QJEEM25Phe170: JEE Main 2025 Physics - Class 11 JEEM25Phe170 Solution: A force is applied on a particle and it undergoes a…

To determine the value of $b$ such that the work done on the particle is zero, follow these steps:

The work done by a force on a displacement is given by the dot product:

$\text{Work} = \overrightarrow{F} \cdot \overrightarrow{d}.$

Given the force

$\overrightarrow{F} = 2\hat{i} + b\hat{j} + \hat{k},$

and the displacement

$\overrightarrow{d} = \hat{i} - 2\hat{j} - \hat{k},$

compute the dot product:

$\overrightarrow{F} \cdot \overrightarrow{d} = (2)(1) + (b)(-2) + (1)(-1).$

Simplify the expression:

$\overrightarrow{F} \cdot \overrightarrow{d} = 2 - 2b - 1 = 1 - 2b.$

Since the work done is zero:

$1 - 2b = 0.$

Solve for $b$ :

$2b = 1,$

$b = \frac{1}{2}.$

Thus, the value of $b$ is $\frac{1}{2}$ , which corresponds to Option B.