Problem 25 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If \(1+x^{4}+x^{5}=\sum_{i=0}^{5} a_{i}(1+x)^{i}\), for all \(x\) in \(R\), then \(a_{2}\) is: (a) \(-4\) (b) 6 (c) \(-8\) (d) 10

Step-by-Step Solution

Answer

The correct answer is not available in the options provided.

Step 1: Expand the Given Equation

Start by understanding the given equation. The expression \(1+x^4+x^5 = \sum_{i=0}^{5} a_{i}(1+x)^{i}\) means the left side of the equation can be represented with a polynomial expansion on the right side where each \(a_i\) represents a coefficient of \((1+x)^i\).

Step 2: Identify Terms of Expansion

Recognize that the expression \(1+x^4+x^5\) is a polynomial expression that should match the expansion \(\sum_{i=0}^{5} a_{i}(1+x)^{i}\). This requires determining how these terms appear in the expansion when \((1+x)^i\) are expanded out and combined.

Step 3: Analyze Each Power of (1 + x) Separately

Consider the powers of \((1+x)^i\) starting from 0, 1, 2 and so on, up to 5. Understanding how each power contributes to the overall expansion will help identify how many terms of each power should match with \(x^4\) and \(x^5\) from the given left-hand side.

Step 4: Consider (1 + x)^2

The term \((1+x)^2 = 1 + 2x + x^2\) contributes to the coefficient of \(x^2\) in the expansion. Since lower powers \((1, x)\) are not supposed to match with \(x^4\) or \(x^5\), the coefficient of \(x^2\) directly comes from here as there are no \(x^2\) terms on the left hand side alone.

Step 5: Substitute Values for Matching Terms

The requirement is to only find \(a_2\) and since neither \(\boldsymbol{x^4}\) nor \(\boldsymbol{x^5}\) contain 2x terms and only occurs directly from \((1+x)^2\), so the coefficient \(a_2\) in this expansion equals the coefficient of \(x^2\) in \((1+x)^2\) which is 1.

Step 6: Review and Final Calculation

After all these analyses, it is evident that \(a_2 = 1\) from our calculations of how \((1+x)^2\) contributes its terms to the overall polynomial.Thus concluding that \(a_2 = 1 \).

Key Concepts

Coefficients Identification

When dealing with polynomial expansions, particularly when expressed in the form \( \sum_{i=0}^{5} a_{i}(1+x)^{i} \), it is crucial to identify the coefficients correctly. Coefficients represent the magnitude attached to each term of the polynomial. In analyzing polynomials, you break them down into simpler parts to ensure each term aligns correctly with the corresponding power of \((1+x)^{i}\).
If you are given an expression like \(1+x^4+x^5\), this should equate to a polynomial expansion such that each coefficient \( a_i \) reflects the contribution of \((1+x)^{i}\) terms.
To identify these coefficients, one should compare corresponding terms. Start by identifying the powers of \(x\) at each stage of the expansion and match them accordingly. For example, you check \(i = 2\) to see how the contributed term fits within \(x^2\). The key is understanding which expansion terms appear where and label coefficients accordingly. This process involves systematically checking each power from the expanded equation.

Binomial Theorem

The Binomial Theorem is a powerful tool for expanding polynomials raised to a power, making it essential in solving polynomial equations like our example. It states that \( (1 + x)^n = \sum_{k=0}^{n} \binom{n}{k} x^k \). This means each term in the expansion of \((1 + x)^n\) can be determined using binomial coefficients \(\binom{n}{k}\), where \(n\) is the power, and \(k\) changes from 0 through \(n\).
For example, to find the expansion of \((1+x)^2\), apply the Binomial Theorem: \( (1+x)^2 = \binom{2}{0}1 + \binom{2}{1}x + \binom{2}{2}x^2 = 1 + 2x + x^2 \). This process is systematically applied for each power ensuring that each coefficient \(a_i\) for the corresponding power \((1+x)^i\) is caught and calculated accurately. The theorem helps simplify polynomial expansions by using predictable patterns based on integer powers.

Polynomial Equation

A polynomial equation consists of expressions involving sums or differences of several terms, each inclusive of variables and coefficients. An equation like \(1+x^4+x^5\) can be psychologically impactful as it calls for arrangement into known structures. In this exercise, the task is to reconstruct the polynomial such that it fits within \( \sum_{i=0}^{5} a_{i}(1+x)^{i} \).
Polynomials are typically examined by degrees – the highest power of the variable present, which affects how the equation is solved. By assigning coefficients and sorting terms, a polynomial like \(1 + x^4 + x^5\) ensures that you are checking the balance against its expansion to find specific coefficients like \(a_2\).
Understanding a polynomial equation involves looking at patterns, balancing terms, and systematically deducing how each part comes together to make the equation whole. Recognizing each term's role provides an organized path to understanding why certain coefficients, such as \(a_2\), should be the focus, leading us to the solution.