Problem 4 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The number of negative integral solutions of \(x^{2} \cdot 2^{x+1}+2^{|x-3|+2}=x^{2} \cdot 2^{(3 x-3 \mid+4)}+2^{x-1}\) is (A) 4 (B) 2 (C) 1 (D) 0

Step-by-Step Solution

Answer

There are 0 negative integral solutions.

Step 1: Analyze the Equation

We start by breaking down the equation: \(x^{2} \cdot 2^{x+1}+2^{|x-3|+2}=x^{2} \cdot 2^{(3x-3)+4}+2^{x-1}\). This equation contains exponential terms with several expressions where the base is 2. We aim to simplify the equation to analyze it better.

Step 2: Simplify the Exponents

Rewrite the equation: \(x^2 \cdot 2^{x+1} + 2^{|x-3|+2} = x^2 \cdot 2^{3x+1} + 2^{x-1}\). This helps show the relationship between the different exponential parts by expressing the multiplication and power terms in a simplified form.

Step 3: Consider Different Cases for \(|x-3|\)

Since \(|x-3|\) affects the range of \(x\), we need to consider the implications:1. If \(x < 3\), then \(|x-3| = 3 - x\).We focus more on the negative integral solutions, so assume \(x < 3\). This implies the secondary expression \(|x-3| + 2 = 5 - x\).

Step 4: Substitute Negative Integers

Substitute small negative integers into the equation to check for solutions. We check \(x = -1, -2, -3, ..., -n\) where \(n\) ensures a reasonable solution to the equation:- Substituting distinct values will yield both unique simplifications in the exponents and test the equation's equality.

Step 5: Calculate for Each Negative Solution

Continue checking specific values for \(x\) to find potential solutions:- Substitute each value separately, simplifying both sides of the equation. The goal is for the left and the right sides to match identically for a valid solution.

Step 6: Determine the Count of Valid Solutions

After testing all possible negative integral values:- Confirm solutions only if \(x\) yields same results on both sides of the equation. Only count solutions where this condition is met.

Key Concepts

Exponential Equation

An exponential equation is one where variables appear in the exponent. In our exercise, the equation contains terms like \(2^{x+1}\) and \(2^{3x+1}\). The base in these terms is 2, which is a constant, while the exponents vary with \(x\). Understanding exponential equations often involves:

• Recognizing the base: Here, 2 is the consistent base allowing a uniform comparison between terms.
• Balancing terms: Ensure like terms on either side of the equation can be simplified equally.

Breaking down these equations into simpler forms can provide insights into their solution, especially when checking for specific types of solutions like negative integral values.

Absolute Value Function

The absolute value function denotes the magnitude of a number without considering its sign. It is illustrated as \(|x - 3|\) in our equation. This affects the expression depending on whether the value inside is positive or negative:

• When \(x \geq 3\): \( |x - 3| \) remains \(x - 3\).
• When \(x < 3\): \( |x - 3| \) becomes \(3 - x\).

For identifying negative integral solutions, we focus on \(x < 3\), leading to necessary adjustments in computations where the absolute terms transform the equation structure, such as \(|x-3| + 2 = 5-x\). Understanding these shifts is key to case analysis.

Mathematical Simplification

Mathematical simplification involves reducing expressions to a simpler form. In the context of our exponential equation, it means rewriting parts to make solving easier:

• Rewriting powers: Convert terms like \(x^2 \cdot 2^{x+1}\) to have unified exponents and bases.
• Combining terms: Similar terms can be brought together on one side for easier comparison.

Simplification allows for a direct evaluation of potential solutions. By reducing expressions, we can seamlessly substitute and verify negative values of \(x\) to find valid solutions. It's a crucial step before considering specific cases.

Case Analysis

Case analysis helps break down complex problems by considering different scenarios. For this equation, it involves checking different values and conditions:

• Focus on negative \(x\), notably \(x < 3\).
• Adjust expressions like \( |x-3| \) accordingly for these scenarios.
• Substitute each negative integer into the simplified equation and solve.

This approach allows for a structured solution process, ensuring that all possible situations are considered. Through substitution, we evaluate if the exponential equation holds true for negative integers, thus determining the valid solutions for each scenario.