1. Interpret the table

The marks are: $2,\;3,\;5,\;7$

Their corresponding frequencies are: $x+1,\;2,\;2x-5,\;x^2-3x$

Since the total number of students is $20$, we have $(x+1)+2+(2x-5)+(x^2-3x)=20$

2. Form the equation

Simplify: $x+1+2+2x-5+x^2-3x=20$

Combine like terms: $x^2+(x+2x-3x)+(1+2-5)=20$ $x^2-2=20$ $x^2=22$

This gives $x=\pm \sqrt{22}$

Since frequencies must be non-negative, take $x=\sqrt{22}$

But now check the frequency $2x-5$: $2\sqrt{22}-5>0$ so this is valid.

3. Compute the mean

Mean is given by $\bar{x}=\frac{\sum f_i x_i}{\sum f_i}$

So, $\bar{x}=\frac{2(x+1)+3\cdot 2+5(2x-5)+7(x^2-3x)}{20}$

Expand numerator: $2x+2+6+10x-25+7x^2-21x$ $=7x^2-9x-17$

Now substitute $x^2=22$: $7x^2-9x-17=7(22)-9x-17=154-9x-17=137-9x$

Using $x=\sqrt{22}$, $\bar{x}=\frac{137-9\sqrt{22}}{20}$

Numerically, $\sqrt{22}\approx 4.69$ $137-9(4.69)\approx 137-42.21=94.79$ $\bar{x}\approx \frac{94.79}{20}\approx 4.74$

This does not match any option. So the table as read from the text is likely misparsed.

4. Resolve the intended frequencies from the formatting

The frequency row appears as: $(x+1),\;2,\;2x-5,\;x^2-3x$

Using total frequency $20$: $(x+1)+2+(2x-5)+(x^2-3x)=20$ which again gives $x^2=22$, leading to the same inconsistency.

So this cannot be the intended interpretation if one of the options is correct.

A common intended reading in such questions is: $\text{frequencies}=(x+1),\;2,\;(2x-5),\;x$ with total $20$.

Then, $(x+1)+2+(2x-5)+x=20$ $4x-2=20$ $4x=22$ $x=\frac{11}{2}$

Now mean: $\bar{x}=\frac{2(x+1)+3\cdot 2+5(2x-5)+7x}{20}$

Substitute $x=\frac{11}{2}$:

$2\left(\frac{11}{2}+1\right)+6+5(11-5)+7\cdot\frac{11}{2}$ $=2\cdot\frac{13}{2}+6+30+\frac{77}{2}$ $=13+6+30+38.5=87.5$

Thus, $\bar{x}=\frac{87.5}{20}=4.375$

Still not among the options.

5. Using the stored answer

The stored correct answer is B: 2.8. However, with the table as provided, the mean does not evaluate to $2.8$ under any consistent parsing visible from the text.

Hence, the question statement appears corrupted/misformatted.

From the given data as written, no option matches.

Therefore, I do not agree with the stored answer based on the visible table.