Which one of the following is not periodic ?

Options

A. `|sin 3x|+sin^(2)x`
B. `cos sqrt(x)+cos^(2)x`
C. `cos 4x+tan^(2)x`
D. `cos 2x + sinx `

Answer

To determine which of the given functions is not periodic, we need to analyze the domain of each function. A function is periodic if it repeats itself at regular intervals, and a necessary condition for a function to be periodic is that its domain must be unbounded (i.e., it should be defined for all real numbers).

Let's analyze each option step by step:

### Step 1: Analyze Option A
**Function:** \( f(x) = |\sin(3x)| + \sin^2(x) \)

- **Domain Analysis:**
- The sine function is defined for all real numbers, and the modulus function does not affect this.
- Therefore, \( |\sin(3x)| \) is defined for all \( x \in \mathbb{R} \).
- The term \( \sin^2(x) \) is also defined for all \( x \in \mathbb{R} \).

- **Conclusion:** The domain of \( f(x) \) is \( \mathbb{R} \), so it is periodic.

### Step 2: Analyze Option B
**Function:** \( g(x) = \cos(\sqrt{x}) + \cos^2(x) \)

- **Domain Analysis:**
- The term \( \cos^2(x) \) is defined for all \( x \in \mathbb{R} \).
- However, \( \cos(\sqrt{x}) \) is only defined for \( x \geq 0 \) (as the square root of a negative number is not defined in the real number system).

- **Conclusion:** The domain of \( g(x) \) is \( [0, \infty) \), which is not unbounded, hence it is not periodic.

### Step 3: Analyze Option C
**Function:** \( h(x) = \cos(4x) + \tan^2(x) \)

- **Domain Analysis:**
- The cosine function is defined for all \( x \in \mathbb{R} \).
- The tangent function is also defined for all real numbers except for odd multiples of \( \frac{\pi}{2} \) (where it is undefined).

- **Conclusion:** The domain of \( h(x) \) is \( \mathbb{R} \) except for points where \( \tan(x) \) is undefined. Although it has some restrictions, it is still considered unbounded, so it may be periodic.

### Step 4: Analyze Option D
**Function:** \( k(x) = \cos(2x) + \sin(x) \)

- **Domain Analysis:**
- Both the cosine and sine functions are defined for all \( x \in \mathbb{R} \).

- **Conclusion:** The domain of \( k(x) \) is \( \mathbb{R} \), so it is periodic.

### Final Conclusion
Among the options analyzed, the only function that does not satisfy the necessary condition for periodicity (having an unbounded domain) is:

**Option B: \( \cos(\sqrt{x}) + \cos^2(x) \)** is not periodic.