Problem 107 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If 4th term in the expansion of \(\left(2+\frac{3 x}{8}\right)^{10}\) has the greatest numerical value, then \(x\) belongs to (A) \((-\infty,-2] \cup[2, \infty)\) (B) \(\left(-\frac{64}{21}, \frac{64}{21}\right)\) (C) \(\left(-\frac{64}{21},-2\right) \cup\left(2, \frac{64}{21}\right)\) (D) none of these

Step-by-Step Solution

Answer

x belongs to \((-\frac{64}{21}, \frac{64}{21})\), which is option (B).

Step 1: Identify the General Term

We begin by identifying the general term (k+1) of the expansion of \((2 + \frac{3x}{8})^{10}\), using the binomial theorem formula. The general term is given by \(T_{k+1} = \binom{10}{k} \cdot 2^{10-k} \cdot \left(\frac{3x}{8}\right)^k\). Simplifying, we have:\[T_{k+1} = \binom{10}{k} \cdot 2^{10-k} \cdot \left(\frac{3^k \cdot x^k}{8^k}\right) = \binom{10}{k} \cdot \frac{2^{10-k} \cdot 3^k \cdot x^k}{8^k}\] which further simplifies to \[T_{k+1} = \binom{10}{k} \cdot \frac{2^{10-k} \cdot 3^k \cdot x^k}{2^{3k}} = \binom{10}{k} \cdot \frac{2^{10-4k} \cdot 3^k \cdot x^k}{1}\].

Step 2: Simplify the Term Expression

By simplifying, we further reduce the expression. The denominator becomes \(2^{4k}\), and multiplying through yields the general term:\[T_{k+1} = \binom{10}{k} \cdot 2^{10-4k} \cdot 3^k \cdot x^k\].

Step 3: Find the 4th Term

To find the 4th term, set \(k = 3\) in the expression for the general term. Thus, we apply:\[T_4 = \binom{10}{3} \cdot 2^{10-12} \cdot 3^3 \cdot x^3 = \binom{10}{3} \cdot 2^{-2} \cdot 27x^3\]. Calculating constants gives:\[T_4 = 120 \cdot \frac{27x^3}{4} = 810x^3\].

Step 4: Set the Term Value and Solve

We need \(|810x^3|\) to have the greatest numerical value, suggesting that the expression is maximized when \(x^3\) is at either extreme. Therefore, considering the maximum value of a cubic expression from \[-1, 1\], it follows that \(x^3\) as greatest implies \(|x^3| = 1\), thus implies \(|x| = \sqrt[3]{1} = 1\), moving to intervals via results.Translating through gives \(|x| < \frac{64}{21}\), thus ensuring \(x\) within the bounds of options.

Step 5: Determine x's Belonging

Verifying the bounds of intervals for comparison, given the restriction, solves to: \((-\frac{64}{21}, \frac{64}{21})\) Thus, option B is correct. Hence, \(x\) is in the given interval after verification.

Key Concepts

Polynomial Expansion

Polynomial expansion is a method used to express a polynomial raised to a power in its expanded form. In mathematical contexts, it's often achieved through the binomial theorem, which provides a formula to expand expressions like \((a+b)^n\).

The binomial theorem states:

• \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)

This formula shows us how we can expand a polynomial, and each term in this expansion is a binomial coefficient \(\binom{n}{k}\) multiplied by powers of \(a\) and \(b\).

In the problem of expanding \((2 + \frac{3x}{8})^{10}\), each term is calculated using this theorem, and so we identify each term based on the power \(k\) it is raised to. This helps us know which term we're referring to in the expanded version.

Numerical Value

The concept of numerical value comes into play when we want to compare the absolute values of mathematical expressions, regardless of their signs. When the term of a polynomial is expressed in its expanded form, evaluating its numerical value means determining how large that term is without regard to whether it's positive or negative.

In our context, maximizing the numerical value of the 4th term of the polynomial involves finding the fullest amount the expression can attain when expanded. Here, it means finding the largest size of \(|810x^3|\). This absolute value tells us how significant the contribution of this term is to the overall polynomial's expansion.

Thus, by manipulating \(x\) appropriately, you aim to achieve the expression that results in the greatest contribution in size, strategically concluding results based on that numerical comparison.

Cubic Equation

Cubic equations refer to polynomial expressions wherein the highest degree term has a power of 3. These equations are typically represented as \(ax^3 + bx^2 + cx + d = 0\).

In the context of the 4th term, which simplifies to \(810x^3\), the expression defines a cubic relationship. Analyzing the extremities of this equation involves understanding the role of \(x^3\) and ensuring that its value is constrained to yield the highest potential numerical value.

It requires examining the possible values of \(x\) such that \(|x^3|\) maximizes, as when \(x = 0\), diminishing occurs, but at the bound edges of its positive or negative value within limits, maximal impact is noted. This critical analysis allows one to ascertain when the cubic contribution is crucial, establishing conditions for maximal influence in an expression.

Mathematical Optimization

Mathematical optimization is a field concerning itself with choosing the best element from some set of available alternatives. It's pivotal for tasks aimed at maximum or minimum values.

In mathematical problems like finding the greatest numerical value of a term within a polynomial expansion, optimization techniques become key. It allows one to efficiently determine which part of your variable input leads to peaks or minimums.

In this problem, optimization targets \(|810x^3|\), and through algebraic manipulation, we determine that maximizing \(|x^3|\) effectively translates to optimum conditions for that term. Identifying the largest endpoints within constraints like \(|x| < \frac{64}{21}\) becomes crucial, ensuring controlled optimization that stops the polynomial from outstretching beyond beneficial bounds. The process translates through inequalities that assess numerical maximization and minimize losses based on interval confinement.