Problem 23 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If the coefficents of \(x^{3}\) and \(x^{4}\) in the expansion of \(\left(1+a x+b x^{2}\right)(1-2 x)^{18}\) in powers of \(x\) are both zero, then \((a, b)\) is equal to: (a) \(\left(14, \frac{272}{3}\right)\) (b) \(\left(16, \frac{272}{3}\right)\) (c) \(\left(16, \frac{251}{3}\right)\) (d) \(\left(14, \frac{251}{3}\right)\)

Step-by-Step Solution

Answer

The correct values are \((a, b) = (16, \frac{272}{3}\)).

Step 1: Understanding the Problem

We have the expression \((1 + ax + bx^2)(1 - 2x)^{18}\). We need to find the values of \(a\) and \(b\) such that the coefficients of \(x^3\) and \(x^4\) in the expansion are both zero.

Step 2: Expanding \((1-2x)^{18}\) using Binomial Theorem

To find the coefficients, we first expand \((1-2x)^{18}\) using the binomial theorem. For \(x^k\) term, the binomial coefficient is \(\binom{18}{k}(-2)^k\). Specifically, \[ c_0 = 1, \, c_1 = -36x, \, c_2 = 306x^2, \, c_3 = -1632x^3, \, c_4 = 6806x^4 \].

Step 3: Finding the Coefficient of \(x^3\)

Multiply the expanded terms of \((1 + ax + bx^2)\) with corresponding terms from \((1-2x)^{18}\) to find the coefficient of \(x^3\):- Use \(1 \times (-1632)x^3\)- Use \(ax \times 306x^2\)- Use \(bx^2 \times -36x\)The coefficient of \(x^3\) is \(-1632 + 306a - 36b\). Set this equal to 0: \[-1632 + 306a - 36b = 0\]

Step 4: Finding the Coefficient of \(x^4\)

Multiply the expanded terms of \((1 + ax + bx^2)\) with corresponding terms from \((1-2x)^{18}\) to find the coefficient of \(x^4\):- Use \(1 \times 6806x^4\)- Use \(ax \times -1632x^3\)- Use \(bx^2 \times 306x^2\)The coefficient of \(x^4\) is \(6806 - 1632a + 306b\). Set this equal to 0: \[6806 - 1632a + 306b = 0\]

Step 5: Solving the System of Equations

We now have the following equations:1. \(306a - 36b = 1632\)2. \(-1632a + 306b = -6806\)Solve this linear system of equations to find values of \(a\) and \(b\).

Step 6: Solution to the Equations

From equation 1: \(a = \frac{1632 + 36b}{306}\). Substitute this in equation 2 to find \(b\). Simplify and solve the system to find:\[a = 16, \quad b = \frac{272}{3}\].

Step 7: Final Verification

Plugging \(a = 16\) and \(b = \frac{272}{3}\) back into the equations to verify if both equations become zero, confirming this is the correct solution.

Key Concepts

Polynomial Expansion

Polynomial expansion is an essential concept in algebra, used to express a polynomial expression in an expanded form. This helps in identifying individual terms, making calculations and finding specific coefficients much easier. The process ensures that you can systematically derive each term of a complex expression like \(1 + ax + bx^2\).

In the original exercise, the challenge was to expand \( (1 + ax + bx^2)(1 - 2x)^{18} \). To do this effectively, one uses the binomial theorem. The binomial theorem provides a way to expand expressions of the form \( (a + b)^n \) into a sum involving terms of the general form \( \binom{n}{k} a^{n-k} b^k \).

In this task, \((1-2x)^{18}\) is expanded to identify terms like \(c_3 = -1632x^3\) and \(c_4 = 6806x^4\). Understanding polynomial expansion through the binomial theorem is fundamental because it allows for the precise location of coefficients, crucial in solving problems involving algebraic expressions and equations.

Linear Systems

The process of solving linear systems involves determining the values of variables that satisfy multiple linear equations simultaneously. In the context of this exercise, the task was to find the values of \(a\) and \(b\) such that the coefficients of \(x^3\) and \(x^4\) are zero.

We formulated a system from the problem:

• \(306a - 36b = 1632\)
• \(-1632a + 306b = 6806\)

Solving such a system requires techniques like substitution, elimination, or using matrices, making the problem solvable analytically. A key insight is that linear systems like these occur frequently in algebra and applied mathematics.

Approaching linear systems requires understanding how to manipulate equations to isolate variables. In our case, isolating \(a\) or \(b\) made it easier to substitute back into the equations and find the exact solution of \(a = 16\) and \(b = \frac{272}{3}\), matching the original problem's criteria.

Algebraic Equations

Algebraic equations are mathematical statements of equality between expressions. They frequently arise in the study of mathematics when comparing algebraic expressions, and solving them is essential for gaining deeper insights into algebraic problems.

For determining whether specific coefficients vanish, we treated the origin of the problem as forming algebraic equations derived from polynomial expansions. Formulating these into the system we solved was crucial to check where they equated to zero.

Working through algebraic equations involves several steps:

• Setting each equation up correctly based on given conditions.
• Using algebraic manipulation, like combining like terms, to simplify.
• Solving the equations, whether independently or within a system, to find unknown values.

Understanding algebraic equations and their dynamic role is vital in many areas of mathematics. By mastering these concepts, students are better equipped to address not just textbook exercises, but also more complex real-world problems.