In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and at most 95 studied Physics; at least 75 and at most 90 studied Chemistry; 30 studied both Physics and Chemistry; 50 studied both Chemistry and Mathematics; 40 studied both Mathematics and Physics and 10 studied none of these subjects. Let m and n respectively be the least and the most number of students who studied all the three subjects. Then m + n is equal to _____
In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and at most 95 studied Physics; at least 75 and at most 90 studied Chemistry; 30 studied both Physics and Chemistry; 50 studied both Chemistry and Mathematics; 40 studied both Mathematics and Physics and 10 studied none of these subjects. Let m and n respectively be the least and the most number of students who studied all the three subjects. Then m + n is equal to _____
Correct Answer: 45
Approach Solution - 1
Given data:
Let:
\(M =\) number of students who studied Mathematics,
\(P =\) number of students who studied Physics,
\(C =\) number of students who studied Chemistry.
Given conditions:
\[ 125 \leq M \leq 130, \quad 85 \leq P \leq 95, \quad 75 \leq C \leq 90. \]
Number of students studying two subjects:
\[ |P \cap C| = 30, \quad |C \cap M| = 50, \quad |M \cap P| = 40. \]
Number of students studying none:
\[ |U| - |M \cup P \cup C| = 10 \implies |M \cup P \cup C| = 210. \]
Using the formula for the union of three sets:
\[ |M \cup P \cup C| = M + P + C - |M \cap P| - |P \cap C| - |C \cap M| + |M \cap P \cap C|. \]
Substituting the values: \[ 210 = M + P + C - 40 - 30 - 50 + x, \]
where \(x\) is the number of students who studied all three subjects.
Simplifying: \[ M + P + C + x = 330. \]
Finding the range for \(x\):
From the given bounds: \[ 125 \leq M \leq 130, \quad 85 \leq P \leq 95, \quad 75 \leq C \leq 90. \]
Therefore: \[ 15 \leq x \leq 30. \]
Calculating \(m + n\):
\[ m = 15, \quad n = 30. \] \[ m + n = 15 + 30 = 45. \]
Approach Solution -2
Total students = 220
10 students studied none of the subjects.
Therefore, number of students who studied at least one subject = 220 - 10 = 210.
Let:
M = number of students who studied Mathematics
P = number of students who studied Physics
C = number of students who studied Chemistry
Let x = number of students who studied all three subjects.
Also given:
At least 125 and at most 130 studied Mathematics → \( 125 \leq M \leq 130 \)
At least 85 and at most 95 studied Physics → \( 85 \leq P \leq 95 \)
At least 75 and at most 90 studied Chemistry → \( 75 \leq C \leq 90 \)
\( n(P \cap C) = 30, \; n(C \cap M) = 50, \; n(M \cap P) = 40. \)
Step 2: Apply the formula for union of three sets.
\[ n(M \cup P \cup C) = M + P + C - [n(M \cap P) + n(P \cap C) + n(C \cap M)] + n(M \cap P \cap C) \] Substitute the given values: \[ 210 = M + P + C - (40 + 30 + 50) + x \] \[ 210 = M + P + C - 120 + x \] \[ x = 330 - (M + P + C) \]
Step 3: Find the least and the greatest possible values of x.
To find the least value of x (m), the expression \( 330 - (M + P + C) \) must be minimum.
This happens when \( M + P + C \) is maximum.
Maximum values are \( M = 130, P = 95, C = 90 \).
\[ M + P + C = 130 + 95 + 90 = 315. \] Thus, \[ m = 330 - 315 = 15. \]
To find the greatest value of x (n), \( 330 - (M + P + C) \) must be maximum.
This happens when \( M + P + C \) is minimum.
Minimum values are \( M = 125, P = 85, C = 75 \).
\[ M + P + C = 125 + 85 + 75 = 285. \] Thus, \[ n = 330 - 285 = 45. \]
Step 4: Final calculation.
\[ m + n = 15 + 30 = 45. \]
Final Answer:
\[ \boxed{45} \]
Top JEE Main Mathematics Questions
- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
- If , then the general value of is:
- The general solution of
- If the constant term in the binomial expansion of $\left(\frac{x^{\frac{3}{2}}}{2}-\frac{4}{x^l}\right)^9$ is $-84$ and the coefficient of $x^{-3 l}$ is $2^\alpha \beta$, where $\beta<0$ is an odd number, then $|\alpha l-\beta|$ is equal to_____
Top JEE Main Set Theory Questions
- 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 ________.
- Let \( A = \{1, 2, 3, \ldots, 7\} \) and let \( P(A) \) denote the power set of \( A \). If the number of functions \( f : A \rightarrow P(A) \) such that \( a \in f(a), \, \forall a \in A \) is \( m^n \), \( m \) and \( n \in \mathbb{N} \) and \( m \) is least, then \( m + n \) is equal to \(\_\_\_\_\_\_\_\_\_\).
- Let \( A \) and \( B \) be two finite sets with \( m \) and \( n \) elements respectively. The total number of subsets of the set \( A \) is 56 more than the total number of subsets of \( B \). Then the distance of the point \( P(m, n) \) from the point \( Q(-2, -3) \) is:
- Let the set \( C = \{(x, y) \mid x^2 - 2^y = 2023, x, y \in \mathbb{N}\} \).Then\[\sum_{(x, y) \in C} (x + y)\]is equal to ______.
- The number of symmetric relations defined on the set {1, 2, 3, 4} which are not reflexive is _____.
Top JEE Main Questions
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
What will be the equilibrium constant of the given reaction carried out in a \(5 \,L\) vessel and having equilibrium amounts of \(A_2\) and \(A\) as \(0.5\) mole and \(2 \times 10^{-6}\) mole respectively?
The reaction : \(A_2 \rightleftharpoons 2A\)- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
A black body is at a temperature of 2880 K. The energy of radiation emitted by this body with wavelength between 499 nm and 500 nm is U1, between 999 nm and 1000 nm is U2 and between 1499 nm and 1500 nm is U3. The Wien's constant, b = 2.88×106 nm-K. Then,