Problem 11 - 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: Understand the Concepts

First, let's recall what a geometric progression (G.P.) is. A G.P. is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (r). Given that \(a, b, c\) are in G.P., we have \(b = ar\) and \(c = ar^2\). Also, it's given that \(a + b + c = xb\).

Step 2: Express Terms in Terms of 'a' and 'r'

Substitute the G.P. terms into the equation. We have: \[ a + ar + ar^2 = x(ar) \]This simplifies to:\[ a(1 + r + r^2) = x(ar) \].

Step 3: Rearrange the Equation

Divide both sides by \(a\) (assuming \(a eq 0\)) to get:\[ 1 + r + r^2 = xr \].This can be rearranged to:\[ r^2 + r(1-x) + 1 = 0 \].

Step 4: Find the Discriminant

This quadratic in \(r\) has a discriminant \(D\). For \(r\) to have real values, the discriminant must be non-negative:\[ D = (1-x)^2 - 4 \times 1 \times 1 \geq 0 \],which simplifies to:\[ (1-x)^2 \geq 4 \].

Step 5: Solve the Inequality for 'x'

Solve \((1-x)^2 \geq 4\):1. \((1-x) \geq 2\) gives \(x \leq -1\).2. \((1-x) \leq -2\) gives \(x \geq 3\).Therefore, the solution is \(x \leq -1\) or \(x \geq 3\).

Step 6: Conclusion

The acceptable range for \(x\) is either less than \(-1\) or greater than \(3\), which matches option (A).

Key Concepts

Real Numbers

Real numbers are the combination of all rational and irrational numbers. These include all numbers on the number line, from negatives to positives, including zero.

• Rational numbers: These can be expressed as a fraction of two integers, like \( \frac{3}{4} \) or 5.
• Irrational numbers: These numbers cannot be expressed as a simple fraction, examples include \( \pi \) and \( \sqrt{2} \).
• Integer numbers: Whole numbers that are not fractions, like -2, 0, or 3.

Understanding real numbers is crucial because they are used in equations, expressions, and models in various fields of mathematics and science.
In this exercise, we deal with real numbers because the terms in the geometric progression, the discriminant, and the inequality must all yield real number solutions.

Quadratic Equation

A quadratic equation is any equation that can be written in the form \( ax^2 + bx + c = 0 \), where \( a, b, \) and \( c \) are constants, and \( a \) is not zero. Solutions to this equation are called the roots or zeros.
You can solve quadratic equations using several methods, including:

• Factoring
• The quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
• Completing the square

In our exercise, the quadratic equation generated during the problem solution is \( r^2 + r(1-x) + 1 = 0 \). This plays a crucial role because it involves the unknowns of the progression and requires finding possible values of \((r)\), the common ratio, using the discriminant.

Discriminant

The discriminant of a quadratic equation \( ax^2 + bx + c = 0 \) is given by the expression \( b^2 - 4ac \). This is a valuable part of solving quadratic equations because it indicates the nature of the roots.

• If \( D > 0 \): The equation has two distinct real roots.
• If \( D = 0 \): The equation has exactly one real root (or a repeated root).
• If \( D < 0 \): The equation has no real roots; the roots are complex numbers.

In the exercise, the discriminant helps determine the values of \( x \) for which the roots are real. The condition \( (1-x)^2 \geq 4 \) arises which eventually helps in analyzing \( x \)'s valid ranges.

Inequalities

Inequalities are mathematical expressions that involve comparisons between values or expressions. They are somewhat like equations but use symbols like \( >, <, \leq, \geq \), instead of inclusively indicating equality.

• Solving inequalities: Similar to equations but often require flipping the inequality sign when multiplying or dividing by a negative number.
• Representation: Solutions are often shown using intervals, like \( x < 3 \) or \( x \geq -1 \).

In the exercise, solving the inequality \( (1-x)^2 \geq 4 \) leads us to determine the acceptable ranges for \( x \). Here, solving the inequality gives the ranges as \( x \leq -1 \) or \( x \geq 3 \), indicating all the possible solutions based on the exercise's criteria.