Problem 157 - 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: \(\ln (\cos \theta)<\cos (\ln \theta)\) where \(e^{-\theta 2}<\theta<\frac{\pi}{2}\) Reason: \(\ln x0\)

Step-by-Step Solution

Answer

(B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A)

Step 1: Analyze the Assertion

The assertion is \( \ln (\cos \theta)<\cos (\ln \theta) \) where \( e^{-\theta^2}<\theta<\frac{\pi}{2} \). We need to verify if this inequality holds for the given range of \( \theta \). First, calculate \( \ln (\cos \theta) \) and \( \cos (\ln \theta) \) for some test values within the given range. Generally, \( \ln(x) \) produces a negative value if \( 0

Step 2: Analyze the Reason

The reason provided is \( \ln x < x \) for all \( x > 0 \). This is generally true because the natural logarithm function \( \ln(x) \) increases slower than the linear function \( x \), crossing it exactly at \( x=1 \). For \( x eq 1 \), \( \ln x < x \) is true when \( x>1 \). Therefore, the reason is accurate for \( x \) greater than 1.

Step 3: Validate the Assertion through Calculation

For \( \theta \) in the range \( e^{-\theta^2}<\theta<\frac{\pi}{2} \), we must validate that \( \ln (\cos \theta)<\cos (\ln \theta) \). At this range, \( \cos \theta < \frac{\pi}{2} \), thus making \( \ln (\cos \theta) \) definitively negative. Similarly, \( \cos (\ln \theta) \) remains positive over the interval since \( \theta < \frac{\pi}{2} \) ensures \( \ln(\theta) \) stays positive and small. Thus, the assertion is likely correct without explicit calculations at points within the interval, but ideally this specific inequality confirmation may require deeper examination or numerical validation.

Step 4: Match Reason to Assertion

Check if the reason \( \ln x < x \) correctly explains the assertion \( \ln (\cos \theta)<\cos (\ln \theta) \). While the reason addresses the general nature of logarithmic growth versus linear, it does not directly explain the relationship in the assertion involving both cosine and logarithmic expressions. Thus, while both statement and reason are true, they seem unrelated in context.

Key Concepts

Inequality Analysis

Inequality analysis is the process of examining mathematical expressions to determine the relation between their sizes or magnitudes. When solving inequalities, we often seek to find the range of values for which the inequality holds true. In this exercise, we focus on comparing two expressions involving trigonometric and logarithmic functions, namely - \( \ln (\cos \theta) \)- \( \cos (\ln \theta) \).
To analyze these values, we have to verify if the assertion holds true in the given interval \( e^{-\theta^2}<\theta<\frac{\pi}{2} \). The inequality process involves taking known properties of logarithmic and trigonometric functions, and testing conditions within the defined interval.
The assertion relates specifically to whether the natural logarithm of a cosine value (which will be less than 1 and thus give a negative logarithm) is truly smaller than the cosine of the logarithm of a positive value (which remains positive if the argument is positive and suitably constrained like in this example). Simplifying inequalities involves testing and understanding not just the calculations, but also using theoretical concepts and properties, making this type of problem a true test of mathematical comprehension.

Logarithmic Functions

Logarithmic functions provide inverse operations to exponentials, with the natural logarithm, \( \ln(x) \), being particularly important when the base is the constant \( e \). The function \( \ln(x) \) is defined for \( x > 0 \), and it is crucial to understand that this function increases at a slower rate than any linear function; essentially, \( \ln x < x \) when \( x eq 1 \). This is important because it explains why natural logarithms of values between 0 and 1 will result in negative values, as seen in this exercise.
By deeply comprehending the behavior of \( \ln(x) \), students can better analyze its interaction with other mathematical functions like cosine, and in other mathematical contexts, as shown in this exercise. Knowing the typical behavior of logarithms in conjunction with other functions opens up better insight into complex mathematical interactions, such as those explored in inequalities and calculus.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in mathematics, particularly within calculus and wave analysis. The cosine function, \( \cos(x) \), outputs values between \(-1\) and \(1\), showing its periodic wave up and down. For the interval between \( x = 0 \) and \( x = \frac{\pi}{2} \), \( \cos(x) \) decreases from 1 to 0. Consequently, \( \cos(x) \) decreases in size, showcasing a clear pattern that can be useful in inequalities.
Understanding these fundamentals is crucial for determining how functions like \( \cos(\ln(\theta)) \) behave when wrapped in expressions involving logarithms. As seen in this exercise, the trigonometric aspect complements the properties of logarithms and provides a richer understanding of how combined functional relationships apply across a defined mathematical range. When presenting these in a mathematical context, the interpretation of trigonometric curves can substantively clarify complex interactions like those present in inequalities.

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Problem 158

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Problem 159

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View solution

Key Concepts

Inequality Analysis

Inequality analysis is the process of examining mathematical expressions to determine the relation between their sizes or magnitudes. When solving inequalities, we often seek to find the range of values for which the inequality holds true. In this exercise, we focus on comparing two expressions involving trigonometric and logarithmic functions, namely - \( \ln (\cos \theta) \)- \( \cos (\ln \theta) \).
To analyze these values, we have to verify if the assertion holds true in the given interval \( e^{-\theta^2}<\theta<\frac{\pi}{2} \). The inequality process involves taking known properties of logarithmic and trigonometric functions, and testing conditions within the defined interval.
The assertion relates specifically to whether the natural logarithm of a cosine value (which will be less than 1 and thus give a negative logarithm) is truly smaller than the cosine of the logarithm of a positive value (which remains positive if the argument is positive and suitably constrained like in this example). Simplifying inequalities involves testing and understanding not just the calculations, but also using theoretical concepts and properties, making this type of problem a true test of mathematical comprehension.

Logarithmic Functions

Logarithmic functions provide inverse operations to exponentials, with the natural logarithm, \( \ln(x) \), being particularly important when the base is the constant \( e \). The function \( \ln(x) \) is defined for \( x > 0 \), and it is crucial to understand that this function increases at a slower rate than any linear function; essentially, \( \ln x < x \) when \( x eq 1 \). This is important because it explains why natural logarithms of values between 0 and 1 will result in negative values, as seen in this exercise.
By deeply comprehending the behavior of \( \ln(x) \), students can better analyze its interaction with other mathematical functions like cosine, and in other mathematical contexts, as shown in this exercise. Knowing the typical behavior of logarithms in conjunction with other functions opens up better insight into complex mathematical interactions, such as those explored in inequalities and calculus.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in mathematics, particularly within calculus and wave analysis. The cosine function, \( \cos(x) \), outputs values between \(-1\) and \(1\), showing its periodic wave up and down. For the interval between \( x = 0 \) and \( x = \frac{\pi}{2} \), \( \cos(x) \) decreases from 1 to 0. Consequently, \( \cos(x) \) decreases in size, showcasing a clear pattern that can be useful in inequalities.
Understanding these fundamentals is crucial for determining how functions like \( \cos(\ln(\theta)) \) behave when wrapped in expressions involving logarithms. As seen in this exercise, the trigonometric aspect complements the properties of logarithms and provides a richer understanding of how combined functional relationships apply across a defined mathematical range. When presenting these in a mathematical context, the interpretation of trigonometric curves can substantively clarify complex interactions like those present in inequalities.