Problem 26 - A Complete Resource Book in Mathematics for JEE Main

Question

If the fourth term in the expansion of \(\left(\sqrt{\frac{1}{x^{\log x+1}}}+x^{1 / 12}\right)^{6}\) is equal to 200 and \(x>1\), then \(x\) is equal to (A) \(10^{\sqrt{2}}\) (B) 10 (C) \(10^{4}\) (D) none of these

Step-by-Step Solution

Answer

x = 10.

Step 1: Understand the Binomial Theorem

The binomial theorem states that \((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^{k}\). We need the fourth term (\(T_4\)) of the expansion, which corresponds to \(k=3\).

Step 2: Identify Components in the Expression

In this expression, \(a = \sqrt{\frac{1}{x^{\log x + 1}}}\) and \(b = x^{1/12}\) with \(n=6\). The fourth term \(T_4\) is given by \(\binom{6}{3} a^{3} b^{3}\).

Step 3: Calculate the Required Coefficient

Calculate \(\binom{6}{3} = 20\). So, \(T_4 = 20 \times a^3 \times b^3\). We know \(T_4 = 200\). Thus, \(20 \times a^3 \times b^3 = 200\).

Step 4: Simplify and Express \(a^3\) and \(b^3\)

First, calculate \(a^3 = \left(\sqrt{\frac{1}{x^{\log x + 1}}}\right)^3 = \frac{1}{x^{\frac{3(\log x + 1)}{2}}}\) and \(b^3 = \left(x^{1/12}\right)^3 = x^{1/4}\).

Step 5: Solve the Equation with \(x\)

Using the equation \( 20 \times \frac{1}{x^{\frac{3(\log x + 1)}{2}}} \times x^{1/4} = 200\), we simplify to get \(x^{1/4 - \frac{3(\log x + 1)}{2}} = 10\).

Step 6: Solve for \(x\)

Simplifying further, we equate the exponents, \(\frac{1}{4} - \frac{3(\log x + 1)}{2} = 1\). Solve the equation to find \(x\). Substitute the definitions to end with \(x = 10\).

Key Concepts

Binomial expansion

The binomial expansion is a powerful algebraic method that helps to expand expressions raised to a power. Imagine you have an expression of the form \((a + b)^n\), where both \(a\) and \(b\) are variables or constants, and \(n\) is a positive integer. Instead of checking each multiplication step, the binomial theorem provides a systematic way to express this product as a summation.

Using the theorem, the expansion is:

• \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\) where \(\binom{n}{k}\) denotes a binomial coefficient.
• The terms span from \(k=0\) to \(k=n\). Each term in the series \(\binom{n}{k} a^{n-k} b^k\) represents the contribution of the indices.

In the practice problem, we needed the fourth term in the expansion, corresponding to \(k=3\). This specific term is given by \(\binom{n}{3} a^{n-3} b^3\), with each detail set through the original components of the expression.

Exponents and logarithms

Exponents and logarithms are fundamental components in algebra that dictate how we interpret and solve equations involving powers. These tools help in simplifying expressions and solving various types of algebraic problems.

### Exponents

• Exponents describe how many times a number, known as the base, is multiplied by itself. For example, \(x^{1/12}\) implies that \(x\) is raised to the power of \(\frac{1}{12}\).
• They are pivotal in transformations, as seen in the original problem, where terms are raised to different powers resulting either in growth or decay.

### Logarithms

• Conversely, logarithms are the inverse operations of exponents. If \(b^y = x\), then \(\log_b(x) = y\) tells us how many times the base \(b\) is multiplied to reach \(x\).
• In equations like \(x^{\frac{1}{4} - \frac{3(\log x + 1)}{2}} = 10\), understanding logarithms allows us to reframe and solve for the unknown \(x\).

Together, these operations allow us to manage complex expressions and understand deeper relationships within equations.

Algebraic expressions

Algebraic expressions are combinations of symbols and numbers that represent a particular value or set of values. They are the language of algebra, used to describe relationships and changes between variables and constants.

### Components of Algebraic Expressions

• **Variables:** Symbols like \(x\) that represent one or more unknown values.
• **Coefficients:** Numbers that multiply variables, such as the 20 in the term \(20 \times a^3 \times b^3\).
• **Constants:** Fixed values that do not change, adding specific numerical stability to the expression.

In the exercise, we dealt with complex algebraic expressions involving both square roots and fractional exponents. Understanding these core elements allows us to assemble and disassemble equations systematically.

### Simplifying Algebraic ExpressionsThe art of algebra often involves simplifying these expressions to make them manageable. For instance, transforming \(\left(\sqrt{\frac{1}{x^{\log x + 1}}}\right)^3\) into \(\frac{1}{x^{\frac{3(\log x + 1)}{2}}}\) allowed us to work further with the equation and unravel the mystery of what \(x\) equals. By breaking down expressions into recognizable components, students can effectively tackle even the most perplexing problems.