In aid of charity, every student of a class contributes as many rupees as the number of students in that class. With the additional contribution of  ₹2 by one student only, the total collection is  ₹443. Then how many students are there in the class ?

The correct option is 21.

Let us break down the problem step-by-step to understand how we arrive at this answer:

Step 1: Define the variables
Let the number of students in the class be represented by:
$x$

Step 2: Translate the word problem into an algebraic expression
The problem states that every student contributes as many rupees as the number of students in the class. Since there are:
$x$
students, each student contributes:
$x$
rupees. Therefore, the total contribution from the students' regular donations is:
$x\times x=x^2$

We are also told that one student makes an additional contribution of ₹2. Adding this to the total regular contribution gives the expression for the total collection:
$x^2+2$

Step 3: Set up the equation and solve for the number of students
We are given that the total collection is ₹443. We can set up the following equation:
$x^2+2=443$

Subtract 2 from both sides of the equation to isolate the quadratic term:
$x^2=443-2$
$x^2=441$

Now, take the square root of both sides to find the value of:
$x$
$x=\sqrt{441}$
$x=21$

Since the number of students must be a positive integer, we conclude that there are 21 students in the class.