1. Given data

The five observations are: $3,\;5,\;7,\;a,\;b$

Mean $=5$ and standard deviation $=2$.

2. Use the mean

For 5 observations, mean is $\frac{3+5+7+a+b}{5}=5$

So, $15+a+b=25$ $a+b=10$

3. Use the standard deviation

For ungrouped data, variance is $\sigma^2=\frac{\sum x_i^2}{n}-\bar{x}^2$

Given standard deviation $=2$, so variance is $\sigma^2=4$

Also, $\bar{x}=5$ and $n=5$. Hence, $4=\frac{3^2+5^2+7^2+a^2+b^2}{5}-25$

Now, $3^2+5^2+7^2=9+25+49=83$

Therefore, $4=\frac{83+a^2+b^2}{5}-25$ $29=\frac{83+a^2+b^2}{5}$ $145=83+a^2+b^2$ $a^2+b^2=62$

4. Find $ab$ using $(a+b)^2$

We know $a+b=10$

So, $(a+b)^2=a^2+b^2+2ab$ $100=62+2ab$ $2ab=38$ $ab=19$

5. Form the quadratic equation whose roots are $a$ and $b$

If roots are $a$ and $b$, then the quadratic is $x^2-(a+b)x+ab=0$

Substitute $a+b=10$ and $ab=19$: $x^2-10x+19=0$

6. Compare with options

The required equation is $x^2-10x+19=0$

This matches Option D.

7. Comparison with stored answer

Stored correct answer: D

Our derived answer: D

So they agree.