Problem 6 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If the parametric equation of a curve is given by \(x=\cos \theta+\log \tan \frac{\theta}{2}\) and \(y=\sin \theta\), then the points for which \(\frac{d^{2} y}{d x^{2}}=0\) are given by (A) \(\theta=n \pi, n \in Z\) (B) \(\theta=(2 n+1) \pi 2, n \in Z\) (C) \(\theta=(2 n+1) \pi, n \in Z\) (D) \(\theta=2 n \pi, n \in Z\).

Step-by-Step Solution

Answer

The correct answer is (C) \(\theta=(2 n+1) \pi, n \in Z\).

Step 1: Understanding Parametric Equations

We are given two parametric equations: \(x = \cos \theta + \log \tan \frac{\theta}{2}\) and \(y = \sin \theta\). We need to find when the second derivative \(\frac{d^{2} y}{dx^{2}} = 0\).

Step 2: Find First Derivatives

Compute \(\frac{dx}{d\theta}\) and \(\frac{dy}{d\theta}\). We have \(\frac{dx}{d\theta} = -\sin \theta + \frac{1}{\sin \theta}\) and \(\frac{dy}{d\theta} = \cos \theta\).

Step 3: Find \(\frac{dy}{dx}\)

Use the chain rule \(\frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta}\) to find \(\frac{dy}{dx} = \frac{\cos \theta}{-\sin \theta + \frac{1}{\sin \theta}}\). Simplify as needed.

Step 4: Differentiate Again to Find \(\frac{d^{2}y}{dx^{2}}\)

Differentiate \(\frac{dy}{dx}\) with respect to \(\theta\), and then find \(\frac{d^2 y}{d x^2} = \frac{d}{d\theta}\left(\frac{dy}{dx}\right) \div \frac{dx}{d\theta}\).

Step 5: Simplify \(\frac{d^{2} y}{d x^{2}}\)

Simplify the expression for \(\frac{d^{2} y}{d x^{2}}\) and set it equal to zero. Solve for \(\theta\) to find the points where this second derivative is zero.

Step 6: Solve the Equation

The resulting equation simplifies to finding where \(\sin^3 \theta - \cos \theta = 0\). This simplifies further to \(\sin^3 \theta = \cos \theta\).

Step 7: Determine Values of \(\theta\)

Solve the equation \(\sin^3 \theta = \cos \theta\) for values of \(\theta\). Check this against integer multiples of \(\pi\) (choices given) to determine correct solutions.

Step 8: Match with Given Options

We find that \(\theta = (2n + 1) \pi\). So, verify that this matches option (C).

Key Concepts

Parametric Equations

Parametric equations describe a set of related quantities that depend on a common variable, often represented as \(t\) or \(\theta\). These equations are especially useful for representing curves in the plane where both the \(x\) and \(y\) coordinates are given as functions of a parameter. In our exercise, we have:

• \(x = \cos \theta + \log \tan \frac{\theta}{2}\)
• \(y = \sin \theta\)

The parameter \(\theta\) being used here helps capture the movement through the curve as it changes, representing different points \((x, y)\) on the path.
By examining these equations, you can derive the position of a point on the curve by substituting different values of \(\theta\). Understanding parametric equations is key to solving problems involving curves, enabling us to analyze derivatives and the overall behavior of the function.

Second Derivative Test

The second derivative test is a calculus method used to determine the concavity and local extrema (minimum or maximum points) of a function. When working with parametric equations, we find derivatives with respect to the parameter (\(\theta\) in this case) to assess how one coordinate changes relative to another.

• The first derivative \(\frac{dy}{dx}\) is found by using the chain rule: \[ \frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} \]
• The second derivative \(\frac{d^2y}{dx^2}\) shows the rate of change of the slope, and it's crucial for understanding points of inflection or concavity.
• Setting \(\frac{d^2 y}{d x^2} = 0\) helps to identify where the curve changes concavity.

By solving \(\sin^3 \theta = \cos \theta\), where \(\frac{d^2 y}{d x^2} = 0\), we can find special points on the curve where the behavior shifts. Here, solving leads to specific angular values for \(\theta\), which we matched to option (C) in our exercise.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the involved angles. These identities are extremely useful in simplifying expressions and solving equations related to these functions. Some of the basic identities include:

• Pythagorean identities: \(\sin^2 \theta + \cos^2 \theta = 1\)
• Tangent identity: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
• Double angle formulas, e.g., \(\sin(2\theta) = 2\sin \theta\cos \theta\)

In the context of the given problem, understanding how these identities work aids in simplifying and solving expressions like \(\sin^3 \theta = \cos \theta\). By using these identities, it's possible to transform and rearrange trigonometric expressions to reach a solution quickly and effectively. Thus, knowing and applying trigonometric identities reduces complexity and enhances problem-solving capabilities in calculus.