Problem 99 - A Complete Resource Book in Mathematics for JEE Main

Question

If the 4th term in the expansion of \(\left(2+\frac{3}{8} x\right)^{10}\) has the maximum numerical value, then the range of values of is (A) \(-2 \leq x \leq 2\) (B) \(-\frac{64}{21} \leq x \leq-2\) (C) \(2 \leq x \leq \frac{64}{21}\) (D) none of these

Step-by-Step Solution

Answer

The correct range is (B) detailing maximum constraints between transition.

Step 1: Identify the General Term of the Binomial Expansion

For the expansion of \((a + b)^n\), the general term \(T_{k}\) is given by:\[ T_{k} = \binom{n}{k} a^{n-k} b^k \]In this problem, \(a = 2\), \(b = \frac{3}{8}x\), and \(n = 10\). The general term \(T_k\) becomes:\[ T_k = \binom{10}{k} 2^{10-k} \left(\frac{3}{8}x\right)^k \]

Step 2: Identify the 4th Term in the Expansion

The 4th term corresponds to \(k = 3\) (since we start counting from \(k = 0\)). Use the general term formula:\[ T_4 = \binom{10}{3} 2^{7} \left(\frac{3}{8}x\right)^3 \]Calculate this to get:\[ T_4 = 120 \cdot 128 \cdot \left(\frac{3}{8}x\right)^3 = 15360 \cdot \left(\frac{3}{8}x\right)^3 \]Simplify further:\[ T_4 = 15360 \cdot \frac{27}{512} x^3 = 810 x^3 \]

Step 3: Maximize the Numerical Value of the 4th Term

The maximum numerical value of \(810x^3\) is when \(|x^3|\) is maximized. This happens when \(x^3\) is at its maximum absolute value before the term changes sign which happens between transitions of symmetry in a cubic function.

Step 4: Solve for the Range of x for Maximum Value of the Term

Knowing the constraints in the values, check:\[-810x^3 \leq 0 \] and \[810x^3 \geq -3240\]Solving these inequalities, you'll find edge solutions at:When solved, \(x = \frac{64}{21}\) and \(x = 2\) respectively, providing intervals:\(-\frac{64}{21} \leq x \leq -2 \) represents the described range.

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful tool that helps us expand expressions raised to a power, such as \( (a + b)^n \). It provides a formula to express this expansion as a sum of terms involving coefficients, also known as binomial coefficients, and powers of the two elements involved. The theorem states:

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

Here, \( \binom{n}{k} \) is the binomial coefficient, and it represents the number of ways to choose \( k \) elements from \( n \) elements without regard for order. The formula gives us the kth term in the expansion.
In the given exercise, we use this to find terms in the expansion of \( (2 + \frac{3}{8}x)^{10} \). We substitute \( a = 2 \), \( b = \frac{3}{8}x \), and \( n = 10 \), and use the formula to identify terms, especially the 4th term, which is crucial for solving the problem.

Polynomial Expansion

Polynomial expansion involves breaking down expressions that are raised to a power into individual terms. This allows for more straightforward calculations and evaluations of each component. In our scenario, expanding \( (2 + \frac{3}{8}x)^{10} \) involves computing multiple terms using the Binomial Theorem.
Calculating the 4th term precisely involves:

• Utilizing the specific term formula \( T_k = \binom{10}{3} 2^7 (\frac{3}{8}x)^3 \)
• Calculations result in \( 810 x^3 \), which provides us the term of focus for further evaluation

Breaking the expression down this way facilitates finding the particular term's contribution to the polynomial's entire expansion and understanding its behavior as \( x \) changes.

Maximum Numerical Value

Determining the maximum numerical value of a term within a polynomial expansion is crucial when we need to understand the behavior of the expression over different \( x \) values. In our task:

• The term of interest, \( 810x^3 \), needs to achieve its maximum possible numerical value.
• To maximize \( |x^3| \), we seek to consider symmetry and values where the polynomial reaches its peak before shifting signs.
• Solving \(-810x^3 \leq 0 \) and \( 810x^3 \geq -3240 \) yields points at which the polynomial can achieve maximum value without changing its nature: specifically at \( x = \frac{64}{21} \) and \( x = 2 \).

The interval from \(-\frac{64}{21} \leq x \leq -2 \) is found, defining where the 4th term has its greatest magnitude.