Problem 4 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The number of all possible triplets \(\left(a_{1}, a_{2}, a_{3}\right)\) such that \(a_{1}+a_{2} \cos 2 x+a_{3} \sin ^{2} x=0\) for all \(x\) is (A) 0 (B) 1 (C) 3 (D) infinite

Step-by-Step Solution

Answer

(D) Infinite

Step 1: Understanding the Expression

The equation given is \(a_1 + a_2 \cos(2x) + a_3 \sin^2(x) = 0\) for all \(x\). Since the expression should hold for any value of \(x\), the expression identifies a functional form that must be zero regardless of \(x\).

Step 2: Simplifying Trigonometric Terms

First, rewrite \(\sin^2(x)\) using the identity \(\sin^2(x) = \frac{1 - \cos(2x)}{2}\). Substituting this, the equation becomes: \(a_1 + a_2 \cos(2x) + \frac{a_3}{2}(1 - \cos(2x)) = 0\).

Step 3: Expanding and Rearranging the Equation

After substitution, expand the equation: \(a_1 + \frac{a_3}{2} + (a_2 - \frac{a_3}{2}) \cos(2x) = 0\). Rearranging for clarity gives us two separate parts: a constant term \(a_1 + \frac{a_3}{2}\) and a variable term \((a_2 - \frac{a_3}{2}) \cos(2x)\).

Step 4: Analyzing Conditions for Zero Functionality

Because \(a_1 + a_2 \cos(2x) + a_3 \sin^2(x) = 0\) must be true for all \(x\), each component must individually be zero. Thus, we need: 1) \(a_1 + \frac{a_3}{2} = 0\), and 2) \(a_2 - \frac{a_3}{2} = 0\).

Step 5: Solving the System of Equations

From the two conditions, solve the equations: \(a_1 = -\frac{a_3}{2}\) and \(a_2 = \frac{a_3}{2}\). These give the solutions in terms of \(a_3\), meaning that \(a_1 = -t\), \(a_2 = t\), and \(a_3 = 2t\) for some parameter \(t\).

Step 6: Determining the Number of Triplets

The solution implies there is a degree of freedom determined by the parameter \(t\). Since \(t\) can be any real number, there is an infinite number of triplets \((a_1, a_2, a_3)\) satisfying the equation.

Key Concepts

Triplets

In mathematics, the concept of triplets usually refers to a group of three elements. In this exercise, the triplet \((a_1, a_2, a_3)\) represents three coefficients in a trigonometric equation. This involves finding specific values that satisfy a given equation for all instances of \(x\).
Exploration in triplets can be found in various mathematical applications, such as solving systems of equations and analyzing vector components.

• Triplets allow us to succinctly express relationships between multiple variables.
• They are instrumental in translating real-life problems into mathematical form.

Understanding how these triplets interact with each other through trigonometric equations is essential to harness their full potential in mathematical modeling.

Solution of Equations

The solution of an equation involves finding the values that satisfy the equation. In this problem, the goal is to determine the triplets that fulfill the equation for every possible value of \(x\).
When dealing with equations that depend on a variable, such as \(x\) in this case, we need to ensure that the expression holds true universally, that is, across all possible scenarios involving \(x\).

• Each component of the equation must independently satisfy the condition of being zero.
• This leads to a set of simultaneous equations that need to be solved.

The approach taken in this exercise simplifies the expression and then employs algebraic techniques to find solutions, ultimately revealing an infinite number of solutions due to the dependency on a parameter \(t\).

Trigonometric Identities

Trigonometric identities, such as \(\sin^2(x) = \frac{1 - \cos(2x)}{2}\) are equations that relate various trigonometric functions to one another.
They are essential tools for simplifying and solving trigonometric equations, as illustrated in this exercise.

• Using identities, complex expressions are simplified into more manageable forms.
• They allow expressions to be factored or rearranged, aiding in the clarity of solving equations.

In this problem, the identity for \(\sin^2(x)\) is used to eliminate the presence of multiple trigonometric functions in the equation, therefore making it easier to solve the underlying system of equations.

System of Equations

A system of equations involves multiple equations that occur simultaneously. Solving such systems requires finding variable values that meet all the conditions set by the equations.
In this exercise, two equations result from the requirement that each term in the trigonometric equation must independently equal zero for all \(x\).

• The solution involves simultaneously addressing both equations: \(a_1 + \frac{a_3}{2} = 0\) and \(a_2 - \frac{a_3}{2} = 0\).
• The solution set demonstrates the dependency of \(a_1, a_2,\) and \(a_3\) on the parameter \(t\).

Such systems are quite common in mathematical problem-solving, and mastering them involves learning to systematically approach each equation and finding a solution that satisfies all, leading to a comprehensive understanding of the problem.