Problem 149 - 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: 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

B

Step 1: Understand the Problem

We are tasked with evaluating whether the given assertion and reason statements are true and to determine the appropriate relationship between them according to the options provided.

Step 2: Evaluate Assertion (A)

The assertion claims that if \(0<\alpha<\beta<\frac{\pi}{2}\), then \(\frac{\tan \beta}{\tan \alpha} > \frac{\alpha}{\beta}\). This stems from the properties of the tangent function, which is increasing in the interval \((0, \frac{\pi}{2})\). Therefore, this assertion is true.

Step 3: Evaluate Reason (R)

The reason states that \(x \tan x\) is increasing for \(0 0\) throughout it, confirming that \(x \tan x\) is increasing. Therefore, the reason is true.

Step 4: Determine Relationship between Assertion and Reason

Both the assertion and the reason are true. However, the reason does not directly explain why \(\frac{\tan \beta}{\tan \alpha} > \frac{\alpha}{\beta}\). The assertion is based on the tangent function's properties, not on the behavior of \(x \tan x\). Thus, reason (R) is not a correct explanation for assertion (A).

Key Concepts

Tangent Function

The tangent function, often denoted as \(\tan(x)\), is a fundamental trigonometric function. It plays a significant role due to its periodic nature and relationship with sine and cosine functions. In general, for any angle \(x\), the tangent is defined as \(\tan(x) = \frac{\sin(x)}{\cos(x)}\). This formula holds wherever the cosine is not zero.

In the interval \((0, \frac{\pi}{2})\), where our problem is focused, the tangent function is increasing. This means as \(x\) gets larger, \(\tan(x)\) increases as well. This property is crucial when comparing angles \(\alpha\) and \(\beta\) in the assertion provided, supporting the inequality that \(\frac{\tan(\beta)}{\tan(\alpha)} > \frac{\alpha}{\beta}\).

Understanding the behavior of the tangent function in this interval helps us infer characteristics about inequalities involving tangent, providing a backbone for logical deductions in trigonometric problems. Its increasing nature is due to its derivative, which is \(\sec^2(x)\), always positive in this interval.

Inequality in Trigonometry

Inequality in trigonometry involves comparing various trigonometric expressions, often over specific intervals. These inequalities are vital in mathematical proofs and problem-solving. They demonstrate how one trigonometric function behaves relative to another.

In the provided exercise, the inequality \(\frac{\tan \beta}{\tan \alpha} > \frac{\alpha}{\beta}\) is under consideration. This arises from the properties of the tangent function, which increase continuously in the interval \((0, \frac{\pi}{2})\). Here's why this matters:

• As \(\beta > \alpha\), and both are within \((0, \frac{\pi}{2})\), we expect \(\tan(\beta) > \tan(\alpha)\).
• The inequality \(\frac{\tan \beta}{\tan \alpha} > 1\) holds, implying a greater change in tangent for \(\beta\) compared to \(\alpha\).

The assertion that \(\frac{\tan \beta}{\tan \alpha} > \frac{\alpha}{\beta}\) uses these properties intuitively. This trigonometric inequality highlights the importance of the domain and behavior of functions in solving such problems.

Function Derivatives

Function derivatives are a core concept in calculus, helping to find the rate at which a function is changing at any point. Understanding derivatives gives insight into the increase or decrease of functions.

In this exercise, the derivative of \(f(x) = x \tan x\) is explored to understand if it is increasing. To find \(f'(x)\), we compute:

• Derivative of \(x\) is 1.
• Derivative of \(\tan x\) is \(\sec^2 x\), using product rule: \(f'(x) = \tan x + x \sec^2 x\).

Since both terms \(\tan x\) and \(x \sec^2 x\) are positive in the interval \((0, \frac{\pi}{2})\), \(f'(x) > 0\). Therefore, \(x \tan x\) is increasing in this interval.

This increase supports the reason statement in the exercise. Although true, it does not directly explain the inequality in the assertion, showcasing how understanding derivatives helps verify function behavior independently from logical inferences needed for solving trigonometric inequalities.