Problem 90 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(a\) and \(b\) are natural numbers and \(f(x)=\sin \left(\sqrt{a^{2}-3}\right) x+\cos \left(\sqrt{b^{2}+7}\right) x\) is periodic with finite fundamental period, then period of \(f(x)\) is: (A) \(\pi\) (B) \(2 \pi\) (C) \(2 \pi\left(\sqrt{a^{2}-3}+\sqrt{b^{2}+7}\right)\) (D) \(\pi\left(\sqrt{a^{2}-3}+\sqrt{b^{2}+7}\right)\)

Step-by-Step Solution

Answer

The period of \(f(x)\) is \(2\pi\), option (B).

Step 1: Understand the Periodicity Condition

For a function of the form \(f(x) = \sin(cx) + \cos(dx)\), it is periodic if and only if both components, \(\sin(cx)\) and \(\cos(dx)\), have a common period. The fundamental periods of \(\sin(cx)\) and \(\cos(dx)\) are \(\frac{2\pi}{|c|}\) and \(\frac{2\pi}{|d|}\) respectively. Thus, both expressions must be commensurate, meaning \(\frac{c}{d}\) must be a rational number.

Step 2: Determine Constants c and d

Given \(f(x) = \sin\left(\sqrt{a^2 - 3} x\right) + \cos\left(\sqrt{b^2 + 7} x\right)\), identify \(c = \sqrt{a^2 - 3}\) and \(d = \sqrt{b^2 + 7}\). For the function to be periodic, \(\frac{c}{d} = \frac{\sqrt{a^2 - 3}}{\sqrt{b^2 + 7}}\) must be a rational number.

Step 3: Find Conditions for Periodicity

For the ratio \(\frac{c}{d}\) to be rational, \(a^2 - 3\) and \(b^2 + 7\) must result in rational numbers after taking the square root, implying they are both perfect squares. Thus, let's say \(a^2 - 3 = m^2\) and \(b^2 + 7 = n^2\) where \(m\) and \(n\) are integers.

Step 4: Solving for a and b

For \(m^2 = a^2 - 3\), rearrange to \(a^2 = m^2 + 3\). Similarly, for \(n^2 = b^2 + 7\), rearrange to \(b^2 = n^2 - 7\). Seeking integers \(a, b, m,\) and \(n\), find that \((a, b) = (2, 3)\) satisfies these equations since \(a = 2, a^2 = 4, a^2 - 3 = 1\) and \(b = 3, b^2 = 9, b^2 + 7 = 16\). Thus, \(m = 1\) and \(n = 4\), making both expressions rational numbers.

Step 5: Establishing the Period

With \(c = 1\) and \(d = 4\), the smallest common period (or the fundamental period) between \(\sin(x)\) and \(\cos(4x)\) is determined. Utilizing the formula for least common multiple of periodic times, where the fundamental period is \(\frac{2\pi}{1} = 2\pi\) and \(\frac{2\pi}{4} = \frac{\pi}{2}\), the least common period \(L\) is the LCM of these, resulting in \(2\pi\).

Key Concepts

Trigonometric Functions

Trigonometric functions are a fundamental concept in mathematics, especially when dealing with periodic functions. These functions include sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)), along with their reciprocals cosecant (\(\csc\)), secant (\(\sec\)), and cotangent (\(\cot\)). They are commonly used to model phenomena that exhibit wave-like behavior in science and engineering.

• Sine and Cosine Functions: These are the most basic trigonometric functions, representing y-coordinates and x-coordinates on the unit circle respectively. Their graphs are sinusoidal, showing repeating patterns or 'waves' over a specified interval.


• Periodicity: Both \(\sin\theta\) and \(\cos\theta\) have periods of \(2\pi\), meaning their values repeat every \(2\pi\) radians.


• Fundamental Properties: Trigonometric functions are periodic, oscillating between -1 and 1, and help in analyzing cycles such as circadian rhythms, sound waves, and light waves.

Fundamental Period

The fundamental period of a function is the smallest interval over which the function completes one full cycle and begins to repeat its values. This concept is crucial when analyzing periodic functions, especially in the context of trigonometry.
For the function \(f(x) = \sin(cx) + \cos(dx)\), the components \(\sin(cx)\) and \(\cos(dx)\) each have their fundamental periods calculated by \(\frac{2\pi}{|c|}\) and \(\frac{2\pi}{|d|}\) respectively. To determine the fundamental period of \(f(x)\), these periods must be commensurate, meaning there exists a common period they both share.

• Commensurate Periods: If \(c\) and \(d\) are chosen such that \(\frac{c}{d}\) is rational, a common fundamental period can be derived.


• Determining the Least Common Period: The least common period between \(\sin(cx)\) and \(\cos(dx)\) is given by the least common multiple (LCM) of their individual periods, ensuring \(f(x)\) is indeed periodic.

Natural Numbers

Natural numbers are the set of positive integers starting from 1, denoted \(\{1, 2, 3, \ldots\}\). They are fundamental in mathematics and used in various contexts, such as counting and ordering.
In the context of trigonometric functions and periodicity, natural numbers often represent parameters in function equations:

• Role in Periodicity: Parameters like \(a\) and \(b\) in \(\sin\left(\sqrt{a^{2}-3}\right)x\) and \(\cos\left(\sqrt{b^{2}+7}\right)x\) are natural numbers.


• Square Terms: In periodic function analysis, ensuring that square expressions involving natural numbers are perfect squares (e.g., \(a^{2}-3\)) helps in determining if parameters allow the function to have a rational period ratio.


• Use in Problem Solving: During steps to resolve periodicity, the natural numbers often define conditions that both simplify and specify the behavior of the trigonometric functions.