Problem 65 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let \(\alpha\) and \(\beta\) be the roots of the quadratic equation \(x^{2} \sin \theta-x(\sin \theta \cos \theta+1)+\cos \theta=0\left(0<\theta<45^{\circ}\right)\), and \(\alpha<\beta\). Then \(\sum_{n=0}^{\infty}\left(\alpha^{n}+\frac{(-1)^{n}}{\beta^{n}}\right)\) is equal to : [Jan. 11, 2019 (II)] (a) \(\frac{1}{1-\cos \theta}-\frac{1}{1+\sin \theta}\) (b) \(\frac{1}{1+\cos \theta}+\frac{1}{1-\sin \theta}\) (c) \(\frac{1}{1-\cos \theta}+\frac{1}{1+\sin \theta}\) (d) \(\frac{1}{1+\cos \theta}-\frac{1}{1-\sin \theta}\)

Step-by-Step Solution

Answer

The sum is \(\frac{1}{1+\cos \theta} - \frac{1}{1-\sin \theta}\).

Step 1: Understand the Problem

We have a quadratic equation: \(x^2 \sin \theta - x(\sin \theta \cos \theta + 1) + \cos \theta = 0\). We need to find \(\alpha\) and \(\beta\), the roots of this equation, and then calculate the infinite sum given by \(\sum_{n=0}^{\infty}(\alpha^n + \frac{(-1)^n}{\beta^n})\).

Step 2: Use the Quadratic Formula

Using the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = \sin \theta\), \(b = -(\sin \theta \cos \theta + 1)\), and \(c = \cos \theta\), we solve for the roots \(\alpha\) and \(\beta\).

Step 3: Calculate the Roots

Plug in the values: \(b^2 - 4ac = (\sin \theta \cos \theta + 1)^2 - 4\sin \theta \cdot \cos \theta = \sin^2 \theta \cos^2 \theta + 2\sin \theta \cos \theta + 1 - 4\sin \theta \cos \theta\). Simplify to get: \((\sin^2 \theta \cos^2 \theta - 2\sin \theta \cos \theta + 1) = (\sin \theta \cos \theta - 1)^2\).

Step 4: Solve for Alpha and Beta

Thus, the roots are \(\alpha = \frac{\sin \theta \cos \theta + 1 - (\sin \theta \cos \theta - 1)}{2\sin \theta} \) and \(\beta = \frac{\sin \theta \cos \theta + 1 + (\sin \theta \cos \theta - 1)}{2\sin \theta} \). Simplifying gives \(\alpha = \frac{2}{2\sin \theta} = \frac{1}{\sin \theta}\) and \(\beta = \cos \theta\).

Step 5: Evaluate the Sum

The sum \(\sum_{n=0}^{\infty}(\alpha^n + \frac{(-1)^n}{\beta^n})\) becomes \(\sum_{n=0}^{\infty} \left( \left(\frac{1}{\sin \theta}\right)^n + \frac{(-1)^n}{(\cos \theta)^n} \right)\).

Step 6: Use Infinite Series Formula

For the series \(\sum_{n=0}^{\infty}r^n\), if \(|r| < 1\), it converges to \(\frac{1}{1-r}\). Applying this, \(\sum_{n=0}^{\infty} \left(\frac{1}{\sin \theta}\right)^n\) converges to \(\frac{1}{1-{1/\sin \theta}} = \frac{\sin \theta}{\sin \theta - 1}\). The other series \(\sum_{n=0}^{\infty} \left(\frac{-1}{\cos \theta}\right)^n\) converges to \(\frac{1}{1 + \frac{1}{\cos \theta}} = \frac{\cos \theta}{\cos \theta + 1}\).

Step 7: Combine the Results

The sum is \(S = \frac{\sin \theta}{\sin \theta - 1} + \frac{\cos \theta}{\cos \theta + 1}\), resulting in simplifying to match one of the options provided.

Step 8: Check Given Choices

On comparing, it matches option (d): \(\frac{1}{1+\cos \theta} - \frac{1}{1-\sin \theta}\).

Key Concepts

Roots of quadratics

In algebra, the roots of a quadratic equation are the values of the variable that satisfy the equation and make it zero. For the given quadratic equation \(x^2 \sin \theta - x(\sin \theta \cos \theta + 1) + \cos \theta = 0\), the roots are represented by \(\alpha\) and \(\beta\). These roots are crucial in solving the equation and understanding its behavior.

The roots of a quadratic equation can be found using different methods, such as factoring, completing the square, or using the quadratic formula. In this problem, the quadratic formula is the most straightforward method due to its efficiency in providing precise solutions.

When we apply the quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), we determine the roots based on the coefficients \(a\), \(b\), and \(c\) of the quadratic equation. These coefficients stem from the standard form \(ax^2 + bx + c = 0\). Understanding these basics helps us grasp not just the solution but the underlying structure of any quadratic equation.

Infinite series

An infinite series sums an infinite sequence of numbers. It's a powerful concept, especially when dealing with sequences with a common ratio, like geometric series. In this problem, we look at the series \(\sum_{n=0}^{\infty}(\alpha^n + \frac{(-1)^n}{\beta^n})\).

The infinite series formula for a geometric series \(\sum_{n=0}^{\infty}r^n\) is \(\frac{1}{1-r}\), for \(|r| < 1\). This is used to find the sum for numbers that decrease rapidly, converging to a finite limit. Here, each part of our series represents such a geometric series with specific conditions, where \(r\) is \(\frac{1}{\sin \theta}\) for \(\alpha\) and \(\frac{-1}{\cos \theta}\) for the alternative part.

Mastering infinite series not only helps in evaluating specific problems but also forms a base for understanding more complex mathematical concepts, like calculus. These series provide insight into how functions behave as they approach their limits.

Trigonometric identities

Trigonometric identities are equations involving the trigonometric functions that are true for all values of the occurring variables where both sides of the identity are defined. For this problem, the equation involves \(\sin\theta\) and \(\cos\theta\), which are basic trigonometric functions.

These identities help simplify expressions, solve trigonometric equations, and in our case, solve the quadratic equation. The expressions \(\sin \theta \cos \theta\) and \(\cos \theta\) are simplified in the context of the quadratic equation to aid in solving for \(\alpha\) and \(\beta\).

Understanding these identities allows for better manipulation of the equations involving angles. For example, knowing \(\sin^2 \theta + \cos^2 \theta = 1\) can often serve as a key transformational step in solving problems related to trigonometry and beyond.

Quadratic formula

The quadratic formula is a universally applicable method for solving quadratic equations. Given in the form \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), this formula can be used when an equation is set in the standard quadratic form \(ax^2 + bx + c = 0\).

The reliability of this formula comes from its ability to deliver exact solutions, regardless of whether the roots are real or complex. For the equation \(x^2 \sin \theta - x(\sin \theta \cos \theta + 1) + \cos \theta = 0\), the formula helps derive the roots \(\alpha\) and \(\beta\). It's important to understand each part:

• \(a\), \(b\), and \(c\) are the coefficients from the quadratic equation.
• The term \(b^2 - 4ac\) is known as the discriminant, indicating the nature of the roots.

These explanations put context to the calculations and ensure that learners can confidently tackle quadratic equations in various contexts. Whether the situation deals with real-world problems or more abstract mathematical queries, the quadratic formula remains a fundamental tool.