Question

Which of the following is an example of a condensation polymer?

• A. Polyethylene
• B. PVC
• C. Nylon 6,6
• D. Polystyrene

Hint:

Remember: polymers like Polyethylene, PVC, and Polystyrene are addition polymers, while Nylon, Terylene, and Bakelite are condensation polymers.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
The problem asks us to calculate the value of a definite integral involving trigonometric functions in the denominator where the limits are from \( 0 \) to \( \pi/2 \).
Step 2: Key Formula or Approach:
We use the "King's Property" of definite integrals:
\[ \int_{0}^{a} f(x) \, dx = \int_{0}^{a} f(a-x) \, dx \] This property is particularly useful for integrals involving \( \sin x \) and \( \cos x \) over the interval \( [0, \pi/2] \).
Step 3: Detailed Explanation:
Let the given integral be \( I \):
\[ I = \int_{0}^{\pi/2} \frac{\sin x}{\sin x + \cos x} \, dx \quad \dots \text{(i)} \]
Applying the property \( \int_{0}^{a} f(x) \, dx = \int_{0}^{a} f(a-x) \, dx \), we replace \( x \) with \( \frac{\pi}{2} - x \):
\[ I = \int_{0}^{\pi/2} \frac{\sin(\frac{\pi}{2} - x)}{\sin(\frac{\pi}{2} - x) + \cos(\frac{\pi}{2} - x)} \, dx \]
Since \( \sin(\frac{\pi}{2} - x) = \cos x \) and \( \cos(\frac{\pi}{2} - x) = \sin x \), we get:
\[ I = \int_{0}^{\pi/2} \frac{\cos x}{\cos x + \sin x} \, dx \quad \dots \text{(ii)} \]
Adding equations (i) and (ii):
\[ I + I = \int_{0}^{\pi/2} \frac{\sin x + \cos x}{\sin x + \cos x} \, dx \]
\[ 2I = \int_{0}^{\pi/2} 1 \, dx \]
\[ 2I = [x]_{0}^{\pi/2} \]
\[ 2I = \frac{\pi}{2} - 0 \]
\[ I = \frac{\pi}{4} \]
Step 4: Final Answer:
The value of the integral is \( \frac{\pi}{4} \).