Let's break down the problem step by step to figure out which answer cannot be determined. Here's the information provided:

• 37 students passed physics.
• 4 students passed only physics and maths.
• 12 students passed only physics (these are the students who passed only physics, not physics with any other subject).
• 30 students passed chemistry.
• 40 students passed maths.
• 29 students passed at least two subjects.

We need to analyze the sets and find out what can and cannot be determined. The following information is needed to make the analysis:

1. The number of students who passed only physics and chemistry.
2. The number of students who passed only maths and chemistry.
3. The number of students who passed all three subjects.
4. The total number of students in the class.

Step 1: Set Analysis

• Physics (P): 37 students passed.

  • Only Physics (P only): 12 students.
  • Physics and Maths (P ∩ M): 4 students passed only these two subjects.
  • Physics and Chemistry (P ∩ C): Unknown, we need to calculate this.

• Chemistry (C): 30 students passed.

• Maths (M): 40 students passed.

• At least two subjects: 29 students passed at least two subjects.

Now, let's use this data to draw some conclusions and identify what cannot be determined.

Step 2: Analysis of "Passed in Physics & Chemistry"

This refers to the students who passed both physics and chemistry but not necessarily maths. We need to find out how many passed only physics and chemistry, and also how many passed all three subjects. These calculations depend on knowing how many students passed only two subjects, and unfortunately, we don't have enough information to precisely calculate this value.

Step 3: Other Questions

• Passed in Chemistry only: This is the number of students who passed chemistry but did not pass physics or maths. This can be determined because we know the total number of chemistry students and can subtract those who passed physics or maths.

• Passed in Maths only: This is the number of students who passed maths but did not pass physics or chemistry. This can also be determined by subtracting the relevant intersections.

Conclusion

The number of students who passed in both Physics and Chemistry (P ∩ C) cannot be determined with certainty because we do not have enough data about the overlap between Physics and Chemistry, specifically the students who passed only these two subjects and the number of students who passed all three subjects.

Thus, the correct answer is:

• Passed in Physics & Chemistry cannot be determined.

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Would you like to dive deeper into any part of the problem or explore similar questions? Here are some related questions:

1. How can we calculate the number of students who passed all three subjects?
2. How do we use Venn diagrams to solve problems involving overlaps like this?
3. If we know the total number of students in the class, how would it affect our ability to determine the unknown values?
4. What is the principle behind counting students who passed exactly two subjects?
5. How would the answer change if we had the total number of students in the class?

Tip: For set problems, always try drawing a Venn diagram to visualize the intersections and avoid confusion!