Problem 33 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If \(f(y)=1-(y-1)+(y-1)^{2}-(y-1)^{3}\) then the coefficient of \(y^{2}\) in it is (a) \({ }^{17} \mathrm{C}_{2}\) (b) \({ }^{17} \mathrm{C}\) (c) \({ }^{18} \mathrm{C}_{2}\) (d) \({ }^{18} \mathrm{C}_{3}\)

Step-by-Step Solution

Answer

The coefficient of \(y^2\) is 4, none of the given options.

Step 1: Simplify the Expression

The function given is \(f(y) = 1 - (y-1) + (y-1)^2 - (y-1)^3\). This expression can be rewritten by expanding each term: \(f(y) = 1 - (y - 1) + (y - 1)^2 - (y - 1)^3\).

Step 2: Expand Each Term

First, expand \((y - 1)\): This remains as \(y - 1\). Expand \((y-1)^2\) using the binomial theorem: \((y-1)^2 = y^2 - 2y + 1\). Expand \((y-1)^3\) using the binomial theorem: \((y-1)^3 = y^3 - 3y^2 + 3y - 1\).

Step 3: Substitute Expansions Back

Substitute the expanded forms back into the function: \(f(y) = 1 - (y-1) + (y^2 - 2y + 1) - (y^3 - 3y^2 + 3y - 1)\).

Step 4: Simplify Expanded Expression

Combine like terms in the equation you've obtained: \(f(y) = 1 - y + 1 + y^2 - 2y + 1 - y^3 + 3y^2 - 3y + 1\). Simplify by combining all terms: \(f(y) = -y^3 + 4y^2 - 6y + 4\).

Step 5: Identify the Coefficient of \(y^2\)

In the expression \(-y^3 + 4y^2 - 6y + 4\), identify the coefficient of \(y^2\), which is 4.

Key Concepts

Polynomial Expansion

To tackle problems like the one given in the exercise, we need to understand what polynomial expansion entails. Polynomial expansion involves expressing a polynomial as a sum of terms, especially when each term is raised to a power. The expression from our exercise, \( f(y) = 1 - (y - 1) + (y - 1)^2 - (y - 1)^3 \), consists of several terms, each involving a power of \((y - 1)\). To solve it, we expand each power in the polynomial.

Using the binomial theorem, we can expand the powers of \((y-1)\). The binomial theorem states that \[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\] where \(\binom{n}{k}\) denotes a binomial coefficient.

This means for an expression like \((y-1)^2\), we calculate the individual terms as if the formula is applied with \((a = y)\) and \((b = -1)\). Expansion helps simplify such complex problems by breaking down powers into simpler terms, such as integers and individual variables, allowing us to handle and simplify the expression further.

Coefficient Calculation

After expanding the polynomial, the next important step involves coefficient calculation. Coefficient calculation helps us find the factor by which a particular term's base (variable part, such as \(y^2\)) is multiplied. This is critical if you are asked to find a specific coefficient like in this exercise, which focuses on finding the coefficient of \(y^2\).

After expanding \((y-1)^2 = y^2 - 2y + 1\) and \((y-1)^3 = y^3 - 3y^2 + 3y - 1\), these terms are substituted back into the function \(f(y)\).

• Substitute: Replace original terms with their expanded powers.
• Add Like Terms: Combine powers of the same degree, which leads to an expression like \(f(y) = -y^3 + 4y^2 - 6y + 4\).

In this expression, the coefficient of \(y^2\) is \(4\). Hence, calculating coefficients involves careful addition and subtraction after expansion.

Algebraic Simplification

Algebraic simplification is the final stage of handling expressions like the one in the given exercise. It transforms complex expanded forms into more manageable expressions by combining and simplifying terms. This simplification is crucial for making sense of polynomial expressions, especially when identifying coefficients.

In simplifying, we go through the steps of combining like terms. For instance, with the substitution \(f(y) = -y^3 + 4y^2 - 6y + 4\), the task here is to simplify the expression further through organized combining of terms.

• Identify Like Terms: Look across the expression for terms that share the same power of \(y\).
• Combine Them: Add or subtract coefficients of like terms, resulting in a cleaner expression.
• Focus on Relevant Terms: Ensure that terms connected to the question (like \(y^2\) for coefficient calculation) are clearly calculated and presented.

Symmetry and order in simplification allow for accurate results, helping to identify key terms and their coefficients, which can endure further algebraic manipulation or serve as final answers.