Problem 52 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The total number is irrational terms in the binomial expansion of \(\left(7^{\frac{1}{5}}-3^{\frac{1}{10}}\right)^{60}\) is: \(\quad\) (a) 55 (c) 48 (b) 49 (d) 54

Step-by-Step Solution

Answer

The total number of irrational terms is 54.

Step 1: Understand the Binomial Expansion

The binomial expansion of \((a-b)^n\) is given by the formula \(\sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\). In our case, the expression is \((7^{\frac{1}{5}} - 3^{\frac{1}{10}})^{60}\) which means \(a = 7^{\frac{1}{5}}\), \(b = 3^{\frac{1}{10}}\), and \(n = 60\).

Step 2: Determine which terms are irrational

For a term \(\binom{60}{k} (7^{\frac{1}{5}})^{60-k} (3^{\frac{1}{10}})^k\) to be rational, the exponents \((60-k)\times \frac{1}{5}\) and \(k \times \frac{1}{10}\) must be integers. This implies that \(60-k\) must be a multiple of 5 and \(k\) must be a multiple of 10.

Step 3: Find the values of k making terms rational

For \((60-k) \times \frac{1}{5}\) to be an integer, \(60 - k\) must be 0, 5, 10, ..., up to 60, giving values of \(k = 0, 5, 10, ..., 60\), which implies \(k\) must be a multiple of 10. Check higher conditions: for \(k \times \frac{1}{10}\) to also be integer, \(k\) must indeed be a multiple of 10.

Step 4: Count the Rational Terms

The possible values of \(k\) that satisfy both conditions are multiple of 10: 0, 10, 20, 30, 40, 50, and 60. Count how many values there are: 7 values: 0 through 60 by 10s.

Step 5: Calculate the Irrational Terms

The total number of terms in the expansion is \(n+1 = 61\). Since there are 7 rational terms, the number of irrational terms is \(61 - 7 = 54\).

Key Concepts

Irrational Terms

In the context of binomial expansion, an irrational term refers to a term that does not simplify to an integer or a rational number. When examining a binomial expression with roots, such as \(7^{\frac{1}{5}} - 3^{\frac{1}{10}}\), terms can include expressions with roots or fractional powers.

For example, a term in this specific exercise can be expressed as \(\binom{60}{k} (7^{\frac{1}{5}})^{60-k} (3^{\frac{1}{10}})^k\). Whether each term is irrational depends on the exponents \(60-k\) and \(k\) when multiplied by their respective fractional powers of \(\frac{1}{5}\) and \(\frac{1}{10}\). If neither exponent becomes an integer, the term remains irrational.

• To be rational, \(60-k\) must be divisible by 5.
• Similarly, \(k\) must be divisible by 10.

All terms not fulfilling these conditions are irrational.

This exercise aims to find how many of the terms in the expansion remain irrational, which in this specific problem is 54.

Rational Exponents

Rational exponents are exponents that are fractions, opposed to whole numbers. Using the expression \(7^{\frac{1}{5}}\), the exponent \(\frac{1}{5}\) signifies the fifth root of 7, while \(3^{\frac{1}{10}}\) indicates the tenth root of 3.

In binomial expansions, determining if a term is rational or irrational involves checking whether these fractional exponents multiply to yield a whole number.

• For \(7^{\frac{1}{5}}\), the exponent \(60-k\) must be a multiple of 5, meaning an integer power of a is formed.
• For \(3^{\frac{1}{10}}\), \(k\) must be a multiple of 10, indicating the exponent resolves to a whole number.

Rational exponents require careful examination because only terms that convert to whole numbers provide rational outcomes. This integral aspect enables us to solve the problem efficiently by delineating between rational and irrational terms.

Binomial Coefficient

The binomial coefficient, represented as \(\binom{n}{k}\), plays a crucial role in binomial expansion. It is a way to determine the number of ways to choose \(k\) elements from a set of \(n\) elements without considering the order.

In any binomial expansion, coefficients are formed using this notation. The term \(\binom{60}{k}\) determines the coefficient for the binomial term \( (7^{\frac{1}{5}})^{60-k} (3^{\frac{1}{10}})^k\).

• This process follows the general formula, \[\binom{n}{k} = \frac{n!}{k!(n-k)!}\], where \(n!\) represents the factorial of \(n\).
• In the problem given, \(n = 60\), illustrating the number of total terms or expansions available in \( (a-b)^{60}\).

Understanding the function and purpose of the binomial coefficient helps delineate the distribution of terms in the expansion, and aids in identifying how each term contributes to the total sum of rational and irrational outputs.