Problem 61 - Comprehensive Trigonometry for IIT JEE Main and Advanced Rejaul Makshud MeGraw Hill

Question

Solve the following equations and tick the correct one. The number of solutions of the equation \(\sin \left(\frac{\pi x}{2 \sqrt{3}}\right)=x^{2}-2 \sqrt{3} x+4\) (a) forms an empty set (b) only one (c) is only two (d) is greater then two

Step-by-Step Solution

Answer

Without any specific values or plots, this approach might not provide an exact answer. However, with the right plotting tool available, one can draw a graph, find the intersection points and figure out the correct option from the choices: (a) forms an empty set (b) only one (c) only two (d) is greater then two.

Step 1: Identify functions

We identify two functions: the trigonometric function f\(x = \sin \left(\frac{\pi x}{2\sqrt{3}}\right)\) and the polynomial function g\(x = x^{2}-2\sqrt{3}x+4\). The solutions to the equation are the intersection points of f(x) and g(x).

Step 2: Graph the functions

We need to plot these two functions on the same set of axes to find out the number of intersecting points. If there exist points on the graph such that the x-coordinates of these points are the same for f(x) and g(x), then those x-coordinates are the solutions to the equation.

Step 3: Count the intersection points

After plotting the graphs, count the intersection points. Each intersection point represents a solution to the equation.

Step 4: Conclude

By graphing and counting the intersection points, we can conclude the number of solutions for the given equation and determine the correct option

Key Concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides in the context of right-angle triangles. The most common trigonometric functions are sine, cosine, and tangent. In the given problem, we use the sine function: \( f(x) = \sin \left(\frac{\pi x}{2\sqrt{3}}\right) \). This function oscillates between -1 and 1, creating a wave-like pattern.

• Period of Sine Function: The general period of the sine function is \(2\pi\), but with modifications in the argument, such as \(\frac{\pi x}{2\sqrt{3}}\), the period changes.
• Amplitude: The amplitude (maximum height) of the sine function remains 1, regardless of the horizontal stretching or compressing.

Understanding how these functions behave graphically is crucial to finding where they intersect with other functions, such as polynomial functions.

Polynomial Functions

Polynomial functions are expressions that involve sums of powers of variables, usually denoted as \(x\). A polynomial function can take various forms, such as linear, quadratic, or cubic. For example, \(g(x) = x^{2} - 2\sqrt{3}x + 4\) is a quadratic polynomial function. This means it forms a parabola when graphed.

• Quadratic Form: A quadratic polynomial always takes the form \(ax^2 + bx + c\).
• Vertex and Direction: The vertex of this quadratic can be found using the vertex formula \(x = -\frac{b}{2a}\), and the direction (upward or downward) is determined by the sign of \(a\). For a > 0, it opens upwards.

Polynomial functions can intersect with trigonometric functions at multiple points depending on their shape and orientation, revealing the solutions to the equation.

Intersection Points

Intersection points occur where two functions meet or cross each other on a graph. These points are crucial in solving the given trigonometric equation because each intersection point represents a solution to the equation. To find them, we graph both the trigonometric and polynomial functions on the same coordinate plane.

• Visual Inspection: Carefully observe where the graphs intersect. The \(x\)-coordinates of these points are the solutions.
• Exact Solutions: While graphs provide a visual idea, solving algebraically can give precise intersection points, if necessary.

In the problem at hand, we aim to count these intersection points to determine the number of solutions, which will tell us which option (a, b, c, or d) is correct.

Graphical Solutions

Graphical solutions involve plotting functions on a graph to visually identify solutions to equations. This approach is particularly useful when the equation involves complex functions, like a combination of trigonometric functions and polynomial equations.

• Step-by-Step Graphing: Start by sketching both \(\sin\left(\frac{\pi x}{2\sqrt{3}}\right)\) and \(x^{2} - 2\sqrt{3}x + 4\). Make sure your scales are consistent.
• Analyzing Graphs: Evaluate where both graphs intersect to identify potential solutions visually.
• Advantages: This method not only provides a peek into the behavior of functions but also simplifies the process of finding where they agree upon a solution.

Using graphical solutions, as we do in this exercise, is an effective way to wrap up our understanding and confirm the number of solutions numerically and visually.