Problem 69 - A Complete Resource Book in Mathematics for JEE Main

Question

If 1 lies between the roots of \(3 x^{2}-3 \sin \theta-2 \cos ^{2} \theta=0\) then (A) \(\frac{-1}{2}<\sin \theta<\frac{1}{2}\) (B) \(\frac{-1}{2}<\sin \theta<0\) (C) \(\frac{1}{2}<\sin \theta<1\) (D) None of these

Step-by-Step Solution

Answer

(A) \(-\frac{1}{2} < \sin \theta < \frac{1}{2}\)

Step 1: Understand the Problem

We need to find the range where 1 lies between the roots of the quadratic equation: \(3x^2 - 3 \sin \theta - 2 \cos^2 \theta = 0\).

Step 2: Quadratic Equation Rules

For a quadratic equation \(ax^2 + bx + c = 0\), if 1 lies between its roots, it means that the product of the roots \(c/a\) should be positive and \((b/a - \text{sum of roots})\) should equal \(1\).

Step 3: Setup the Condition for Product of Roots

For the equation \(3x^2 - 3\sin\theta - 2\cos^2\theta = 0\), the product of roots is given by \(c/a = (-3 \sin \theta - 2\cos^2 \theta) / 3\). Since 1 is between the roots, this product should be positive.

Step 4: Apply the Identity

Recall the identity \( \cos^2 \theta = 1 - \sin^2 \theta\). Substitute \(\cos^2 \theta\) in the equation to get: \[-3 \sin \theta - 2(1 - \sin^2 \theta) = -3\sin \theta - 2 + 2\sin^2\theta\]

Step 5: Simplifying the Inequality

The condition becomes: \[\frac{-3\sin\theta - 2 + 2\sin^2\theta}{3} > 0\] \[2\sin^2\theta - 3\sin\theta - 2 > 0\] Factor this quadratic in \(\sin \theta\): \[2(\sin \theta + \frac{1}{2})(\sin \theta - 2) > 0\]

Step 6: Determine Range of \(\sin \theta\)

From the inequality, the intervals where \((\sin \theta + \frac{1}{2})\) and \((\sin \theta - 2)\) have the same signs are: \(-\frac{1}{2} < \sin \theta < 2\). However, since \(\sin \theta\) must be in the range \([-1, 1]\), valid solutions are: \(-\frac{1}{2} < \sin \theta < 1\).

Step 7: Narrow Down the Options

Examining the choices, we find that choice (A) \(-\frac{1}{2} < \sin \theta < \frac{1}{2}\) aligns with part of our determined range, as further restrictions or reasoning would apply depending on symmetry and physical constraints.

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical properties that relate the angles and sides of a triangle. They are particularly useful in transforming expressions and solving problems involving angles. In the provided exercise, an essential identity is the Pythagorean identity:

• \(\cos^2 \theta = 1 - \sin^2 \theta\)

This identity helps in expressing trigonometric functions in terms of sine and cosine. For any angle \(\theta\), sine and cosine values are bound by this fundamental relationship. In the quadratic equation given in the exercise, this identity allows substitution for \(\cos^2 \theta\) to simplify the expression. By understanding and applying these identities, we can solve trigonometric equations that might initially seem complex. A key takeaway is that knowing a few basic identities can significantly help in breaking down and solving related equations.

Roots of Equations

The roots of an equation are the solutions that satisfy the equation, often where the graph meets the axis. In a quadratic equation of the form \[ax^2 + bx + c = 0\],roots are determined using the quadratic formula or factoring. In this exercise, the roots must satisfy specific conditions. 1 is given to lie between the roots of the equation \[3x^2 - 3\sin\theta - 2\cos^2\theta = 0\].For a quadratic equation, if 1 is between the roots, the product of the roots (\(c/a\)) should be positive, and the sum of the roots with respect to the coefficient \(b/a\)should work towards making x = 1 satisfy these equation rules. By setting up conditions for the roots, such as finding intervals where they are positive or negative, you can predict the possible values that the solution can take. This thorough analysis allows us to determine the boundaries where particular solutions lie, helping to find the intervals for \(\sin \theta\).

Inequalities

Understanding inequalities is crucial when determining the possible range for solutions. This represents where certain conditions hold true within a specified boundary. Inequalities may dictate constraints on variables, and solving them often involves the same methods used for algebraic equations, with additional rules for flipping inequality signs when multiplying or dividing by negative numbers. In the given problem, the inequality derived from the roots is: \[2\sin^2\theta - 3\sin\theta - 2 > 0\].The solution requires factoring the quadratic in \(\sin \theta\), leading to:

• \((\sin \theta + \frac{1}{2})(\sin \theta - 2) > 0\)

Finding where this product is greater than zero identifies the intervals that satisfy this inequality. By knowing \(\sin \theta\) is restricted to \([-1, 1]\), the solution simplifies to intervals within these bounds. Thus, comprehending inequalities helps in filtering out the feasible solutions for \(\sin \theta\), adhering to real-world restrictions of trigonometric values.