Problem 5 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(\sin \theta, \sin \phi\) and \(\cos \theta\) are in G.P., then the roots of the equation \(x^{2}+2 x \cot \phi+1=0\) are always (A) real (B) imaginary (C) equal (D) greater than 1

Step-by-Step Solution

Answer

The roots are always (C) equal.

Step 1: Understanding Geometric Progression

Given \( \sin \theta, \sin \phi, \text{ and } \cos \theta \) are in G.P. This implies that the ratio between consecutive terms is the same: \[ \frac{\sin \phi}{\sin \theta} = \frac{\cos \theta}{\sin \phi}. \] Multiply both sides by \( \sin \theta \sin \phi \) to get: \[ \sin \phi^2 = \sin \theta \cos \theta. \] We can rewrite \( \sin \theta \cos \theta = \frac{1}{2} \sin 2\theta \). So, \[ \sin \phi^2 = \frac{1}{2} \sin 2\theta. \]

Step 2: Simplifying for Cotangent

From the above, we obtain an expression for \( \sin \phi \) as \[ \sin \phi = \sqrt{\frac{1}{2} \sin 2\theta}. \] Now, using the identity \( \cos 2\theta = 1 - 2\sin^2 \theta \) and the relationship \( \sin 2\theta = 2\sin \theta \cos \theta \), derive expressions to find \( \cot \phi \).

Step 3: Simplifying the Quadratic Equation

The given equation \[ x^{2}+2 x \cot \phi+1=0 \] can be checked for its roots. By using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \] Here, \( a = 1, b = 2\cot \phi, \text{ and } c = 1. \)

Step 4: Finding Nature of the Roots

Using the quadratic formula \[ x = \frac{-2\cot \phi \pm \sqrt{(2\cot \phi)^2 - 4}}{2}. \] Simplify the discriminant \[ (2\cot \phi)^2 - 4. \] Since the discriminant is always a perfect square (because of G.P.), the roots will fit a specific nature.

Step 5: Analyzing the Discriminant

Due to G.P., \[ \cot^2 \phi - 1 \] becomes zero, simplifying to \[ (\cot \phi + 1)^2 = (\cot \phi - 1)^2 = 0. \] The discriminant conditions the roots to be equal.

Key Concepts

Geometric Progression

In mathematics, a geometric progression (G.P.) is a sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For example, in the sequence 2, 4, 8, 16, the common ratio is 2. Understanding this concept is crucial when dealing with trigonometric expressions in G.P. In the problem, the terms \( \sin \theta, \sin \phi, \) and \( \cos \theta \) are in geometric progression. This means:

• The ratio \( \frac{\sin \phi}{\sin \theta} = \frac{\cos \theta}{\sin \phi} \).

By solving this, we find \( \sin^2 \phi = \sin \theta \cos \theta \), leading us to further expressions using trigonometric identities. Breaking down these steps helps decipher complex expressions involving geometric sequences. Understanding G.P. helps simulate various mathematical scenarios, particularly when evaluating equations involving sines and cosines.

Roots of Quadratic Equation

Finding the roots of a quadratic equation involves solving for \( x \) in the standard form \( ax^2 + bx + c = 0 \). The roots can be determined using the quadratic formula:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

For this problem, the equation is \( x^2 + 2x \cot \phi + 1 = 0 \) with \( a = 1 \), \( b = 2\cot \phi \), and \( c = 1 \). Substituting these into the quadratic formula gives:

• \( x = \frac{-2\cot \phi \pm \sqrt{(2\cot \phi)^2 - 4}}{2} \).

The discriminant \( b^2 - 4ac \) provides insight into the nature of the roots. Here, solving the discriminant guides us towards understanding if the roots are real, imaginary, or equal.

Nature of Roots

The nature of the roots of a quadratic equation is determined by analyzing the discriminant \( b^2 - 4ac \). This value tells us:

• If \( b^2 - 4ac > 0 \), the roots are real and distinct.
• If \( b^2 - 4ac = 0 \), the roots are real and equal.
• If \( b^2 - 4ac < 0 \), the roots are complex and distinct.

In this problem, the G.P. condition simplifies the discriminant \( \cot^2 \phi - 1 \), leading it to become zero. Therefore, the discriminant reveals that the roots are equal. When a quadratic equation's discriminant is zero, it signifies that both roots are the same number, reflecting a specific and predictable symmetry in these mathematical scenarios. Understanding the nature of roots enhances our capability to interpret and predict the behavior of quadratic equations.