Problem 129 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(x\) is so small that \(x^{3}\) and higher powers of \(x\) may be \(\begin{aligned}&\text { neglected, then } \\\&\text { mated as } & (1+x)^{3 / 2}-\left(1+\frac{1}{2} x\right)^{3} \\\&\text { may be approxi- } \\\&(1-x)^{1 / 2} & {[2005]}\end{aligned}\) (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 answer is (C) \(-\frac{3}{8}x^2\).

Step 1: Understand the problem

We need to approximate the expression \((1+x)^{3/2}-\left(1+\frac{1}{2} x\right)^3\) assuming that \(x\) is very small, meaning that terms like \(x^3\) and higher can be neglected.

Step 2: Expand using Binomial Approximation

When \(x\) is small, the binomial expansion can be used for approximation: \((1+x)^n \approx 1 + nx + \frac{n(n-1)}{2}x^2\). We'll use this to expand \((1+x)^{3/2}\) and \((1+\frac{1}{2}x)^3\).

Step 3: Expand \((1+x)^{3/2}\)

Using binomial approximation:\[ (1+x)^{3/2} \approx 1 + \frac{3}{2}x + \frac{3}{8}x^2 \].

Step 4: Expand \((1+\frac{1}{2}x)^3\)

Similarly, applying the binomial expansion:\[ (1+\frac{1}{2}x)^3 \approx 1 + \frac{3}{2}x + \frac{3}{4}x^2 \].

Step 5: Subtract the two expanded expressions

Subtract the expression from Step 4 from the expression in Step 3:\[ (1 + \frac{3}{2}x + \frac{3}{8}x^2) - (1 + \frac{3}{2}x + \frac{3}{4}x^2) \].

Step 6: Simplify the subtraction

After subtracting, we cancel out the matching terms and get:\[ 0 + 0 + \left(\frac{3}{8}x^2 - \frac{3}{4}x^2\right) = -\frac{3}{8}x^2 \].

Step 7: Match to the given options

The simplified expression is \(-\frac{3}{8}x^2\), which matches option (C).

Key Concepts

Small Angle Approximation

The Small Angle Approximation is a powerful mathematical technique used when angles or values are sufficiently small, making it possible to simplify complex expressions. This approach is especially useful in situations where dealing with powers of a variable can become cumbersome. - The key idea is that if a variable, say \(x\), is very small, higher powers like \(x^3\), \(x^4\), etc., have less impact and can often be neglected to simplify calculations.- For example, when expanding expressions like \((1+x)^n\) using Taylor or Binomial series, only the first few terms are kept: \(1 + nx + \frac{n(n-1)}{2}x^2\). This assumes that terms involving \(x^3\) and higher are quite small and can thus be disregarded.In physics and engineering, this approximation finds frequent use, allowing us to work with simpler expressions that are close approximations of more complex realities.

Mathematical Proof

A mathematical proof is a logical process that verifies the validity of a given statement using previously established facts, such as theorems, axioms, and lemmas. The goal of a proof is to demonstrate that if certain premises are true, then a particular conclusion must also be true. This provides a solid foundation for mathematical concepts and ensures consistency within the field. When approximating expressions using the binomial theorem, proof methods often: - Establish base facts and build upon these through deduction to show that approximations are logically consistent. - Use substitution and simplification techniques to derive results that naturally follow from basic principles. Proofs in mathematics not only confirm results but also help build a deeper understanding of why certain properties hold, through a series of logical steps and reasoning.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions or equations to make them more manageable. It is often employed for expression simplification, solving equations, or transforming functions. This process often requires a mix of rules and techniques to ensure that the manipulations maintain the equality or properties initially present.Key techniques in algebraic manipulation include:- **Simplification**: Canceling out like terms by recognizing patterns within expressions (e.g., subtracting similar terms like in the solution: \(\frac{3}{8}x^2 - \frac{3}{4}x^2\)).- **Factorization**: Breaking down expressions into products of simpler expressions, which can often make subsequent calculations easier.These techniques are crucial for effectively working with mathematical expressions, particularly when aiming to reach a desired form or simplify complex problems. The ability to manipulate algebraic structures is foundational in both pure and applied mathematics.