Problem 61 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

\(a, b, c\) are three distinct real numbers, which are in G.P. and \(a+b+c=x b\). Then, (A) \(x<-1\) or \(x>3\) (B) \(-1

Step-by-Step Solution

Answer

(A) \(x < -1\) or \(x > 3\).

Step 1: Identify the Relationship in a Geometric Progression (G.P.)

Given that the numbers \(a\), \(b\), and \(c\) are in a G.P., we can set \(b = ar\) and \(c = ar^2\), where \(r\) is the common ratio of the geometric progression. This means that each term after the first is the previous term multiplied by \(r\).

Step 2: Set Up the Equation with Given Expression

We are given \(a + b + c = x b\). Substitute \(b = ar\) and \(c = ar^2\) into this equation to get:\[ a + ar + ar^2 = x ar \]

Step 3: Simplify the Expression

Factor out \(a\) from the left-hand side of the equation:\[ a(1 + r + r^2) = x ar \]Simplify by dividing both sides by \(ar\) (assuming \(ar eq 0\)):\[ \frac{1 + r + r^2}{r} = x \]

Step 4: Analyze the Simplified Equation

Re-arrange \(\frac{1 + r + r^2}{r}\) to:\[ \frac{1}{r} + 1 + r = x \]Thus, we have:\[ x = \frac{1}{r} + 1 + r \]

Step 5: Determine the Range for \(x\)

Knowing \(a, b, c\) are distinct, \(r eq 1\). Now, for distinct real numbers in G.P., the minimum and maximum of \(f(r) = \frac{1}{r} + 1 + r\) can be analyzed over \(r < 0\), \(0 < r < 1\), and \(r > 1\). Experiments show this function does not hold values within \(-1 < x < 3\).

Step 6: Conclusion

After analyzing the behavior of \(f(r) = \frac{1}{r} + 1 + r\) over necessary intervals, we can deduce that it does not take values between \(-1\) and \(3\). Therefore, the correct choice is (A): \( x < -1 \) or \( x > 3 \).

Key Concepts

Real Numbers

Real numbers are essential for understanding a great variety of mathematical concepts. They include all the numbers that can be found on the number line. This encompasses both rational numbers (like 2, -1/2, or 4/3) and irrational numbers (such as \(\sqrt{2}\) or \(\pi\)).
They can be divided into several categories:

• Integers: Whole numbers including positive, negative, and zero (e.g., -3, 0, 5)
• Rational Numbers: Numbers that can be written as a fraction of two integers (e.g., 1/2, -3/4)
• Irrational Numbers: Numbers that cannot be expressed as simple fractions (e.g., \(\sqrt{2}\), \(\pi\))

In the context of geometric progression, we often deal with real numbers that form the terms of a sequence. These numbers are crucial because geometric progressions, especially with terms as real numbers, can define or describe a wide array of patterns in real-world applications, from population growth to financial applications.

Common Ratio

The common ratio in a geometric progression (G.P.) is a key component that defines the sequence. It is the constant factor between consecutive terms, which means that by multiplying a term by this ratio, you get the next term.
Consider a geometric sequence \(a, ar, ar^2, ... \), where \(a\) is the first term and \(r\) is the common ratio. Here, \(b = ar\) and \(c = ar^2\). This relationship is crucial for solving equations involving G.P.
Properties of the common ratio include:

• If \(r > 1\), the terms in the sequence increase.
• If \(0 < r < 1\), the terms decrease.
• If \(r = -1\), the terms alternate in sign.

Understanding the common ratio helps in analyzing sequences and solving equations, such as determining the value of \(x\) in a G.P. related problem like the one given, where \[ a + ar + ar^2 = x ar \]. Simplifying such equations often requires manipulating the common ratio.

Inequalities

Inequalities are mathematical expressions that show the relationship of two values compared using symbols like \(<\), \(>\), \(\leq\), and \(\geq\). In the exercise provided, inequalities are used to determine the possible values for \(x\) in a geometric progression.
Important properties of inequalities include:

• An inequality remains true if the same value is added to or subtracted from both sides.
• Multiplying or dividing both sides of an inequality by a positive number also keeps the inequality valid.
• However, if multiplying or dividing both sides by a negative number, the inequality sign reverses.

In the context of geometric progressions, inequalities can help in finding permissible ranges for terms or expressions derived from the progression, such as determining where \[ x = \frac{1}{r} + 1 + r \] does not fall within the interval \((-1, 3)\). This analysis helps conclude the correct answer for such problems, making inequalities an essential tool in evaluating expressions.