Consider the following statements:
I. If 3`theta` is an acute angle such that sin3`theta` = cos 2`theta`, then the measurement of `theta` in radian equals to `(pi)/(10)`.
II. One radian is the angle subtended at the centre of a circle by an arc of the same circle chose length is equal to the diameter of that circle.
Which of the above statements is/are correct?

Options

A. A) Only I
B. B) Only II
C. C) Both I and II
D. D) Neither I nor II

Answer

To solve the given problem, we need to analyze both statements and determine their correctness step by step.

### Step 1: Analyze the first statement
The first statement claims that if \(3\theta\) is an acute angle such that \(\sin 3\theta = \cos 2\theta\), then the measurement of \(\theta\) in radians equals \(\frac{\pi}{10}\).

1. We start with the equation:
\[ \sin 3\theta = \cos 2\theta \]

2. We can use the co-function identity, which states that \(\cos x = \sin\left(\frac{\pi}{2} - x\):
\[ \cos 2\theta = \sin\left(\frac{\pi}{2} - 2\theta\right) \]

3. Therefore, we can rewrite the equation as:
\[ \sin 3\theta = \sin\left(\frac{\pi}{2} - 2\theta\right) \]

4. Since \(3\theta\) is an acute angle, we can set the angles equal to each other:
\[ 3\theta = \frac{\pi}{2} - 2\theta \]

5. Rearranging gives:
\[ 3\theta + 2\theta = \frac{\pi}{2} \]
\[ 5\theta = \frac{\pi}{2} \]

6. Solving for \(\theta\):
\[ \theta = \frac{\pi}{10} \]

### Step 2: Analyze the second statement
The second statement claims that one radian is the angle subtended at the center of a circle by an arc of the same circle whose length is equal to the diameter of that circle.

1. By definition, one radian is the angle subtended at the center of a circle by an arc whose length is equal to the **radius** of that circle, not the diameter.

2. Therefore, the statement is incorrect.

### Conclusion
- The first statement is correct: \(\theta = \frac{\pi}{10}\).
- The second statement is incorrect.

Thus, the correct answer is that only the first statement is true.

### Summary of the solution:
- Statement I is correct: \(\theta = \frac{\pi}{10}\).
- Statement II is incorrect.