Identify the incorrect statement

Options

A. (1)The period of f(x) `= sin cos((x)/(2)) + cos (sin x)` is `2pi`
B. (2)The period of f(x) = cos x cos 2x cos 3x is `2pi`
C. (3)Let `n in Z` and the period of `f(x) = (sin nx)/(sin((x)/(n)))` is `4pi` then n = 2
D. (4)If the period of `f(x) = cos sqrt((a))x` is `pi` and () denotes the least integer function then `a in [2, 4)`

Answer

To identify the incorrect statement among the given options, we will analyze each statement step by step.

### Step 1: Analyze Statement 1
**Statement:** \( f(x) = \sin x \cdot \cos x + \cos \sin x \) has a period of \( 2\pi \).

1. The period of \( \sin x \cdot \cos x \) is \( 2\pi \).
2. The period of \( \cos \sin x \) is also \( 2\pi \).
3. To find the overall period, we take the least common multiple (LCM) of the periods:
- LCM of \( 2\pi \) and \( 2\pi \) is \( 2\pi \).
4. However, the function \( \sin x \cdot \cos x \) can be rewritten using the identity \( \sin x \cdot \cos x = \frac{1}{2} \sin(2x) \), which has a period of \( \pi \).
5. Therefore, the overall period is \( \pi \), not \( 2\pi \).

**Conclusion:** Statement 1 is incorrect.

### Step 2: Analyze Statement 2
**Statement:** \( f(x) = \cos x \cdot \cos 2x \cdot \cos 3x \) has a period of \( 2\pi \).

1. The period of \( \cos x \) is \( 2\pi \).
2. The period of \( \cos 2x \) is \( \frac{2\pi}{2} = \pi \).
3. The period of \( \cos 3x \) is \( \frac{2\pi}{3} \).
4. To find the overall period, we take the LCM of \( 2\pi, \pi, \frac{2\pi}{3} \):
- The LCM is \( 2\pi \).

**Conclusion:** Statement 2 is correct.

### Step 3: Analyze Statement 3
**Statement:** \( f(x) = \frac{\sin x}{n} \) has a period of \( 2n\pi \).

1. The period of \( \sin x \) is \( 2\pi \).
2. The period of \( \frac{\sin x}{n} \) does not change the period of the sine function; it remains \( 2\pi \).
3. Therefore, the overall period is \( 2\pi \), not \( 2n\pi \).

**Conclusion:** Statement 3 is incorrect.

### Step 4: Analyze Statement 4
**Statement:** \( f(x) = \cos(\sqrt{a}x) \) has a period of \( \frac{2\pi}{\sqrt{a}} \).

1. The period of \( \cos(kx) \) is \( \frac{2\pi}{k} \).
2. Here, \( k = \sqrt{a} \), so the period is \( \frac{2\pi}{\sqrt{a}} \).
3. If \( a \) is between 2 and 3, then the period will be \( \frac{2\pi}{\sqrt{a}} \) which is not fixed at \( 2\pi \).

**Conclusion:** Statement 4 is correct.

### Final Conclusion
The incorrect statements are:
- Statement 1
- Statement 3