Problem 36 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(f\left(2 x+\frac{y}{8}, 2 x-\frac{y}{8}\right)=x y\), then \(f(m, n)+f(n, m)=0\) (A) only when \(m=n\) (B) only when \(m \neq n\) (C) only when \(m=-n\) (D) for all \(m\) and \(n\).

Step-by-Step Solution

Answer

(D) for all \(m\) and \(n\).

Step 1: Understand the Problem

We are given a function \( f \) defined as \( f(2x + \frac{y}{8}, 2x - \frac{y}{8}) = xy \). We need to find when \( f(m, n) + f(n, m) = 0 \).

Step 2: Substitute Values into the Function

Substitute \( m = 2x + \frac{y}{8} \) and \( n = 2x - \frac{y}{8} \) to express \( x \) and \( y \) in terms of \( m \) and \( n \).

Step 3: Solve for x and y in Terms of m and n

Add the equations \( m = 2x + \frac{y}{8} \) and \( n = 2x - \frac{y}{8} \) to find \( x \). Subtract them to find \( y \).

Step 4: Validate Function Operation

From the equations in Step 3, use \( x = \frac{m+n}{4} \) and \( y = 2(m-n) \) to verify \( f(m, n) = \left(\frac{m+n}{4}\right) (2(m-n)) \).

Step 5: Simplifying f(m, n)

Simplify \( f(m, n) \) to observe \( f(m,n) = \frac{(m+n)(m-n)}{2} = \frac{m^2 - n^2}{2} \).

Step 6: Evaluate f(m, n) + f(n, m)

Using the expression from Step 5, calculate \( f(n, m) = \frac{n^2 - m^2}{2} \). Then, find \( f(m, n) + f(n, m) = \frac{m^2 - n^2}{2} + \frac{n^2 - m^2}{2} = 0 \).

Step 7: Analyze Conditions for Zero Sum

Since \( f(m, n) + f(n, m) = 0 \) independently of \( m \) and \( n \), verify if it holds for all cases without any additional conditions.

Key Concepts

Function Operations

Function operations involve combining or manipulating different functions to achieve a particular outcome. In this exercise, the function is expressed as \( f(2x + \frac{y}{8}, 2x - \frac{y}{8}) = xy \). The operations performed on this function include substitution, simplification, and transformation.

• Substitution: We substitute into the function by expressing specific variables, like \( m \) and \( n \), in terms of other variables, \( x \) and \( y \).
• Simplification: Simplification is applied to break the function into a recognizable and more approachable form, making it easier to collaborate with these expressions.

Function operations are crucial, as understanding these enables handling more complex function scenarios by identifying patterns or relationships among variables.

Algebraic Manipulation

Algebraic manipulation consists of rearranging and simplifying equations to achieve a clearer form. In this problem, we are addressing expressions like \( m = 2x + \frac{y}{8} \) and \( n = 2x - \frac{y}{8} \).

• By adding the equations, we can express \( x = \frac{m+n}{4} \). This action helps to consolidate separate variables into one simplified form.
• Similarly, by subtracting the equations, we can find \( y = 2(m-n) \), giving a clear expression for \( y \) in terms of \( m \) and \( n \).

These algebraic moves are essential because they transform obtuse relationships into familiar arithmetic, making subsequent calculations easier and more intuitive.

Equations

Equations are mathematical statements that assert the equivalence of two expressions, using an equal sign. They form the core of this exercise as they are used to calculate and verify variables, thereby solving the problem.In the exercise, after substituting for \( m \) and \( n \), equations are restructured and manipulated to derive new expressions and simplify to find meaningful results:

• The expression for \( f(m, n) \) is simplified to \( \frac{m^2 - n^2}{2} \), and vice versa for \( f(n, m) \).
• These equations enable the calculation and verification that \( f(m, n) + f(n, m) = 0 \) holds true.

Equations serve as the roadmap in this exercise and lead us to seek understanding by focusing on relationships between variables, ultimately solving the exercise.

Problem-Solving Steps

Problem-solving steps are systematic methods used to solve mathematical problems systematically. The original solution outlines these steps sequentially. Here's a recap with some insights:- **Understand the Problem**: Identify what is being asked. In this case, find when \( f(m, n) + f(n, m) = 0 \) holds true.- **Substitute and Rearrange**: Replace and manipulate expressions to isolate and solve for unknowns \( x \) and \( y \).- **Validate your Steps**: By inserting newly derived expressions for \( x \) and \( y \), ensure that the function \( f(m, n) \) matches expectations.- **Conclusion and Analysis**: After confirming calculations, analyze the results to understand under what conditions they hold true, recognizing that for this problem, the function operation's sum is zero for all values of \( m \) and \( n \). These steps help structure an approach to tackle similar challenging problems reliably and effectively, creating a blueprint for finding solutions.