Problem 2 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

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

Step-by-Step Solution

Answer

(C) -1

Step 1: Understanding the Problem

We are given three successive terms of a geometric progression (G.P.) that form the sides of a triangle \( \Delta ABC \). These terms are \( a = A \), \( ar = B \), and \( ar^2 = C \) where \( r > 1 \) is the common ratio. We need to find \( [r] + [-r] \) where \( [r] \) denotes the greatest integer less than or equal to \( r \) and \( [-r] \) is the greatest integer less than or equal to \( -r \).

Step 2: Using the Triangle Inequality

For these three terms to form a triangle, they must satisfy the triangle inequality: \( a + ar > ar^2 \), \( a + ar^2 > ar \), and \( ar + ar^2 > a \). Simplifying these inequalities, we get: \( a(1 + r) > ar^2 \), \( a(1 + r^2) > ar \), and \( ar(1 + r) > a \). These simplify further to \( 1 + r > r^2 \), \( 1 + r^2 > r \), and \( r(1 + r) > 1 \).

Step 3: Analyzing the Inequalities

Now, let's analyze these inequalities:- \( 1 + r > r^2 \) simplifies to \( r^2 - r - 1 < 0 \). The roots of the equation \( r^2 - r - 1 = 0 \) are \( \frac{1 \pm \sqrt{5}}{2} \). For \( r > 1 \), \( \frac{1 + \sqrt{5}}{2} < r < 1 \), which is a contradiction.- \( r(1 + r) > 1 \) is satisfied since \( r > 1 \).Since the first inequality \( r^2 - r - 1 < 0 \) is a contradiction for \( r > 1 \), such a triangle cannot exist with sides in a G.P. with \( r > 1 \). So, we need to examine the greatest integer function instead.

Step 4: Evaluating Greatest Integer Functions

Since the formation of a triangle is not possible under these conditions, we essentially rely on manipulating the greatest integer functions directly: - As \( r > 1 \), we have \([r] = 1\) since it is the greatest integer less than or equal to \( r \).- For \([-r]\), we evaluate it assuming that \( r > 1 \). Here, \(-r < -1\), hence \([-r] = -2\) (since -2 is the greatest integer less than or equal to -r).Thus \([r] = 1\) and \([-r] = -2\). Their sum is \([r] + [-r] = 1 - 2 = -1\).

Step 5: Conclusion

From the calculations, we conclude that since \([r] = 1\) and \([-r] = -2\), the value of \([r] + [-r]\) is \(-1\). Therefore, the answer is (C) \(-1\).

Key Concepts

Geometric Progression

A geometric progression (G.P.) is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In mathematical notation, if the first term is denoted by \(a\), and the common ratio by \(r > 1\), the sequence can be expressed as \(a, ar, ar^2, ar^3, \ldots\).

In the context of this problem, the terms are \(a, ar, ar^2\), representing the sides of a triangle \(\Delta ABC\). This setup enables us to analyze the relationship between the terms through other mathematical concepts like the Triangle Inequality. Understanding a G.P. is crucial because it lays a foundation for analyzing how sequences behave in different mathematical contexts, such as within triangles.

When dealing with sequences, make sure to:

• Identify the first term and the common ratio clearly.
• Recognize the multiplicative nature of the sequence.
• Consider the implications of the common ratio being greater than one, especially when they outline properties related to geometry or algebra.

Triangle Inequality

The Triangle Inequality is a fundamental principle in geometry, stating that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Given sides \(a, ar, ar^2\), as in our problem, the inequalities are:

\[ a + ar > ar^2 \]
\[ a + ar^2 > ar \]
\[ ar + ar^2 > a \]

Simplifying these inequalities involves reviewing each equation under the condition \(r > 1\). While attempting to simplify, each equation needs checking and rearrangement for the common ratio \(r\):

• Verify logical simplification of the inequalities.
• Ensure a contradiction doesn't arise from simplifying the inequalities, as seen with \(r^2 - r - 1 < 0\) which isn't valid for \(r > 1\).
• Understand that a contradiction indicates that the set of numbers in question cannot form a triangle under the given conditions.

Greatest Integer Function

The greatest integer function, often symbolized as \([x]\), returns the largest integer less than or equal to \(x\). For any real number, \(x\), \([x]\) is easily calculated by truncating \(x\) down to the nearest whole number.

In the problem at hand, we apply this function to both \(r\) and \(-r\) where \(r > 1\):

• Since \(r > 1\), \([r] = 1\) because 1 is the greatest integer not exceeding \(r\).
• For \(-r\), since \(-r < -1\), \([-r] = -2\) because -2 is the greatest integer less than \(-r\).

By calculating \([r] + [-r]\) as in the exercise, we conclude:
\[ 1 + (-2) = -1 \]
This solution reflects how the greatest integer function helps in discrete mathematics contexts, especially in problems where integer boundaries are relevant.