Problem 2 - A Complete Resource Book in Mathematics for JEE Main

Question

If three successive terms of a G.P. with common ratio \(r(r>1)\) form the sides of a \(\triangle A B C\) and \([r]\) denotes greatest integer function, then \([r]+[-r]=\) (A) 0 (B) 1 (C) \(-1\) (D) None of these

Step-by-Step Solution

Answer

([r] + [-r]) = -1.

Step 1: Understanding the Problem

We are given three successive terms of a geometric progression (G.P.) with a common ratio greater than 1. These terms form the sides of a triangle. We need to evaluate the expression \([r] + [-r]\). The function \([x]\) denotes the greatest integer less than or equal to \(x\).

Step 2: Representing the G.P Terms

Assume the first term of the G.P. is \(a\). Then, the terms are \(a, ar, ar^2\), where \(r > 1\). These terms form the sides of the triangle \(\triangle ABC\).

Step 3: Applying the Triangle Inequality

For these terms to form a triangle, they must satisfy the triangle inequalities. Since these are already in increasing order, we ensure:1. \(a + ar > ar^2\) simplifies to \(r + 1 > r^2\).2. \(a + ar^2 > ar\) simplifies to \(1 + r^2 > r\).3. \(ar + ar^2 > a\) simplifies to \(r + r^2 > 1\).

Step 4: Solving the Inequalities

From \(r + 1 > r^2\), we get \(r^2 - r - 1 < 0\). Solving the quadratic equation \(r^2 - r - 1 = 0\) using the quadratic formula gives roots \(r = \frac{1 \pm \sqrt{5}}{2}\). Since \(r > 1\), the inequality \(r^2 - r - 1 < 0\) holds for \(1 < r < \frac{1 + \sqrt{5}}{2}\).

Step 5: Evaluating Greatest Integer Function

With \(1 < r < \frac{1 + \sqrt{5}}{2} \approx 1.618\), the greatest integer less than or equal to \(r\) is 1, so \([r] = 1\). Similarly, \([-r] = [-1.618] = -2\) because the greatest integer less than or equal to -1.618 is -2.

Step 6: Calculate the Expression

Now calculate \([r] + [-r]\). With \([r] = 1\) and \([-r] = -2\), we find\[[r] + [-r] = 1 + (-2) = -1. \]

Key Concepts

Triangle Inequality

In a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. This is known as the Triangle Inequality Theorem. This principle is crucial for determining whether three given lengths can form a triangle.

For instance, if we have three sides represented by terms of a geometric progression (G.P.), we must verify that:

• The sum of the first two terms is greater than the third term.
• The sum of the last two terms is greater than the first term.
• The sum of the first and last terms is greater than the second term.

These conditions ensure the lengths can form a triangle.

When applying the triangle inequality to terms of a G.P., like in this exercise, we start with sides given by a common ratio greater than one: typically represented as \(a\), \(ar\), and \(ar^2\). Each pair of sides added together must be greater than the third, resulting in three inequalities that can be used to validate the sides of a triangle.

Greatest Integer Function

The Greatest Integer Function, also known as the floor function, is denoted by \([x]\). It represents the greatest integer that is less than or equal to a given real number \(x\). This function is pivotal in problems where integer values are needed from floating-point numbers.

In mathematical terms, for any real number \(x\), \([x]\) stands for the largest integer not greater than \(x\). Here are some examples:

• For \(x = 2.3\), \([x] = 2\).
• For \(x = -1.7\), \([x] = -2\).

In the context of this exercise, the function helps us determine \([r]\), where \(r\) is the common ratio of the G.P and is a number with specific bounds due to the inequalities derived from the triangle inequality.

The problem also requires computing \([-r]\), which involves assessing the greatest integer less than or equal to the negative value of \(r\). By understanding the behavior of this function, complex mathematical calculations become manageable, as we convert real numbers into simpler integer values.

Quadratic Equations

Quadratic Equations are equations of the form \(ax^2 + bx + c = 0\), where \(a eq 0\). These equations are vital in various areas of mathematics, including geometric progressions and inequalities, to find the roots or solutions that satisfy them.

In our exercise, solving the quadratic equation derived from the inequality \(r^2 - r - 1 < 0\) requires us to find where this quadratic expression is less than zero.

To find the roots, we use the quadratic formula:
\[ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Inserting \(a = 1\), \(b = -1\), and \(c = -1\) into the formula yields:
\[ r = \frac{1 \pm \sqrt{5}}{2}\]
The solutions give us critical points on the number line, which divide it into intervals that can be tested against the inequality. Testing indicates that \(1 < r < \frac{1 + \sqrt{5}}{2}\) satisfies the inequality, meaning \(r\) falls within this range if the terms indeed form a triangle.

This understanding helps when solving mathematical problems involving G.P.s, where the greatest integer function is applied to real solutions obtained from quadratic equations.