Problem 147 - 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

Step-by-Step Solution

Answer

C: Assertion is true, Reason is false.

Step 1: Evaluate the Assertion Condition

We need to check if the assertion is true: For \(0 < x < \frac{\pi}{2}\), we have \( \frac{2}{\pi} < \frac{\sin x}{x} < 1 \). This inequality can be verified by calculus or known inequalities. Both bounds are well-known properties of \( \frac{\sin x}{x} \) which is a decreasing function in this interval, hence \( \frac{\sin x}{x} \) does satisfy these conditions.

Step 2: Evaluate the Reason Condition

The reason states \( \tan x < x \) for \(0 < x < \frac{\pi}{2}\). To verify, notice that for small values of \(x\), \( \tan x = \frac{\sin x}{\cos x} \) and since \(\sin x\) and \(\cos x\) are less than or equal to 1, it's known that in general \( \tan x > x \) for \(0 < x < \frac{\pi}{2}\), making the reason false.

Step 3: Determine if Reason Explains Assertion

Since the assertion is true and the reason is false, the reason cannot explain the assertion no matter the condition of the assertion itself.

Key Concepts

Trigonometric Inequalities

Understanding trigonometric inequalities such as \( \frac{2}{\pi}<\frac{\sin x}{x}<1 \) requires a solid grasp of the behavior of trig functions. Trigonometric inequalities compare the values of sine, cosine, tangent, etc., in relation to one another, across different intervals. In our example, we focus on the interval \(0 < x < \frac{\pi}{2}\). During this range, \( \sin x/x \) is known as a decreasing function. This means as \(x\) increases from 0 towards \(\frac{\pi}{2}\), the value of \( \sin x/x \) drops. It's bounded between 1 and a smaller fraction, \(2/\pi\). This property is widely acknowledged within mathematical circles, thanks to calculus and trigonometric insights.

• This inequality is pivotal in understanding how trigonometric functions behave in small intervals.
• It illustrates the fundamental nature of \( \sin x/x\) as a sequence approaching zero.
• Helps in approximations and estimative computations involving sinusoidal functions.

Grasping these inequalities empowers students to analyze, estimate, and predict trigonometric outcomes across specific domains.

Calculus

Calculus is the mathematical study that allows us to understand changes and motion, specifically through derivatives and integrals. It’s invaluable for validating the trigonometric inequality \(\frac{\sin x}{x}\) lies between \(\frac{2}{\pi}\) and 1 when \(0 < x < \frac{\pi}{2}\).

• Start by considering that the function \( f(x) = \frac{\sin x}{x} \) can be analyzed using derivatives. This helps us to see how the function increases or decreases.
• The derivative shows the rate at which \( f(x) \) changes as \( x \) changes, indicating \( f(x) \) is indeed decreasing over the interval \(0 < x < \frac{\pi}{2}\).
• Moreover, calculus allows us to establish bounds for functions, confirming well-known properties such as \( \frac{2}{\pi}<\frac{\sin x}{x}<1 \).

By utilizing derivative tests and integral bounds, calculus provides robust frameworks to ensure the accuracy of these inequalities. Thus, calculus isn't just a tool for mathematicians; it's a powerful framework that underpins much of modern science and engineering.

Logical Reasoning

Logical reasoning is the process of using rational, systematic steps to arrive at a conclusion. In the context of assertions and reasons, it’s about determining whether provided statements align and support each other logically. Here, we evaluate two claims: the assertion about \( \frac{\sin x}{x} \) and the reason \( \tan x < x \).

• Logical reasoning requires verifying the truth of each statement independently. The assertion holds true within the given range.
• The reason, however, \( \tan x < x \) is false for \(0 < x < \frac{\pi}{2}\), as opposed to \( \tan x > x \).
• Once verified, logical constructs are used to understand the relationship: if a reasoning can substantiate an assertion.

Evaluating each component through a logical lens ensures clarity. Incorrect reasoning cannot rationalize correct assertions, highlighting the necessity for precise critical analysis in mathematics. Logical reasoning strengthens critical thinking, allowing students to dissect and understand complex ideas systematically.