Problem 53 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

A ratio of the \(5^{\text {th }}\) term from the begining to the 5 th term from the end in the binomial expansion of \(\left(2^{\frac{1}{3}}+\frac{1}{2(3)^{\frac{1}{3}}}\right)^{10}\) is: (a) \(1: 2(6)^{\frac{1}{3}} \quad\) (b) \(1: 4(16)^{\frac{1}{3}}\) (c) \(4(36)^{\frac{1}{3}}: 1\) (d) \(2(36)^{\frac{1}{3}}: 1\)

Step-by-Step Solution

Answer

Option (d) \(2(36)^{\frac{1}{3}}: 1\).

Step 1: Identify the Formula for Binomial Expansion

The binomial expansion of \((a + b)^n\) is given by the formula: \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \] where \(\binom{n}{k}\) is the binomial coefficient.

Step 2: Determine the Terms from the Formula

In this exercise, the terms in the binomial expansion are:\[ T_k = \binom{10}{k-1} (2^{\frac{1}{3}})^{10-(k-1)} \left(\frac{1}{2(3)^{\frac{1}{3}}}\right)^{k-1} \] where \(k\) is the term number, beginning from \(k=1\).

Step 3: Find the 5th Term from the Beginning

The 5th term from the beginning corresponds to \(k=5\). Use the term formula:\[ T_5 = \binom{10}{4} (2^{\frac{1}{3}})^6 \left(\frac{1}{2(3)^{\frac{1}{3}}}\right)^4 \]Calculate:\[ T_5 = \binom{10}{4} \times (2^{2}) \times \frac{1}{16(3)^{\frac{4}{3}}} \] \[ T_5 = 210 \times 4 \times \frac{1}{16(3)^{\frac{4}{3}}} \]\[ T_5 = \frac{210}{4(3)^{\frac{1}{3}}} \]

Step 4: Find the 5th Term from the End

With a binomial theorem, the 5th term from the end is identical to the term obtained with the reflected positions in \((k)\) indices. So:\[ T_6 = \binom{10}{5} (2^{\frac{1}{3}})^5 \left(\frac{1}{2(3)^{\frac{1}{3}}}\right)^5 \]Calculate:\[ T_{6} = \binom{10}{5} \times (2^{\frac{5}{3}}) \times \frac{1}{32(3)^{\frac{5}{3}}} \]\[ T_{6} = \frac{252 \times 32}{32(3)^{\frac{5}{3}}} \]\[ T_{6} = \frac{252 \times 2^{2}}{8(3)^{\frac{1}{3}}} \]

Step 5: Compute the Ratio of the Terms

The ratio of the 5th term from the beginning to the 5th term from the end is given by:\[ \text{Ratio} = \frac{T_5}{T_6} = \frac{\frac{210}{4(3)^{\frac{1}{3}}}}{\frac{252 \times 2^{2}}{8(3)^{\frac{1}{3}}}} \]This simplifies to:\[ \frac{T_5}{T_6} = 2(36)^{\frac{1}{3}} \]

Step 6: Identify the Matching Option

The ratio we calculated is equivalent to option (d) \(2(36)^{\frac{1}{3}}: 1\). Therefore, the correct answer is option (d).

Key Concepts

Binomial Coefficient

In the realm of binomial expansions, understanding the binomial coefficient is essential. The binomial coefficient, denoted as \( \binom{n}{k} \), represents the number of ways to choose \( k \) elements from \( n \) elements without regard to the order of selection. It is a crucial component when expanding binomial expressions.

It is calculated using the formula:

• \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \)

where \( n! \) (read as "n factorial") signifies the product of all positive integers up to \( n \). For example, in the binomial expansion of \( (a + b)^{10} \), \( \binom{10}{4} \) calculates how many ways four elements can be chosen from ten, functioning as a multiplier for the term it represents.

The binomial coefficient is inherent in each term's computation within the expansion. Recognizing its role simplifies the process of determining specific terms and ultimately understanding the structure of the entire polynomial expression.

Term Calculation

Determining specific terms in a binomial expansion is grounded in understanding the pattern and formula offered by the Binomial Theorem. Each term in the expansion of \( (a + b)^n \) is given by:

• \( T_k = \binom{n}{k-1} a^{n-(k-1)} b^{k-1} \)

where \( k \) refers to the term number in the sequence. Here, each term is a product of a binomial coefficient, and variable powers determined by the position of the term.

For example, consider the 5th term from the beginning in the expression \( \left(2^{\frac{1}{3}} + \frac{1}{2(3)^{\frac{1}{3}}}\right)^{10} \). Using the formula, the term would be \( T_5 = \binom{10}{4} \times (2^{\frac{1}{3}})^6 \times \left(\frac{1}{2(3)^{\frac{1}{3}}}\right)^4 \). Term calculation means translating the abstract formula into a specific answer, which involves calculating each part's power and the binomial coefficient.

This part of the process ensures you correctly identify and evaluate the specific term you are looking for.

Exponentiation in Binomial Theorem

Exponentiation plays a significant role in the binomial theorem as it dictates the behaviour of terms in an expansion. When expanding \( (a + b)^n \), each term involves raising the binomial's two components to specific powers that always sum to \( n \).

The pattern follows that the first component \( a \) starts with the highest power \( n \) and reduces by one in each term, while the second component \( b \) starts with power zero and increases by one in each term. This predictable shift of exponents is what maintains the balance across the terms in the expansion.

In practical application, such as in \( \left(2^{\frac{1}{3}}+\frac{1}{2(3)^{\frac{1}{3}}}\right)^{10}, \) careful attention to exponentiation is required. Calculating the powers of the components correctly ensures that the resulting individual terms reflect the structure dictated by the theorem. This is crucial for finding terms, understanding their sizes, and preparing them for further manipulations like comparing or constructing ratios.