Problem 156 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

In the following questions an Assertion (A) is given, followed by a Reason (R). Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason(R) is False (D) Assertion(A) is False, Reason(R) is True Assertion: \(303^{202}<202^{303}\) Reason: The function \(f(x)=\frac{\ln x}{x}\) strictly increases in \((e, \infty)\).

Step-by-Step Solution

Answer

C: Assertion is true; reason is false.

Step 1: Understand the Assertion

The assertion given is \(303^{202} < 202^{303}\). This means we need to determine whether raising 303 to the power of 202 results in a smaller number compared to raising 202 to the power of 303.

Step 2: Understand the Reason

The reason given is about the behavior of the function \( f(x) = \frac{\ln x}{x} \), stating it strictly increases in the interval \((e, \infty)\). To evaluate this, we need to understand the derivative of \( f(x) \) to determine its behavior.

Step 3: Analyze the Function

To analyze \( f(x) = \frac{\ln x}{x} \), we find its derivative \( f'(x) \). Using the quotient rule: \( f'(x) = \frac{d}{dx}\left(\frac{\ln x}{x}\right) = \frac{x \cdot \frac{1}{x} - \ln x \cdot 1}{x^2} = \frac{1 - \ln x}{x^2} \). This derivative is positive when \( 1 > \ln x \) or \( x < e \). Hence, \(f(x)\) is not strictly increasing in \((e, \infty)\), it decreases instead.

Step 4: Evaluate the Comparison via Logarithms

To compare \(303^{202}\) and \(202^{303}\), take logarithms: \(202 \ln 303\) and \(303 \ln 202\). Compute both and compare. Let us simplify: if \(202 \ln 303 < 303 \ln 202 \), then \(202/303 < \ln 202/\ln 303 \), implying \(303^{202} < 202^{303}\). This requires additional numerical verification but generally, 303 raised fewer times is smaller than 202 raised more times.

Step 5: Conclusion on Assertion and Reason

The assertion \(303^{202} < 202^{303}\) is true since numerically and conceptually one side grows larger by having higher power with lower base. The reason, however, \( f(x) = \frac{\ln x}{x}\) does not correctly explain this increase as the function does not strictly increase over the given interval, rather decreases.

Key Concepts

Inequalities

Inequalities play a crucial role in comparing numbers and mathematical expressions. In the context of the given problem, we deal with exponential inequalities such as whether \(303^{202} < 202^{303}\). To solve such inequalities, a common strategy is to use logarithms. Logarithms translate multiplicative relationships into additive ones, making comparisons more straightforward. In this particular exercise, we use logarithms to infer:

• That \(202 \ln 303 < 303 \ln 202\), allowing us to compare the size of expressions without directly computing gigantic powers.

By using properties of logarithms, we can effectively simplify and resolve inequalities involving exponential terms.
Understanding and manipulating inequalities is vital, as they frequently emerge in various math and science fields. Whether dealing with simple algebra, calculus, or beyond, mastering inequalities can offer deeper insights into the relationships between different mathematical structures.

Exponential Functions

Exponential functions are powerful tools for modeling growth and decay in various real-life phenomena. The exercise at hand involves exponential expressions, \(303^{202}\) and \(202^{303}\), which are particular examples of exponential functions.

• Exponential terms grow based on the value of the base and the exponent.

In general, the larger the base or exponent, the faster the function grows. However, comparative growth can become complex due to the interplay between base size and exponent magnitude.
This scenario requires evaluating which factor—base or exponent—dominates. When comparing such expressions, employing logarithmic transformations is essential. This action simplifies calculations, enabling us to judge which term ultimately exceeds the other. Exponential functions not only enrich mathematical reasoning but also offer essential insights into fields like finance, physics, and biology.

Derivative Analysis

Derivative analysis is indispensable for understanding the behavior of functions. In this exercise, it is crucial for analyzing the function \(f(x) = \frac{\ln x}{x}\), presented in the reason statement. By finding the derivative through the quotient rule, we determine:\[f'(x) = \frac{1 - \ln x}{x^2}\]

• This derivative is positive when \(1 > \ln x\), or \(x < e\), indicating where the function increases.

However, it is negative when \(x > e\), which shows that the function decreases over \((e, \infty)\). This analysis reveals that the reason statement—asserting the function strictly increases within \((e, \infty)\)—is incorrect.
Derivative analysis uncovers the nuances of function behaviors, providing insights into critical points, increasing or decreasing intervals, and the general shape of the function's graph. It serves as a foundational tool in calculus, enabling a deeper understanding of both simple and complex functions.