Answer

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**Hint:**

Modal class is the class interval which has the maximum or the highest frequency. Here frequency is represented by the number of the students and the class interval is represents by the marks intervals which can be written as $0 - 10,10 - 20,20 - 30.........$

**Complete step by step solution:**

Here we are given the distribution of the number of students and their marks which is as follows:

Marks less than ${\text{10 20 30 40 50 60}}$

No. of students $3{\text{ 12 27 57 75 80}}$

Now here we are given the intervals in the form of the marks less than but we need to convert it into the interval form. So we can write the ${\text{less than 10 as }}0 - 10$ and similarly the other ones.

So in the interval $0 - 10$ we have only $3$ students and similarly as given in the interval $10 - 20$ we have total of $12 - 3 = 9$students because the students whose marks are less than $10$ are also included in the students whose marks are less than $20$ and therefore we need to subtract the values to get the desired range frequency.

So for $20 - 30$ we have $27 - 12 = 15$ students.

For $30 - 40$ we have $57 - 27 = 30$ students.

For $40 - 50$ we have $75 - 57 = 18$ students.

For $50 - 60$ we have $80 - 75 = 5$ students.

So we have got all the values of the students who have secured the marks in the desired range. Hence we can form the table as follows of the above distribution:

Marks (class interval) ${\text{0 - 10 10 - 20 20 - 30 30 - 40 40 - 50 50 - 60}}$

No. of students $3{\text{ 9 15 30 18 5}}$

Hence here we can see that the maximum frequency is $30$ which is of the class interval $30 - 40$

Hence $30 - 40$ is the modal class

**Hence option C is the correct option.**

**Note:**

If we are given the modal class as $(a - b)$then its mode lies between $a{\text{ and }}b$ and for example: If we are given the modal class as $(30 - 40)$ then the value of mode can be any number between $30{\text{ and }}40$ and therefore mode cannot be less than $30$ and not greater than $40$

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