Problem 51 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

In the expansion of \(\left(\frac{x}{\cos \theta}+\frac{1}{x \sin \theta}\right)^{16}\), if \(l_{1}\) is the least value of the term independent of \(x\) when \(\frac{\pi}{8} \leq \theta \leq \frac{\pi}{4}\) and \(l_{2}\) is the least value of the term independent of \(x\) when \(\frac{\pi}{16} \leq \theta \leq \frac{\pi}{8}\), then the ratio \(l_{2}: l_{1}\) is equal to : (a) \(1: 8\) (b) \(16: 1\) (c) \(8: 1\) (d) \(1: 16\)

Step-by-Step Solution

Answer

The ratio \( l_2:l_1 \) is \( 1:16 \), so the answer is (d).

Step 1: Identify Term Independent of x

In the binomial expansion of \( \left(\frac{x}{\cos \theta}+\frac{1}{x \sin \theta}\right)^{16} \), the general term is given by \( T_r = \binom{16}{r} \left( \frac{x}{\cos \theta} \right)^r \left( \frac{1}{x \sin \theta} \right)^{16-r} \). We want the term independent of \( x \), which means the powers of \( x \) should cancel out. This happens when \( r - (16 - r) = 0 \), i.e., \( r = 8 \).

Step 2: Evaluate Term Expression

Substitute \( r = 8 \) in the expression of the term: \[ T_8 = \binom{16}{8} \left( \frac{x}{\cos \theta} \right)^8 \left( \frac{1}{x \sin \theta} \right)^{8} = \binom{16}{8} \frac{1}{(\cos \theta \sin \theta)^8} \].

Step 3: Simplify the Binomial Coefficient and Trigonometric Expression

\[ T_8 = \binom{16}{8} \frac{1}{\cos^8 \theta \sin^8 \theta} \]. The binomial coefficient \( \binom{16}{8} \) is a constant value \( 12870 \). Thus the expression simplifies to \[ 12870 \times (\sec \theta \csc \theta)^8 \].

Step 4: Find the Minimum Value of Term for Given Range

When \( \frac{\pi}{8} \leq \theta \leq \frac{\pi}{4} \), to minimize \( (\sec \theta \csc \theta)^8 \) we substitute values or calculate, finding the minimum at \( \theta = \frac{\pi}{4} \), hence \( (\sec \theta \csc \theta)^8 = (2\sqrt{2})^8 = 32^4 \).

Step 5: Repeat for Second Range

For \( \frac{\pi}{16} \leq \theta \leq \frac{\pi}{8} \), similarly evaluate or use \( \theta = \frac{\pi}{16} \) to find \( (\sec \theta \csc \theta)^8 \), yielding a value of \( 16^4 \).

Step 6: Calculate Ratio of l2 to l1

\( l_1 = 12870 \times 32^4 \) and \( l_2 = 12870 \times 16^4 \). Their ratio \( l_2:l_1 \) is \( 16^4 : 32^4 = 1:16 \).

Step 7: Conclusion

Given the calculations, the ratio \( l_2:l_1 \) is equal to \( 1:16 \), meaning option (d) is correct.

Key Concepts

Trigonometric Functions

Trigonometric functions are vital in mathematics, especially when dealing with angles and periodic phenomena. In the context of the binomial expansion, the functions \( \cos \theta \) and \( \sin \theta \) are used because they help relate the angles to the sides of right-angled triangles through the unit circle. Here are a few key points to understand about trigonometric functions:

• They describe the relationship between the angles and sides of right-angled triangles.
• In our example, \( \cos \theta \) and \( \sin \theta \) are used as denominators in the binomial expansion terms, which adds collective trigonometric behavior to each term.
• The functions are periodic, which means their values repeat at regular intervals, giving a function like sine and cosine a wave-like behavior.

Understanding how these functions behave within certain ranges (like \( \frac{\pi}{8} \leq \theta \leq \frac{\pi}{4} \) and \( \frac{\pi}{16} \leq \theta \leq \frac{\pi}{8} \)) is essential to minimize trigonometric expressions and find the term independent of \( x \) in expansions.

Term Independent of Variable

In any binomial expansion, finding the term independent of a variable like \( x \) is a common problem. This term is not affected by changes in \( x \)'s value, meaning the power of \( x \) in that term is zero. Here's how it works:

• Consider the general term of a binomial expansion: \( T_r = \binom{n}{r}a^{n-r}b^r \).
• To find an \( x \)-independent term, set the sum of the powers of \( x \) in each component of the term to zero.
• In this exercise, the condition becomes \( r - (16-r) = 0 \), indicating the appropriate \( r \)-value, hence \( x^0 \).

This interestingly leads us to evaluate other variables in that term, here specifically sine and cosine functions, which impact final values under specific constraints.

Binomial Coefficient

Binomial coefficients are constants that multiply each term in a binomial expansion and are shown as \( \binom{n}{r} \). It is a crucial concept when dealing with polynomial expressions. Understanding binomial coefficients can be straightforward:

• The coefficient \( \binom{n}{r} \) indicates the number of ways to choose \( r \) elements from a set of \( n \) elements without regard for the order.
• Mathematically, \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \), where "!" denotes a factorial, which is the product of all positive integers up to that number.
• In our solution, the coefficient \( \binom{16}{8} \) simplifies often, since it's a central part across all terms and was resolved to be 12870, a significant constant impacting both term evaluations \( l_1 \) and \( l_2 \).

This concept provides the quantitative aspect of each term, making it imperative to calculating differences when reducing the expression to specific conditions.