JEE PYQ: Units & Measurements - Question ID 3c14278119bf (JEE Main 2024)

Core Formula & Concept:

In physics and engineering, a quantity is often expressed in scientific notation as $x = a \times 10^b$ where

• $a$ is a real number satisfying $1 \leq a < 10$, and
• $b$ is an integer (the exponent).

The order of magnitude of $x$ is defined as the power of ten that most closely approximates $x$. Concretely, the order of magnitude is the integer $b$ if $a < 5$, and $b+1$ if $a \geq 5$. In other words, $$\text{Order of magnitude} = \begin{cases} b, & \text{if } a < 5,\\ b+1, & \text{if } a \geq 5. \end{cases}$$

Step-by-Step Derivation:

Step 1: Recall the definition.
For $x = a \times 10^b$ with $1 \leq a < 10$, the order of magnitude is the integer $n$ such that $10^n \leq x < 10^{n+1}.$

Step 2: Split the range of $a$.

• If $1 \leq a < 5$, then $a \times 10^b < 5 \times 10^b \leq 10^b \times 10^{0.7} \approx 10^{b+0.7}$. Since $10^{b+0.7} < 10^{b+1}$, the interval $[10^b,\ 10^{b+1})$ is not crossed. Hence the order of magnitude is $b$.
• If $5 \leq a < 10$, then $a \times 10^b \geq 5 \times 10^b = 10^{b+0.7}$. This value already exceeds $10^b$ and reaches up to $10^{b+1}$. Therefore the order of magnitude jumps to $b+1$.

Step 3: Match with the given options.
The question asks when $b$ itself is the order of magnitude. From the above, this happens precisely when $a < 5$. Among the choices, only option B states “$b$ is order of magnitude for $a \leq 5$.” Although the strict inequality is $a < 5$, the option $a \leq 5$ is the closest match and is the one marked correct in the official key.

Common Traps & Exam Tip:

Trap 1: Many students confuse the condition $a \leq 5$ with $a < 5$. The official answer key uses $\leq$, so in the exam one must select B even though the strict mathematical condition is $a < 5$.
Trap 2: Another frequent error is to think that $b$ is always the order of magnitude regardless of $a$. Remember: the order of magnitude increases by one when $a$ crosses 5.
Exam Tip: Always write down the definition of order of magnitude before reading the options. This prevents misreading the question and helps you spot the correct condition quickly.