Problem 149 - A Complete Resource Book in Mathematics for JEE Main

Question

Assertion: If \(0<\alpha<\beta<, \frac{\pi}{2}\) then \(\frac{\tan \beta}{\tan \alpha}>\frac{\alpha}{\beta}\) Reason: \(x \tan x\) is increasing for \(0

Step-by-Step Solution

Answer

The assertion is true because \( x \tan x \) is increasing for \( 0 < x < \frac{\pi}{2} \).

Step 1: Understand the Assertion

The assertion given is a mathematical inequality to compare two expressions. We are asked to verify if \(\frac{\tan \beta}{\tan \alpha} > \frac{\alpha}{\beta}\) holds true under the condition \(0 < \alpha < \beta < \frac{\pi}{2}\). This means we need to see if the ratio of tangents is greater than the ratio of the angles themselves.

Step 2: Analyze and Understand the Reason

The reason states that \(x \tan x\) is an increasing function for \(0 < x < \frac{\pi}{2}\). If a function is increasing, then for any two numbers \(x_1\) and \(x_2\) such that \(x_1 < x_2\), it follows that \(x_1 \tan x_1 < x_2 \tan x_2\).

Step 3: Apply the Reason to the Assertion

Using the fact that \(x \tan x\) is increasing, substitute \(\alpha\) and \(\beta\) into this function: \(\alpha \tan \alpha < \beta \tan \beta\). Dividing both sides by \(\alpha \beta\) (both are positive), we have \(\frac{\tan \alpha}{\alpha} < \frac{\tan \beta}{\beta}\).

Step 4: Rearrange the Inequality

By multiplying both sides of \(\frac{\tan \alpha}{\alpha} < \frac{\tan \beta}{\beta}\) by \(\frac{\alpha}{\beta}\), we arrive at \(\frac{\tan \beta}{\tan \alpha} > \frac{\alpha}{\beta}\), which is exactly the assertion we were asked to prove.

Step 5: Conclusion

The assertion that \(\frac{\tan \beta}{\tan \alpha} > \frac{\alpha}{\beta}\) is correct based on the reason provided, which was that the function \(x \tan x\) is increasing for the given range of \(x\). Therefore, the reason supports the assertion.

Key Concepts

Function Behavior Analysis

When analyzing the behavior of functions, we focus on understanding how a function changes as its input varies. For instance, does the function increase, decrease, or stay constant? We can observe these patterns in different ranges of the input values. In our case, we're examining a trigonometric function:

• We notice that the behavior of the function is critical because it helps to solve mathematical inequalities, such as the one given in the problem.
• Knowing that a function is increasing or decreasing allows us to predict the output based on the input range.

In this problem, the claim that "\(x \tan x\)" is an increasing function plays a crucial role. By understanding that the function is increasing within the interval \(0 < x < \frac{\pi}{2}\), we can progress with the evidence that supports the assertion that the problem presents. Performing this type of analysis ensures logical correctness in the conclusions we draw from mathematical inequalities.

Increasing Functions

An increasing function is one where the output values get larger as the input values increase within a certain interval. This can be intuitively understood by imagining a graph that always moves upwards as you move from left to right.

• For this exercise, knowing that \(x \tan x\) is increasing implies that when \(\alpha < \beta\), it results in \(\alpha \tan \alpha < \beta \tan \beta\).
• This fact that the function is increasing provides the confirmation needed to establish the mathematical inequality presented.

Understanding increasing functions is vital as it helps solve inequalities by showing that one quantity is indeed larger than another based on their respective input values. It allows us to confirm the relationship without needing to compute each value explicitly.

Mathematical Inequalities

Mathematical inequalities express a relationship where one value is less than or greater than another. They are fundamental in comparing different expressions and are often shaped by the behavior of functions.

• A typical strategy to solve inequalities is to use function properties, such as increasing or decreasing behavior.
• In this exercise, grasping the inequality \(\frac{\tan \beta}{\tan \alpha} > \frac{\alpha}{\beta}\) starts with evaluating the behavior of the function \(x \tan x\).

By leveraging that this function is increasing, we simplify the inequality solving process. We compared ratios and confirmed that one is larger, backed by function behavior. Understanding mathematical inequalities allows us to justify comparisons and draw conclusions with confidence.