Problem 129 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(x\) is so small that \(x^{3}\) and higher powers of \(x\) may be neglected, then \(\frac{(1+x)^{3 / 2}-\left(1+\frac{1}{2} x\right)^{3}}{(1-x)^{1 / 2}}\) may be approximated as (A) \(1-\frac{3}{8} x^{2}\) (B) \(3 x+\frac{3}{8} x^{2}\) (C) \(-\frac{3}{8} x^{2}\) (D) \(\frac{x}{2}-\frac{3}{8} x^{2}\)

Step-by-Step Solution

Answer

The correct answer is (C) \(-\frac{3}{8} x^{2}\).

Step 1: Simplify Using Binomial Expansion

Starting with the expression \((1+x)^{3/2}\), use binomial expansion: \[(1+x)^{3/2} \approx 1 + \frac{3}{2}x + \frac{3}{8}x^2\]Only terms up to \(x^2\) are considered since higher powers of \(x\) are neglected.

Step 2: Expand the Denominator

For the denominator \((1-x)^{1/2}\), use binomial expansion:\[(1-x)^{1/2} \approx 1 - \frac{1}{2}x\]Again, we only consider terms up to \(x^2\).

Step 3: Expand the Inner Parentheses

For \((1 + \frac{1}{2} x)^{3}\), expand using the binomial theorem:\[(1 + \frac{1}{2}x)^3 = 1 + \frac{3}{2}x + \frac{3}{4}x^2 + \frac{1}{8}x^3\]Neglecting the \(x^3\) term, we simplify to:\[1 + \frac{3}{2}x + \frac{3}{4}x^2\]

Step 4: Compute the Numerator

Compute the numerator: \((1+x)^{3/2} - (1+\frac{1}{2}x)^3\).Substituting from previous steps, it becomes:\[(1 + \frac{3}{2}x + \frac{3}{8}x^2) - (1 + \frac{3}{2}x + \frac{3}{4}x^2)\]Simplify the expression:\[-\frac{3}{8}x^2\]

Step 5: Divide the Numerator by the Denominator

Now divide \(-\frac{3}{8}x^2\) by \(1-\frac{1}{2}x\):\[\frac{-\frac{3}{8}x^2}{1-\frac{1}{2}x} \approx -\frac{3}{8}x^2 \times (1 + \frac{1}{2}x)\]Simplify the expression:\[-\frac{3}{8}x^2 - \frac{3}{16}x^3\]Since \(x^3\) and higher powers are neglected, the resulting expression is:\[-\frac{3}{8}x^2\]

Step 6: Finalize the Approximation

The final simplified expression is:\[-\frac{3}{8}x^2\]Compare this with given options, which is identical to option (C).

Key Concepts

Approximation Techniques

In mathematics, when dealing with complex expressions, approximation techniques simplify the process of finding solutions. A common way of achieving this simplification is by using binomial expansion. This method breaks down a binomial expression raised to a power into a polynomial series. For example, with expressions like \(1+x\) raised to non-integral powers, we can use the binomial theorem to approximate the outcome by a series expansion:

• We often only retain terms up to a certain power, which is sufficient for approximations when \(x\) is small.
• For higher powers, the terms become increasingly insignificant.

By approximating complex expressions, you reduce the amount of work needed, making calculations manageable and faster when higher precision is not required.

Neglecting Higher Powers

When engaging with expressions that include powers of a variable, particularly small ones, it quite often becomes necessary to set aside high power terms. This concept is pivotal when using binomial expansions where variables are small enough that their high powers have minimal impact. Here's why we do it:

• High power terms have decreasing effects on the value of an expression, especially when the base (like \(x\) in \(x^n\)) is less than 1. They shrink rapidly, making their contribution negligible.
• This allows us to focus solely on the lower power terms that meaningfully affect the outcome.

By neglecting terms like \(x^3\), \(x^4\), etc., we make problems easy to solve, especially in situations requiring swift, manageable computations. This approach is especially useful in educational settings to help students grasp the behavior of functions without the burden of overwhelming complexity.

Simplifying Algebraic Expressions

Simplifying algebraic expressions is a crucial skill for solving mathematics problems more effectively. It involves reducing expressions to their most straightforward form so that their characteristics are easier to understand and compute. During simplification, especially while using approximation techniques and neglecting higher powers:

• One typically combines like terms where possible, managing coefficients for terms of the same degree.
• It's essential to handle expressions step-by-step, systematically converting complex parts into simpler ones to avoid mistakes.

For instance, in the problem we are discussing, expressions like \(1+x\) raised to a power are expanded and then simplified by removing higher-order terms. By organizing terms, algebraic expressions become less complicated, serving as a foundation for more advanced problem-solving activities in both academic and practical mathematical applications.