QJEEM_2025_6: JEE Main 2025 Physics - Class 11 JEEM_2025_6 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, we use the definition of work in terms of vector dot products.

1. Physical Principle

Work done ($W$) by a constant force $\vec{F}$ over a displacement $\vec{d}$ is given by the dot product:

2. Given Vectors

1. Force: $\vec{F} = 2\hat{i} + b\hat{j} + \hat{k}$
2. Displacement: $\vec{d} = \hat{i} - 2\hat{j} - \hat{k}$

3. Dot Product Calculation

The dot product is calculated by multiplying the corresponding components ($\hat{i}, \hat{j}, \hat{k}$) and summing them:

Simplify the expression:

4. Solve for $b$

Since the problem states the work done is zero ($W = 0$):

Conclusion

The value of $b$ is $\displaystyle \frac{1}{2}$, which corresponds to Option (B).