Problem 60 - 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 the function is \( \pi(\sqrt{a^2 - 3} + \sqrt{b^2 + 7}) \) (Option D).

Step 1: Analyze the periodicity condition for trigonometric functions

The function \( f(x) = \sin(u)x + \cos(v)x \) is periodic if both \( \sin(u)x \) and \( \cos(v)x \) are periodic. The general period of \( \sin(kx) \) and \( \cos(kx) \) is \( \frac{2\pi}{|k|} \). For \( \sin(\sqrt{a^2 - 3} \cdot x) \), its period will be \( \frac{2\pi}{\sqrt{a^2 - 3}} \). Similarly, for \( \cos(\sqrt{b^2 + 7} \cdot x) \), its period will be \( \frac{2\pi}{\sqrt{b^2 + 7}} \).

Step 2: Find common period

For \( f(x) \) to be periodic, the periods of \( \sin(\sqrt{a^2 - 3} \cdot x) \) and \( \cos(\sqrt{b^2 + 7} \cdot x) \) must be commensurable, meaning they must have a least common multiple. The least common multiple of \( \frac{2\pi}{\sqrt{a^2 - 3}} \) and \( \frac{2\pi}{\sqrt{b^2 + 7}} \) is \( \pi\left(\sqrt{a^2 - 3} + \sqrt{b^2 + 7}\right) \), requires both \( \sqrt{a^2 - 3} \) and \( \sqrt{b^2 + 7} \) to be equal. Hence, their sum becomes the effective factor for the period calculation.

Step 3: Verify the periodic solution

For simplicity and comparative understanding, we set \( T = \pi(\sqrt{a^2 - 3} + \sqrt{b^2 + 7}) \). According to the general rules of trigonometric periodicity, both functions will repeat themselves completely in this interval, confirming it is the least period that satisfies conditions for \( f(x) \) repeating identically.

Step 4: Conclude the period of the function

The fundamental period of the function \( f(x) \) is therefore the least common multiple derived in Step 2. Checking the provided options, \( \pi(\sqrt{a^2 - 3} + \sqrt{b^2 + 7}) \) corresponds to option D.

Key Concepts

Trigonometric Functions

Trigonometric functions like sine and cosine are some of the most important functions in mathematics, especially because they describe periodic phenomena such as waves and oscillations. They are defined using the unit circle on the Cartesian plane and have a few key properties:

• Periodicity: Both sine ( \( \sin(x) \) ) and cosine ( \( \cos(x) \) ) are periodic functions with a fundamental period of \( 2\pi \) when the argument is in radians.
  In simpler terms, this means that their values repeat every \( 2\pi \) . For example, \( \sin(x) = \sin(x + 2\pi) \) and \( \cos(x) = \cos(x + 2\pi) \) .
• Range: These functions can only produce results between -1 and 1.
  This is crucial when analyzing their graphs and determining periodic properties.
• Amplitude: For the standard sine and cosine functions, the amplitude is 1.
  This represents the peak deviation from zero.

In the case of our function \( f(x) = \sin(u)x + \cos(v)x \), the specific forms \( \sin(u) \) and \( \cos(v) \) depend on the parameters inside these trigonometric functions. Understanding how these parameters affect periodicity is key to solving problems involving periodic functions.

Fundamental Period

The fundamental period of a trigonometric function is the smallest positive interval over which the function completes one full cycle and begins to repeat itself. For the standard sine and cosine functions, this is \( 2\pi \) , meaning that if we move along the x-axis for a length of \( 2\pi \) , the function's graph will look the same as it did at the start.
If we modify the function by changing its argument (the variable inside the trigonometric functions), this period might change. For example, the period of \( \sin(kx) \) and \( \cos(kx) \) is determined by the expression \( \frac{2\pi}{|k|} \) . The factor \( k \) changes how rapidly the function oscillates. Larger values of \( |k| \) mean more cycles fit into the same interval, thus reducing the visible length of each cycle:

• When \( k = 1 \) , the period is \( 2\pi \) , the standard case.
• If \( k = 2 \) , the period is \( \frac{2\pi}{2} = \pi \) , indicating that the function completes two full cycles over the interval \( 0 \) to \( 2\pi \) .

In the problem, we see periods like \( \frac{2\pi}{\sqrt{a^2 - 3}} \) for \( \sin(\sqrt{a^2 - 3}\cdot x) \). This parameter-driven approach requires us to think about how these periods interact when dealing with summing or combining functions.

Least Common Multiple

Finding the least common multiple (LCM) concerns discovering the smallest value that two (or more) numbers can divide into without leaving a remainder. In the context of periodic functions, the LCM becomes important when determining the period of a sum of functions, like \( f(x) = \sin(u)x + \cos(v)x \).
For \( f(x) \) to be periodic, there's a key requirement: the periods of its constituent functions, \( \sin(u)x \) and \( \cos(v)x \), must line up at regular intervals.

• Calculating the LCM of two periods (\( p_1 \) and \( p_2 \)) involves finding a shared period where both functions repeat simultaneously.
• This concept often applies when the periods aren't integers, such as \( \frac{2\pi}{\sqrt{a^2 - 3}} \) and \( \frac{2\pi}{\sqrt{b^2 + 7}} \). We must compute their LCM to see when both return to their initial values.

In the exercise, since the period of the overall function \( f(x) \) equals the LCM of the individual periods, it partly selects the outcome based on the need for these periods to be commensurable, syncing their cycles every \( T = \pi(\sqrt{a^2 - 3} + \sqrt{b^2 + 7}) \), making sure all trigonometric cycles align perfectly.