1. Given data

• Mass of body: $m=4\,\text{kg}$
• Initial position: $P(3,4)$
• Final position after $4\,$s: $Q(6,10)$
• Force: $\vec F=(2\hat i+3\hat j)\,\text{N}$
• Time taken: $t=4\,$s

We need the ratio: $\text{average power} : \text{instantaneous power at } t=4\text{ s}$

2. Displacement from $P$ to $Q$

$\Delta \vec r=(6-3)\hat i+(10-4)\hat j=3\hat i+6\hat j$

3. Average power

Average power is $P_{\text{avg}}=\frac{\text{work done}}{\text{time}}$

Work done by constant force: $W=\vec F\cdot \Delta \vec r$

So, $W=(2\hat i+3\hat j)\cdot(3\hat i+6\hat j)=2\cdot 3+3\cdot 6=6+18=24\,\text{J}$

Hence, $P_{\text{avg}}=\frac{24}{4}=6\,\text{W}$

4. Find acceleration

Using Newton's second law, $\vec a=\frac{\vec F}{m}$

Thus, $\vec a=\frac{1}{4}(2\hat i+3\hat j)=\left(\frac12\hat i+\frac34\hat j\right)\,\text{m/s}^2$

5. Find initial velocity

Use $\Delta \vec r=\vec u t+\frac12\vec a t^2$

Given $t=4$ s, $3\hat i+6\hat j=4\vec u+\frac12\vec a(16)$ $3\hat i+6\hat j=4\vec u+8\vec a$

Now, $8\vec a=8\left(\frac12\hat i+\frac34\hat j\right)=4\hat i+6\hat j$

So, $3\hat i+6\hat j=4\vec u+4\hat i+6\hat j$

Therefore, $4\vec u=-\hat i$ $\vec u=-\frac14\hat i$

6. Velocity at the end of 4 s

$\vec v=\vec u+\vec a t$

So, $\vec v=-\frac14\hat i+4\left(\frac12\hat i+\frac34\hat j\right)$ $\vec v=-\frac14\hat i+2\hat i+3\hat j$ $\vec v=\frac74\hat i+3\hat j$

7. Instantaneous power at $t=4$ s

Instantaneous power is $P_{\text{inst}}=\vec F\cdot \vec v$

Thus, $P_{\text{inst}}=(2\hat i+3\hat j)\cdot\left(\frac74\hat i+3\hat j\right)$ $P_{\text{inst}}=2\cdot\frac74+3\cdot 3$ $P_{\text{inst}}=\frac72+9=\frac{25}{2}\,\text{W}$

8. Required ratio

$P_{\text{avg}}:P_{\text{inst}}=6:\frac{25}{2}=12:25$

So the required ratio is $\boxed{12:25}$

9. Comparison with options

Given options are:

• A: $4:3$
• B: $13:6$
• C: $1:2$
• D: $6:13$

Our derived ratio $12:25$ does not match any option.

10. Check stored answer

Stored correct answer: D ($6:13$)

But our calculated answer is $12:25$, so I do not agree with the stored answer.

A likely reason is that the question/options may contain a typo or some intended assumption is missing. Under the standard interpretation with constant force and motion under that force, the ratio is definitively $12:25$.