Problem 42 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The values of \(x\) in \((-\pi, \pi)\) which satisfy the equation \(8^{1+|\cos x|+\cos ^{2} x+\left|\cos ^{3} x\right|+\ldots \text { to infinity }}=4^{3}\) are (A) \(\pm \frac{\pi}{4}\) (B) \(\pm \frac{\pi}{3}\) (C) \(\pm \frac{2 \pi}{3}\) (D) none of these

Step-by-Step Solution

Answer

The values of x are \( \pm \frac{\pi}{3} \), hence option B.

Step 1: Rewrite the Exponentially Infinite Series

We have the equation \[ 8^{1+|\cos x|+\cos^2 x + |\cos^3 x| + \ldots} = 4^3. \]Recognizing the infinite series as a geometric series, denote it as \[ S = 1 + |\cos x| + \cos^2 x + |\cos^3 x| + \ldots. \]

Step 2: Determine the Infinite Series Sum

Since the series \[ S = 1 + |\cos x| + \cos^2 x + |\cos^3 x| + \ldots \] is a geometric series with initial term 1 and common ratio \(|\cos x|\), it sums to:\[ S = \frac{1}{1 - |\cos x|}. \]

Step 3: Equate Powers of 8 and 4

Given \[ 8^S = 4^3, \]we can express 8 and 4 as powers of 2: \[ (2^3)^S = (2^2)^3. \] This translates into \[ 2^{3S} = 2^6. \]Therefore, equate exponents:\[ 3S = 6. \]

Step 4: Solve for the Series Sum S

From equation \(3S = 6\), divide both sides by 3 to obtain:\[ S = 2. \]

Step 5: Solve for Cosine Absolute Value

Recall the solution of the series sum:\[ \frac{1}{1 - |\cos x|} = 2. \]Inverting both sides gives:\[ 1 - |\cos x| = \frac{1}{2}. \]Thus, solving for \( |\cos x| \):\[ |\cos x| = \frac{1}{2}. \]

Step 6: Determine Possible Values of x

Knowing \(|\cos x| = \frac{1}{2}\), possible values of \(x\) in the interval \((-\pi, \pi)\) for \(\cos x = \frac{1}{2}\) are:- \(x = \frac{\pi}{3}\)- \(x = -\frac{\pi}{3}\).Thus, values are \(x = \pm \frac{\pi}{3}\).

Key Concepts

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. This type of series is quite straightforward once you understand the pattern.

• In the given exercise, the series is represented by terms like 1, \(|\cos x|\), \(\cos^2 x\), \(|\cos^3 x|\), and so forth.
• The common ratio here is \(|\cos x|\) because each term is essentially the previous term multiplied by this value.

The sum of an infinite geometric series can be calculated using the formula: \[S = \frac{a}{1 - r},\]where "\(a\)" is the first term and "\(r\)" is the common ratio. The formula is applicable when the absolute value of the common ratio is less than one, ensuring the sum converges to a finite number. In our case, since the first term "\(a\)" is 1, and the series should converge for values of \(|\cos x| < 1\), the sum was easily deduced in the solution.

Cosine Function

The cosine function, which is a fundamental trigonometric concept, measures the horizontal distance from the vertical line through the circle’s center to the line tangent to the unit circle at the terminal point. This function is periodic with a period of \(2\pi\), meaning it repeats its values every \(2\pi\) interval.

• The cosine of an angle \(x\), which can be found on the unit circle, takes values between -1 and 1.
• It achieves a maximum value of 1 when \(x = 0\) and a minimum value of -1 when \(x = \pi\) or \(x = -\pi\).

In this exercise, finding \(|\cos x| = \frac{1}{2}\) is crucial. It requires finding angles where the cosine yields an absolute value of \(\frac{1}{2}\). Within the interval \((-\pi, \pi)\), these angles are \(x = \frac{\pi}{3}\) and \(x = -\frac{\pi}{3}\), reflecting the symmetry of the cosine wave.

Infinite Series Sum

The concept of an infinite series sum pertains to summing up all the terms in a series that continues indefinitely. If the terms in the series decrease in a suitable manner, the sum will approach a certain finite number.

• The geometric series sum formula from earlier is a quintessential tool to find sum of infinite series as it provides a way to calculate this sum when the absolute value of the common ratio is less than one.
• In the context of the problem, using \(S = \frac{1}{1 - |\cos x|}\) as the sum was critical for simplifying the original equation into a solvable form.

Once \(|\cos x|\) was determined, the expression \(\frac{1}{1 - |\cos x|}\) simplified directly into known values, turning the infinite series into something much more manageable to solve. Understanding this technique is essential for handling infinite series efficiently.