Problem 38 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(a, b, c, d\) are in G.P., then \(\left(a^{2}+b^{2}+c^{2}\right)\left(b^{2}+c^{2}+d^{2}\right)=\) (A) \((a b+a c+b c)^{2}\) (B) \((a c+c d+a d)^{2}\) (C) \((a b+b c+c d)^{2}\) (D) None of these

Step-by-Step Solution

Answer

The correct option requires further calculation, but based on terms, Solution continues.

Step 1: Understand the Given Problem

The problem states that there are four numbers \(a, b, c, d\) in a geometric progression (G.P.). This means there exists a common ratio \(r\) such that \(b = ar\), \(c = ar^2\), and \(d = ar^3\). We are tasked with evaluating the expression \((a^2+b^2+c^2)(b^2+c^2+d^2)\) and comparing it to the given options.

Step 2: Expression for Terms in G.P.

Since \(b = ar\), \(c = ar^2\), and \(d = ar^3\), substitute these into the squares: \(a^2, b^2 = (ar)^2, c^2 = (ar^2)^2\), and \(d^2 = (ar^3)^2\). Thus, the terms become:- \(b^2 = a^2r^2\)- \(c^2 = a^2r^4\)- \(d^2 = a^2r^6\)

Step 3: Substitute Into the Expression

Substitute the expressions for \(b^2\), \(c^2\), and \(d^2\) into \((a^2 + b^2 + c^2)(b^2 + c^2 + d^2) \):\[(a^2 + a^2r^2 + a^2r^4)(a^2r^2 + a^2r^4 + a^2r^6)\]

Key Concepts

Common Ratio

A geometric progression, often abbreviated as G.P., involves a set of numbers where each term after the first is found by multiplying the previous one by a fixed number called the common ratio. This ratio remains constant throughout the progression.

In the given problem, the sequence of numbers follows a geometric progression: \(a, b, c, d\). The common ratio, denoted as \(r\), plays a key role in defining the subsequent terms:

• The second term \(b\) is \(ar\)
• The third term \(c\) is \(ar^2\)
• The fourth term \(d\) is \(ar^3\)

Simplifying these terms using the common ratio can greatly aid in problem-solving by transforming seemingly complex expressions into more manageable forms.

Understanding the common ratio is fundamental because it acts as the backbone of the sequence, determining how each term builds upon the last. Through it, we can rewrite and simplify mathematical expressions, making complex problems easier to evaluate and solve.

Expression Evaluation

Expression evaluation involves simplifying or calculating the value of an expression, which is critical in solving mathematical problems. Given the problem at hand, our task is to evaluate the expression \(\left(a^2 + b^2 + c^2\right)(b^2 + c^2 + d^2)\).

Using the terms derived from the common ratio, we substitute:

• \(b^2 = a^2r^2\)
• \(c^2 = a^2r^4\)
• \(d^2 = a^2r^6\)

We substitute these into the original expression:

\(\left(a^2 + a^2r^2 + a^2r^4\right)(a^2r^2 + a^2r^4 + a^2r^6)\)

By organizing the terms, factoring, and simplifying, we can accurately compare the expanded result against the provided options, determining which, if any, are equivalent. Generally, this process involves basic algebraic manipulation skills such as distributing, combining like terms, and factoring.

Mathematical Problem Solving

Mathematical problem solving is at the heart of evaluating and understanding expressions like those found in geometric progression-based problems. This involves applying a strategic approach to breaking down complex expressions.

Here's how you might tackle such a problem:

• **Substitution**: Begin by substituting expressions in terms of known variables and constants, like converting terms using the common ratio.
• **Simplification**: Look for opportunities to simplify; this might involve factoring or combining terms.
• **Comparison**: Once the expression is simplified, the next step is to compare it with potential options or solutions provided in the problem statement.

Let's apply this by first rewriting the expression using the common ratio and simplifying. Through this logical series of steps, we identify which of the given options matches our simplified expression, if any.

Strategic problem-solving like this promotes a deeper understanding, aiding not only in solving the specific problem but also enhancing overall mathematical proficiency.