 survey was conducted among 240 students in a school to find out their applications in JEE, NEET and CUET It was found that the number of students who applied for at most two of these tests was three times that of those who applied for at least 2 of these tests. The number of students who applied to only NEET was more than that of those who applied to only JEE, which, in turn, was more than that of those who applied to only CUET. Find number of only jee

Let’s define the variables for the number of students applying to each of the three tests:

- \(x_J\): Number of students who applied for only JEE
- \(x_N\): Number of students who applied for only NEET
- \(x_C\): Number of students who applied for only CUET

Additionally, we need to define variables for the number of students who applied for combinations of tests:

- \(x_{JN}\): Number of students who applied for both JEE and NEET (but not CUET)
- \(x_{JC}\): Number of students who applied for both JEE and CUET (but not NEET)
- \(x_{NC}\): Number of students who applied for both NEET and CUET (but not JEE)
- \(x_{JNC}\): Number of students who applied for all three tests.

### Key Information:
1. The total number of students surveyed is 240.
   \[
   x_J + x_N + x_C + x_{JN} + x_{JC} + x_{NC} + x_{JNC} = 240
   \]

2. The number of students who applied for at most two of the tests is three times that of those who applied for at least two tests.

   Students who applied for **at least two tests** include:
   \[
   x_{JN} + x_{JC} + x_{NC} + x_{JNC}
   \]
   Students who applied for **at most two tests** include:
   \[
   x_J + x_N + x_C + x_{JN} + x_{JC} + x_{NC}
   \]
   Given that the number of students who applied for at most two tests is three times those who applied for at least two tests:
   \[
   x_J + x_N + x_C + x_{JN} + x_{JC} + x_{NC} = 3(x_{JN} + x_{JC} + x_{NC} + x_{JNC})
   \]

3. The number of students who applied for only NEET is greater than the number of students who applied for only JEE, which is greater than the number of students who applied for only CUET:
   \[
   x_N > x_J > x_C
   \]

We now have a system of equations based on these conditions. Let’s proceed to solve this system for \(x_J\).The system of equations leads us to the following relationships:

1. The sum of the students who applied for only one test (JEE, NEET, or CUET) is related to those who applied for combinations of tests by:
   \[
   x_J + x_N + x_C = 2(x_{JN} + x_{JC} + x_{NC}) + 3x_{JNC}
   \]

2. The number of students who applied for at least two tests (those who applied to combinations) follows:
   \[
   x_{JN} + x_{JC} + x_{NC} + x_{JNC} = 240 - 2(x_{JN} + x_{JC} + x_{NC}) - 3x_{JNC}
   \]

Now, we need to apply the additional condition \(x_N > x_J > x_C\) and use this system to solve for \(x_J\). Would you like to proceed with any assumptions on the values of combinations (such as equal distribution), or would you like further exploration of the relationships?

A survey was conducted among 240 students in a school to find out their applications in JEE, NEET, and CUET. It was found that the number of students who applied for at most two of these tests was three times that of those who applied for at least 2 of these tests. The number of students who applied to only NEET was more than that of those who applied to only JEE, which, in turn, was more than that of those who applied to only CUET. Find the number of only JEE applicants.

Discover how to solve a complex set theory problem involving a survey of 240 students applying to JEE, NEET, and CUET. Learn key concepts like algebraic equations and set theory with detailed step-by-step instructions. Perfect for students in Grades 11-12.

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