To solve this problem, we will use the given information and the formula related to the mode in a frequency distribution.

Given:

• The mode is 28.
• Marks (below): 10, 20, 30, 40, 50
• No. of students (frequencies): 2, 10, 30, ?, ?

The mode formula for a grouped frequency distribution is:

$\text{Mode} = L + \left( \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \right) \times h$

Where:

• $L$ is the lower boundary of the modal class,
• $f_1$ is the frequency of the modal class,
• $f_0$ is the frequency of the class preceding the modal class,
• $f_2$ is the frequency of the class succeeding the modal class,
• $h$ is the class width.

From the problem:

• The modal class (based on Mode = 28) lies in the class interval for marks below 30, i.e., the class with frequency 30.
• So:
  • $L = 20$ (lower boundary of the modal class),
  • $f_1 = 30$ (frequency of the modal class),
  • $f_0 = 10$ (frequency of the class before the modal class),
  • $f_2$ is the missing frequency for the class interval 30-40.

We will now solve for $f_2$, the missing frequency for the 40 marks class.

Given that $h$, the class width, is constant and can be taken as 10 (since the class intervals are uniform, i.e., 10-20, 20-30, etc.), we plug the values into the mode formula:

$28 = 20 + \left( \frac{30 - 10}{2(30) - 10 - f_2} \right) \times 10$

Simplify the equation:

$28 = 20 + \left( \frac{20}{60 - f_2} \right) \times 10$ $28 - 20 = \left( \frac{200}{60 - f_2} \right)$ $8 = \frac{200}{60 - f_2}$ $8(60 - f_2) = 200$ $480 - 8f_2 = 200$ $480 - 200 = 8f_2$ $280 = 8f_2$ $f_2 = \frac{280}{8} = 35$

Thus, the missing frequency for the 40 marks class is 35.

Next, the frequency for the 50 marks class remains unknown, but if the total number of students is known or if the problem provides more context, we could solve it. However, based on the mode information, we have determined $f_2 = 35$.

Would you like to discuss further or clarify anything? Here are 5 questions that expand on this problem:

1. How is the mode calculated for different types of data distributions?
2. Why is the mode important in understanding frequency distributions?
3. How do we determine the class width when the class intervals are not uniform?
4. Can we find the missing frequency for the 50 marks class if more information is given?
5. What are the similarities and differences between mean, median, and mode?

Tip: In grouped data, the class containing the mode is not necessarily the class with the highest frequency; it's determined by how the frequencies of the neighboring classes compare.