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

Related Formula

Given functional property equation:

Core Logic

Substitute x = y = 0$x = y = 0$ into the property equation: f(0) = 2f(0) + 1 f(0) = -1$f(0) = 2f(0) + 1 \implies f(0) = -1$. Since f(0) = b$f(0) = b$, we instantly find b = -1$b = -1$.

Step 1: Extracting Parameter Values

Substitute y = -x$y = -x$ into the property equation:

Matching coefficients for x²$x^2$ gives:

Therefore, the absolute functional identity is:

Step 2: Computing the Target Series

Rewriting using common denominators:

Evaluating for i=1$i=1$ to 5$5$:

Pattern Recognition

Substituting standard points like 0$0$ and -x$-x$ decouples symmetric multi-variable systems with maximum efficiency.

Chapter Mix

Class 12 Maths: Relations and Functions

Related Formula

An equivalence relation must be simultaneously reflexive, symmetric, and transitive.

Core Logic

Let's check each property sequentially:

• Reflexive: For any x in Z$x \in \mathbb{Z}$, x + x = 2x$x + x = 2x$, which is always even. Thus, (x, x) in R$(x, x) \in R$.
• Symmetric: If x + y$x + y$ is even, then y + x$y + x$ must also be even due to commutative addition. Thus, if (x, y) in R (y, x) in R$(x, y) \in R \implies (y, x) \in R$.
• Transitive: If x + y$x + y$ is even and y + z$y + z$ is even, then adding them gives (x + y) + (y + z) = x + 2y + z = even x + z = even - 2y = even$(x + y) + (y + z) = x + 2y + z = \text{even} \implies x + z = \text{even} - 2y = \text{even}$. Thus, (x, z) in R$(x, z) \in R$.

Step 1: Final Property Summary

Since all three criteria are satisfies simultaneously, R$R$ is an equivalence relation.

Pattern Recognition

Parity relation properties (even/odd checking sums) over integer sets universally form clean modular equivalence systems.

Chapter Mix

Class 12 Maths: Relations and Functions

Related Formula

• Elements in a relation satisfy the exact range constraint.
• Reflexive criteria: For every x in A$x \in A$, (x, x) in R$(x, x) \in R$.

Core Logic

Rewrite the inequality to isolate variables systematically:

Test every valid value of y in A$y \in A$ to discover valid integer values for x$x$

• y = -3 6 ≤ x² ≤ 10 x in -3, 3$y = -3 \implies 6 \le x^2 \le 10 \implies x \in \{-3, 3\}$
• y = -2 4 ≤ x² ≤ 8 x in -2, 2$y = -2 \implies 4 \le x^2 \le 8 \implies x \in \{-2, 2\}$
• y = -1 2 ≤ x² ≤ 6 x in -2, 2$y = -1 \implies 2 \le x^2 \le 6 \implies x \in \{-2, 2\}$
• y = 0 0 ≤ x² ≤ 4 x in -2, -1, 0, 1, 2$y = 0 \implies 0 \le x^2 \le 4 \implies x \in \{-2, -1, 0, 1, 2\}$
• y = 1 -2 ≤ x² ≤ 2 x in -1, 0, 1$y = 1 \implies -2 \le x^2 \le 2 \implies x \in \{-1, 0, 1\}$
• y = 2 -4 ≤ x² ≤ 0 x in 0$y = 2 \implies -4 \le x^2 \le 0 \implies x \in \{0\}$
• y = 3 -6 ≤ x² ≤ -2 No real x exists$y = 3 \implies -6 \le x^2 \le -2 \implies \text{No real } x \text{ exists}$

Step 1: Listing set elements and counting

Compile all distinct matching coordinate pairs (x,y)$(x,y)$ into set R$R$:

Counting elements gives l = 15$l = 15$ To make the relation reflexive, the pairs (-3,-3), (-2,-2), (-1,-1), (0,0), (1,1), (2,2), (3,3)$(-3,-3), (-2,-2), (-1,-1), (0,0), (1,1), (2,2), (3,3)$ must all belong to R$R$. Checking missing elements:

Sum of variables

Pattern Recognition

Isolating terms explicitly via a variable-by-variable bounded testing grid avoids missing distinct coordinate boundary values.

Chapter Mix

Class 11 Mathematics: Relations and Functions

Related Formula

• For (g(x))$\log(g(x))$, we require g(x) > 0$g(x) > 0$.
• For ⁻¹(h(x))$\sin^{-1}(h(x))$, we require -1 ≤ h(x) ≤ 1$-1 \le h(x) \le 1$.

Core Logic

Evaluate constraints independently:

Constraint 1 (Logarithmic Argument):

Constraint 2 (Arcsine Argument):

Step 1: Solving the Arcsine inequalities

Split inequality into separate conditional frames: Left frame:

Right frame:

Intersecting both sets gives:

Step 2: Final Intersection and Value Solving

Intersect Log constraint with Arcsine constraint solution range:

Thus, identify parameters:

Compute the requested expression value:

Pattern Recognition

When dealing with fractional variables inside boundaries, flipping inequalities according to denominator signs prevents fatal zone misinterpretations.

Chapter Mix

Class 12 Mathematics: Relations and Functions

Core Logic

The relation condition is 2x - y = 0$2x - y = 0$ or 2x - y = 1$2x - y = 1$ where x, y in A$x, y \in A$.

Case 1: 2x - y = 0 y = 2x$2x - y = 0 \implies y = 2x$. Possible pairs in A × A$A \times A$ are:

Case 2: 2x - y = 1 y = 2x - 1$2x - y = 1 \implies y = 2x - 1$. Possible pairs in A × A$A \times A$ are:

Combining both subsets, the total relation set R$R$ contains:

Hence, the number of existing elements l = 7$l = 7$.

Step 1: Elements to add for Reflexivity

For a relation to be reflexive on set A$A$, it must contain (x,x)$(x,x)$ for all 7 elements of A$A$.

Currently, R$R$ contains (0,0), (1,1)$\{(0,0), (1,1)\}$.

Missing diagonal elements are (-3,-3), (-2,-2), (-1,-1), (2,2), (3,3)$\{(-3,-3), (-2,-2), (-1,-1), (2,2), (3,3)\}$.

Therefore, the minimum number of elements to add for reflexivity is m = 5$m = 5$.

Step 2: Elements to add for Symmetry

For a relation to be symmetric, if (x,y) in R$(x,y) \in R$, then (y,x)$(y,x)$ must also belong to R$R$.

Let's check the non-diagonal elements currently in R$R$:

• (1,2) in R$(1,2) \in R \implies$ need (2,1)$(2,1)$
• (-1,-2) in R$(-1,-2) \in R \implies$ need (-2,-1)$(-2,-1)$
• (0,-1) in R$(0,-1) \in R \implies$ need (-1,0)$(-1,0)$
• (2,3) in R$(2,3) \in R \implies$ need (3,2)$(3,2)$
• (-1,-3) in R$(-1,-3) \in R \implies$ need (-3,-1)$(-3,-1)$

None of these reverse pairs are currently in R$R$. Thus, we must add exactly 5 elements to ensure symmetry, giving n = 5$n = 5$.

Step 3: Final Computation

Based on the official valuation tracking, the required evaluation metric simplifies to:

Pattern Recognition

To quickly count elements needed for reflexivity, subtract the number of identity pairs already present from the total cardinality of the set. For symmetry, find all elements where x ≠ y$x \neq y$ and check if their mirrors are absent.

Chapter Mix

Class 12 Mathematics: Relations and Functions