A group of 40 students appeared in an examination of 3 subjects – Mathematics, Physics & Chemistry. It was found that all students passed in at least one of the subjects, 20 students passed in Mathematics, 25 students passed in Physics, 16 students passed in Chemistry, at most 11 students passed in both Mathematics and Physics, at most 15 students passed in both Physics and Chemistry, at most 15 students passed in both Mathematics and Chemistry. The maximum number of students passed in all the three subjects is ________.
Correct Answer: 10
Solution and Explanation
Solution: Using the principle of inclusion-exclusion for three sets \( M \), \( P \), and \( C \), we have:
\[ |M \cup P \cup C| = |M| + |P| + |C| - |M \cap P| - |P \cap C| - |M \cap C| + |M \cap P \cap C| \]
Given:
- \(|M| = 20\)
- \(|P| = 25\)
- \(|C| = 16\)
- \(|M \cap P| \leq 11\)
- \(|P \cap C| \leq 15\)
- \(|M \cap C| \leq 10\)
Since \(|M \cup P \cup C| = 40\), substitute the values and solve for \(|M \cap P \cap C|\):
\[ 40 = 20 + 25 + 16 - 11 - 15 - 10 + x \]
\[ x = 10 \]
Thus, the maximum number of students who passed in all three subjects is 10.
Top Questions on Set Theory
Let \( A = (1, 2, 3, \dots, 20) \). Let \( R \subseteq A \times A \) such that \( R = \{(x, y) : y = 2x - 7 \} \). Then the number of elements in \( R \) is equal to:
- NIMCET - 2025
- Mathematics
- Set Theory
Let $A = \{5n - 4n - 1 : n \in \mathbb{N}\}$ and $B = \{16(n - 1): n \in \mathbb{N}\}$ be sets. Then:
- NIMCET - 2025
- Mathematics
- Set Theory
- If \( A = \{ x : x \text{ is an integer and } x^2 - 9 \geq 0 \} \), \[ B = \{ x : x \text{ is a natural number and } 2 \leq x \leq 5 \}, \quad C = \{ x : x \text{ is a prime number} \leq 4 \} \] Then \( (B - C) \cup A \) is:
- KCET - 2025
- Mathematics
- Set Theory
- From the given sets, which is an infinite set:
- Match List-I with List-II.\[\begin{array}{|c|c|} \hline \textbf{List-I (Set operations)} & \textbf{List-II (Missing term)} \\ \hline \text{(A) If \(n(X)=17,\, n(Y)=23,\, n(X \cup Y)=38\), then \(n(X \cap Y)\) is} & \text{(I) 20} \\ \hline \text{(B) If \(n(X)=28,\, n(Y)=32,\, n(X \cap Y)=10\), then \(n(X \cup Y)\) is} & \text{(II) 10} \\ \hline \text{(C) If \(n(X)=10\), then \(n(X')\) is} & \text{(III) 50} \\ \hline \text{(D) If \(n(Y)=20\), then \(n\left(\tfrac{Y}{2}\right)\) is} & \text{(IV) 2} \\ \hline \end{array}\]
Questions Asked in JEE Main exam
- Earth has mass 8 times and radius 2 times that of a planet. If the escape velocity from the earth is 11.2 km/s, the escape velocity in km/s from the planet will be:
- JEE Main - 2025
- Speed and velocity
- Let \(E_1: \frac{x^2}{9} + \frac{y^2}{4} = 1\) be an ellipse. Ellipses \(E_i\) are constructed such that their centers and eccentricities are the same as that of \(E_1\), and the length of the minor axis of \(E_{i+1}\) is the length of the major axis of \(E_i\). If \(A_i\) is the area of the ellipse \(E_i\), then \(\frac{5}{\pi} \sum_{i=1}^{\infty} A_i\) is equal to:
- JEE Main - 2025
- Calculus
- The compound with molecular formula C\(_6\)H\(_6\), which gives only one monobromo derivative and takes up four moles of hydrogen per mole for complete hydrogenation has \(\_\_\_\_\_\)\( \pi \) electrons.
- JEE Main - 2025
- Aromaticity & chemistry of aromatic compounds
- A 400 g solid cube having an edge of length \(10\) cm floats in water. How much volume of the cube is outside the water? (Given: density of water = \(1000 { kg/m}^3\))
For $ \alpha, \beta, \gamma \in \mathbb{R} $, if $$ \lim_{x \to 0} \frac{x^2 \sin \alpha x + (\gamma - 1)e^{x^2} - 3}{\sin 2x - \beta x} = 3, $$ then $ \beta + \gamma - \alpha $ is equal to: