$
C. $(50)/(11)$ $\frac{50}{11}$
D. $(5)/(11)$ $\frac{5}{11}$

Solution

Related Formula

Least Count (LC) definition:

LC = 1 MSD - 1 VSD

Core Logic

From the problem:

10 MSD = 11 VSD 1 VSD = (10)/(11) MSD

Substitute into the Least Count formula:

LC = 1 MSD - (10)/(11) MSD = (1)/(11) MSD

Step 1: Calculate with Main Scale Units

Given that $1 MSD = 5 units$ $1\text{ MSD} = 5\text{ units}$:

LC = (1)/(11) \times 5 = (5)/(11) units

Pattern Recognition

Watch out for unconventional scale arrangements where $VSD > MSD$ $\text{VSD} > \text{MSD}$. The fundamental difference formula $1 MSD - 1 VSD$ $1\text{ MSD} - 1\text{ VSD}$ safely determines magnitudes without needing sign corrections.

Chapter Mix

Class 11 Physics: Units and Measurements

Q44 jee_main_2024_01_february_morning Error Analysis

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)~Ω

l = (15 \pm 0.2)~cm

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

A. 25.6%
B. 39.9%
C. 37.3%
D. 35.6%

Solution

Related Formula

Resistivity formula:

ρ = R(A)/(l) = R(π r²)/(l)

Maximum relative error equation:

(Δρ)/(ρ) = (Δ R)/(R) + 2(Δ r)/(r) + (Δ l)/(l)

Core Logic

Substitute the measured fractions:

(Δρ)/(ρ) = (10)/(100) + 2((0.05)/(0.35)) + (0.2)/(15)

(Δρ)/(ρ) = (1)/(10) + 2((1)/(7)) + (1)/(75)

(Δρ)/(ρ) = 0.1 + 0.2857 + 0.0133 = 0.399

Step 1: Convert to Percentage

% severance = 0.399 \times 100% = 39.9%

Pattern Recognition

The power of 2 in $r²$ $r^2$ doubles its fractional error impact, making it the most dominant source of error in this setup.

Chapter Mix

Class 11 Physics: Units and Measurements Class 12 Physics: Current Electricity

Q45 jee_main_2024_01_february_morning Dimensional Analysis

The dimensional formula of angular impulse is:

A. $[M L⁻²T⁻¹]$
B. $[M L²T⁻²]$
C. $[M L T⁻¹]$
D. $[M L²T⁻¹]$

Solution

Related Formula

Angular Impulse equation:

Angular Impulse = \int τ dt = Δ L

Angular Momentum (L) = mvr

Core Logic

By the impulse-momentum theorem for rotation, angular impulse equals the net change in angular momentum.

Extract dimensions from $[mvr]$ $[mvr]$:

[m] = [M]

[v] = [LT⁻¹]

[r] = [L]

Step 1: Combine Dimensions

[Angular Impulse] = [M] \times [LT⁻¹] \times [L] = [ML²T⁻¹]

Pattern Recognition

Angular impulse shares the exact same dimensional signature as Planck's constant ( $h$ $h$) and angular momentum ( $L$ $L$).

Chapter Mix

Class 11 Physics: Units and Measurements Class 11 Physics: System of Particles and Rotational Motion

Q33 jee_main_2024_29_january_evening Error Analysis

A physical quantity

Q

is found to depend on quantities

a, b, c

by the relation

Q = (a⁴ b³)/(c²)

. The percentage error in

a, b

and

c

are

3%

4%

and

5%

respectively. Then, the percentage error in

Q

is:

A. $66%$ $66\%$
B. $43%$ $43\%$
C. $34%$ $34\%$
D. $14%$ $14\%$

Solution

Related Formula

For a quantity $Q = (a^x b^y)/(c^z)$ $Q = \frac{a^x b^y}{c^z}$, the maximum relative error is:

(Δ Q)/(Q) = x (Δ a)/(a) + y (Δ b)/(b) + z (Δ c)/(c)

Multiplying by $100$ $100$ gives the percentage error equation:

% error in Q = x(% error in a) + y(% error in b) + z(% error in c)

Core Logic

Given formula:

Q = (a⁴ b³)/(c²)

Applying the error propagation rule:

(Δ Q)/(Q) \times 100 = 4 ((Δ a)/(a) \times 100) + 3 ((Δ b)/(b) \times 100) + 2 ((Δ c)/(c) \times 100)

Step 1: Substitute the Percentage Errors

Substitute the given values:

$% error in a = 3%$ $\%\text{ error in } a = 3\%$
$% error in b = 4%$ $\%\text{ error in } b = 4\%$
$% error in c = 5%$ $\%\text{ error in } c = 5\%$

% error in Q = 4(3%) + 3(4%) + 2(5%)

% error in Q = 12% + 12% + 10% = 34%

Pattern Recognition

Error percentages always add up, weighted by their exponents in the mathematical expression, regardless of whether the variable is in the numerator or denominator.

Chapter Mix

Class 11 Physics: Units and Measurements

Consider two physical quantities A and B related to each other as E = (B - x²)/(At)$E = \frac{B - x^2}{At}$ where E, x and t have dimensions of energy, length and time respectively. The dimension of AB is

Solution & Explanation

Related Formula

Core Logic

Step 1: Dimension of B

Step 2: Dimension of A

Step 3: Dimension of AB

Pattern Recognition

Chapter Mix

Reference Study Guides

More Units and Measurements Previous-Year Questions — Page 8

Solution

Related Formula

Core Logic

Step 1: Calculate Dimensions of ABC

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 1: Calculate with Main Scale Units

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 1: Convert to Percentage

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 1: Combine Dimensions

Pattern Recognition

Chapter Mix

Solution

Related Formula

Core Logic

Step 1: Substitute the Percentage Errors

Pattern Recognition

Chapter Mix

More Units and Measurements Questions — jee_main_2024_31_jan_evening

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Consider two physical quantities A and B related to each other as $E = (B - x²)/(At)$ $E = \frac{B - x^2}{At}$ where E, x and t have dimensions of energy, length and time respectively. The dimension of AB is