JEE PYQ: Units & Measurements - Question ID 90a065751921 (JEE Main 2021)

Core Formula & Concept:

In dimensional analysis, every physical quantity can be expressed in terms of the fundamental dimensions: mass ($M$), length ($L$), and time ($T$). The key principle is that the arguments of trigonometric functions (like $\cos$ and $\sin$) must be dimensionless. This ensures that the functions yield pure numbers, which is essential for mathematical consistency.

Given the force equation: $F = A \cos(Bx) + C \sin(Dt)$

• The term $A \cos(Bx)$ implies that $Bx$ must be dimensionless.
• The term $C \sin(Dt)$ implies that $Dt$ must be dimensionless.

Force ($F$) has the dimensional formula $[M^1 L^1 T^{-2}]$. The coefficients $A$ and $C$ must also have the same dimensions as force since they multiply dimensionless trigonometric functions.

Step-by-Step Derivation:

Step 1: Determine the dimensions of $B$
The argument $Bx$ must be dimensionless. Since $x$ is displacement, its dimensions are $[L^1]$. Thus: $[B] \cdot [L^1] = [M^0 L^0 T^0] \implies [B] = [L^{-1}]$ Step 2: Determine the dimensions of $D$
The argument $Dt$ must be dimensionless. Since $t$ is time, its dimensions are $[T^1]$. Thus: $[D] \cdot [T^1] = [M^0 L^0 T^0] \implies [D] = [T^{-1}]$ Step 3: Determine the dimensions of $A$ and $C$
Since $F = A \cos(Bx) + C \sin(Dt)$ and $\cos(Bx)$ and $\sin(Dt)$ are dimensionless, the dimensions of $A$ and $C$ must match those of force: $[A] = [C] = [M^1 L^1 T^{-2}]$ Step 4: Compute the dimensions of ${{AD} \over B}$
Substitute the dimensions of $A$, $D$, and $B$: $\left[ \frac{AD}{B} \right] = \frac{[A] \cdot [D]}{[B]} = \frac{[M^1 L^1 T^{-2}] \cdot [T^{-1}]}{[L^{-1}]}$ Simplify the expression: $= \frac{[M^1 L^1 T^{-3}]}{[L^{-1}]} = [M^1 L^{1+1} T^{-3}] = [M^1 L^2 T^{-3}]$ Step 5: Match with the given options
The derived dimensional formula is $[M^1 L^2 T^{-3}]$, which corresponds to Option B.

Common Traps & Exam Tip:

1. Ignoring the dimensionless nature of trigonometric arguments: Students often forget that $\cos(Bx)$ and $\sin(Dt)$ must be dimensionless, leading to incorrect dimensions for $B$ and $D$.
2. Miscalculating the dimensions of $A$ and $C$: Some assume $A$ and $C$ are dimensionless or have different dimensions than force, which is incorrect.
3. Algebraic errors in dimensional simplification: When computing ${{AD} \over B}$, students may incorrectly add or subtract exponents, especially for length ($L$). For example, dividing by $[L^{-1}]$ is equivalent to multiplying by $[L^1]$, not subtracting.
4. Confusing the order of operations: Ensure that the dimensions of $A$, $D$, and $B$ are substituted correctly before simplifying. A common mistake is to misplace the exponents during multiplication or division.

Exam Tip: Always verify that the arguments of trigonometric, exponential, or logarithmic functions are dimensionless. This is a golden rule in dimensional analysis and will help you avoid critical errors.