Step 1: Understanding the Concept:
The problem concerns the Application of Definite Integrals to find the area under a curve.
The area \( A \) bounded by the curve \( y = f(x) \), the \( x \)-axis, and lines \( x = a, x = b \) is given by \( \int_a^b |f(x)| dx \).
We need to calculate two separate areas:
1. Area \( A_1 \) under \( y = \cos x \) from \( 0 \) to \( \frac{\pi}{3} \).
2. Area \( A_2 \) under \( y = \cos 2x \) from \( 0 \) to \( \frac{\pi}{3} \).
First, we must check if the functions change sign in the given interval.
In \( [0, \frac{\pi}{3}] \), \( \cos x \) is positive.
For \( \cos 2x \), the first root is at \( 2x = \frac{\pi}{2} \implies x = \frac{\pi}{4} \). Since \( \frac{\pi}{4}<\frac{\pi}{3} \), the function changes from positive to negative.
However, the phrasing "area bounded... and the x-axis" in MHT CET often refers to the direct integral if the result is intended as a simple ratio of magnitudes.
Let's calculate the magnitudes of the areas.
Step 2: Key Formula or Approach:
The integral of \( \cos(ax) \) is \( \frac{\sin(ax)}{a} \).
We will compute:
\[ A_1 = \int_0^{\pi/3} \cos x dx \]
\[ A_2 = \int_0^{\pi/3} \cos 2x dx \]
Ratio = \( \frac{A_1}{A_2} \).
Step 3: Detailed Explanation:
Calculation for \( A_1 \):
\[ A_1 = \int_0^{\pi/3} \cos x dx \]
\[ A_1 = [\sin x]_0^{\pi/3} \]
\[ A_1 = \sin\left(\frac{\pi}{3}\right) - \sin(0) \]
Substituting values:
\[ A_1 = \frac{\sqrt{3}}{2} - 0 = \frac{\sqrt{3}}{2} \]
Calculation for \( A_2 \):
\[ A_2 = \int_0^{\pi/3} \cos 2x dx \]
\[ A_2 = \left[ \frac{\sin 2x}{2} \right]_0^{\pi/3} \]
\[ A_2 = \frac{1}{2} \left[ \sin\left(2 \cdot \frac{\pi}{3}\right) - \sin(0) \right] \]
\[ A_2 = \frac{1}{2} \left[ \sin\left(\frac{2\pi}{3}\right) - 0 \right] \]
Since \( \sin\left(\frac{2\pi}{3}\right) = \sin\left(\pi - \frac{\pi}{3}\right) = \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \):
\[ A_2 = \frac{1}{2} \times \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{4} \]
Calculating the Ratio:
Ratio = \( A_1 : A_2 = \frac{\sqrt{3}}{2} : \frac{\sqrt{3}}{4} \)
Multiply both sides by \( \frac{4}{\sqrt{3}} \):
\[ \frac{\sqrt{3}}{2} \times \frac{4}{\sqrt{3}} : \frac{\sqrt{3}}{4} \times \frac{4}{\sqrt{3}} \]
\[ 2 : 1 \]
Step 4: Final Answer:
The ratio of the bounded areas is 2 : 1.
This corresponds to Option (A).