1. Given velocity of the particle

   $\vec v(t)=2t\,\hat i+3t^2\,\hat j$

   Mass of the particle:

   $m=500\text{ g}=0.5\text{ kg}$

2. Find acceleration

   Acceleration is the time derivative of velocity:

   $\vec a=\frac{d\vec v}{dt}$

   Differentiate each component:

   $a_x=\frac{d}{dt}(2t)=2$ $a_y=\frac{d}{dt}(3t^2)=6t$

   So,

   $\vec a=2\hat i+6t\hat j$

3. Acceleration at $t=1\text{ s}$

   $\vec a(1)=2\hat i+6\hat j$

4. Use Newton's second law

   $\vec F=m\vec a$

   Therefore,

   $\vec F=0.5(2\hat i+6\hat j)$

   $\vec F=\hat i+3\hat j$

5. Compare with given force

   Given:

   $\vec F=(\hat i+x\hat j)\text{ N}$

   Matching components,

   $x=3$

6. Option check

   • A: $2$ ❌
   • B: $4$ ❌
   • C: $6$ ❌
   • D: $3$ ✅

Hence, the correct answer is D.