Problem 58 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Given that, for all real \(x\), the expression \(\frac{x^{2}-2 x+4}{x^{2}+2 x+4}\) lies between \(\frac{1}{3}\) and \(3 .\) The values between which the expression \(\frac{9 \cdot 3^{2 x}+6 \cdot 3^{x}+4}{9 \cdot 3^{2 x}-6 \cdot 3^{x}+4}\) lies are (A) 0 and 2 (B) \(-\) and 1 (C) \(-2\) and 0 (D) \(\frac{1}{3}\) and 3 .

Step-by-Step Solution

Answer

The values for the second expression lie between \( \frac{1}{3} \) and 3, which is Option D.

Step 1: Analyze the given expression

We are provided with a condition that \( \frac{x^2 - 2x + 4}{x^2 + 2x + 4} \) lies between \( \frac{1}{3} \) and 3 for all real numbers \( x \). This implies there is a common characteristic or transformation that makes the new expression remain within the specified bounds.

Step 2: Define the new variable substitution

To simplify the problem, let \( y = 3^x \). This substitution transforms the expression \( \frac{9 \, \cdot \, 3^{2x} + 6 \, \cdot \, 3^x + 4}{9 \, \cdot \, 3^{2x} - 6 \, \cdot \, 3^x + 4} \) into \( \frac{9y^2 + 6y + 4}{9y^2 - 6y + 4} \).

Step 3: Compare transformed expressions

Analyze the relationship between the expressions: newly transformed \( \frac{9y^2 + 6y + 4}{9y^2 - 6y + 4} \) compared to the original known bound condition \( \frac{x^2 - 2x + 4}{x^2 + 2x + 4} \). We note that the transformation and variable substitution keep the expressions behaving similarly.

Step 4: Match the bounds to given options

From our analysis, the behavior of the transformed expression is similar to that of the initial provided expression, which lies between \( \frac{1}{3} \) and 3. Check given options: Option D, \( \frac{1}{3} \) and 3, matches with our interpretation.

Key Concepts

Expression Evaluation

Mathematical expression evaluation is the process of determining the value of an expression. First, it's important to understand the components of your expression. Essentially, we're combining numbers, variables, and operations to create a mathematical statement that we can evaluate. In our exercise, we begin with the fraction \( \frac{x^2 - 2x + 4}{x^2 + 2x + 4} \). This is an algebraic expression because it contains variables raised to a power, coefficients, and constants.

• The numerator \( x^2 - 2x + 4 \) and the denominator \( x^2 + 2x + 4 \) are quadratic polynomials.
• Each polynomial can be expanded, simplified, or factored to better understand their behavior with the variable \( x \).

To find where the expression lies, we analyze its behavior as \( x \) takes on different values. Despite being a complex expression, it simplifies the task when we identify that the quotient remains between \( \frac{1}{3} \) and 3 for all real \( x \). This understanding of expression evaluation is foundational for solving inequalities.

Variable Substitution

Variable substitution is a vital technique in mathematical problem-solving. Substituting a variable can transform a complex expression into something simpler, making it more manageable to evaluate or compare. In our exercise, we employ this by letting \( y = 3^x \).

• This changes our expression from \( \frac{9 \cdot 3^{2x} + 6 \cdot 3^x + 4}{9 \cdot 3^{2x} - 6 \cdot 3^x + 4} \) to \( \frac{9y^2 + 6y + 4}{9y^2 - 6y + 4} \).
• A major benefit of substitution is simplifying the calculations and analysis of the expression's behavior.

When performing variable substitution, always ensure the substitution is valid for all values considered and clearly describe any new restrictions imposed by the substitution. In our case, \( y = 3^x \) means \( y > 0 \) since the exponential function is always positive for any real \( x \). This technique helps us see that the transformed expression behavior mirrors the conditions placed on the original expression.

Problem Solving Strategies

Solving mathematical problems requires strategic thinking. One effective method is breaking down the problem into smaller, more manageable parts. This approach was utilized in our exercise as follows:

• Start by understanding the conditions given in the problem—recognizing the boundaries of the expression \( \frac{x^2 - 2x + 4}{x^2 + 2x + 4} \).
• Apply a useful technique, such as variable substitution, to simplify the expression.
• Lastly, compare the behavior of the transformed expression \( \frac{9y^2 + 6y + 4}{9y^2 - 6y + 4} \) with the initial conditions to determine the solution range.

The ability to match the essence of an unknown expression with known boundaries is crucial. Here, the key strategy was identifying that the behavior of the expressions remained bounded between the same values. Problem-solving includes strategies such as reading the question thoroughly, planning, and verifying potential solutions to ensure that the strategic path leads logically and correctly to the conclusion.