Problem 59 - 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 \(\Delta A B C\) 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: Define the terms of G.P.

Let the three successive terms of the G.P. be \(a, ar, ar^2\). These represent the lengths of the sides of the triangle \(\Delta ABC\).

Step 2: Apply the triangle inequality theorem

For three sides to form a triangle, they must satisfy the triangle inequality theorem. Let's check the inequalities:1. \(a + ar > ar^2\) simplifies to \(1 + r > r^2\) or \(r^2 - r - 1 < 0\).2. \(ar + ar^2 > a\) simplifies to \(r(1 + r) > 1\).3. \(ar^2 + a > ar\) simplifies to \(a(1 + r^2) > ar\), which is always true for \(r > 1\).

Step 3: Analyze inequality \(r^2 - r - 1 < 0\)

Let's solve the inequality for roots:The quadratic \(r^2 - r - 1 = 0\) has roots \(\frac{1 \pm \sqrt{5}}{2}\). Since \(r > 1\), analyze the inequality in the interval \((1, \infty)\) considering it is a downward parabola in this form, we get that it holds for \(r \in \left( 1, \frac{1 + \sqrt{5}}{2} \right)\).

Step 4: Solve for greatest integer function conditions

The value of \(r\) should be more than 1 and less than \(\frac{1 + \sqrt{5}}{2}\). Approximating \(\frac{1 + \sqrt{5}}{2} \approx 1.618\), so 1 < \(r < 1.618\). If \(1 < r < 1.618\), then \([r] = 1\). Since \([-r]\) for these values of \(r\) would be \(-2\) because \(-1.618 < -r < -1\). Therefore, \([-r] = -2\).

Step 5: Compute the sum \([r] + [-r]\)

Add the two greatest integer results: \([r] + [-r] = 1 + (-2) = -1\).

Key Concepts

Triangle Inequality Theorem

The triangle inequality theorem is an essential principle in geometry. It states that for three sides to form a triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. This condition must hold for all three combinations of the sides. In simpler terms:

• The sum of the lengths of two sides must always be more than the third side.
• This has to be true for all combinations of picking two sides out of three.

In the exercise, the triangle inequality theorem is applied to the sides given by a geometric progression (G.P.) as: \[ \begin{align*} 1. & \quad a + ar > ar^2 \ 2. & \quad ar + ar^2 > a \ 3. & \quad ar^2 + a > ar \end{align*} \] These inequalities are used to ensure the sides can actually form a triangle, which is necessary for further calculations.

Quadratic Inequality

Solving a quadratic inequality like \(r^2 - r - 1 < 0\) is crucial for determining valid values of \(r\). This inequality represents a parabola opening upwards, and we want to find where this parabola is below the \(x\)-axis.
To do this effectively, we first find the roots of the quadratic equation by using the quadratic formula: \[ r = \frac{1 \pm \sqrt{5}}{2} \] These roots determine the intervals we need. Since we need \(r > 1\), our interval of interest is \( (1, \frac{1 + \sqrt{5}}{2} ) \).
In this range, \(r^2 - r - 1\) is less than zero because it lies under the parabola's vertex on the \(x\)-axis. This interval helps us ascertain the possible values of \(r\) needed for the triangle inequality condition.

Greatest Integer Function

The greatest integer function, also known as the floor function, takes a real number and gives the greatest integer less than or equal to that number. It is denoted as \([r]\). Understanding this function is vital for certain types of math problems, especially those involving non-integer solutions.
In our exercise, we've learned that for \(r\) in the range \(1 < r < \frac{1 + \sqrt{5}}{2}\), approximately \(1.618\), we can break it down with the following logic:

• Since \(r\) is between 1 and \(1.618\), \([r] = 1\). This is because the greatest integer below \(1.618\) is 1.
• Considering the negative, \([-r]\) would be \(-2\). This happens as \(-1.618 < -r < -1\) resulting in the greatest integer less than \(-r\) being \(-2\).

Adding these values, we have \([r] + [-r] = 1 + (-2) = -1\). This computation shows how changes within a specific region affect the integer values derived from the floor function.