Let `f:R rarr R` be a function defined by `f(x)={abs(cosx)}`, where {x} represents fractional part of x. Let S be the set containing all real values x lying in the interval `[0,2pi]` for which `f(x) ne abs(cosx)`. The number of elements in the set S is

Options

A. (a)0
B. (b)1
C. (c)3
D. (d)infinite

Answer

To solve the problem, we need to analyze the function \( f(x) = \{ \cos x \} \), where \( \{x\} \) represents the fractional part of \( x \). We want to find the set \( S \) containing all real values \( x \) in the interval \([0, 2\pi]\) for which \( f(x) \neq |\cos x| \).

### Step-by-step Solution:

1. **Understanding the Function**:
The function \( f(x) = \{ \cos x \} \) means we take the fractional part of \( \cos x \). The fractional part of a number \( y \) is defined as \( y - \lfloor y \rfloor \), where \( \lfloor y \rfloor \) is the greatest integer less than or equal to \( y \).

2. **Condition for \( f(x) \neq |\cos x| \)**:
The condition \( f(x) \neq |\cos x| \) holds true when \( \cos x \) is an integer. This is because the fractional part \( \{ \cos x \} \) will be zero if \( \cos x \) is an integer (i.e., \( \cos x = 0, 1, -1 \)).

3. **Finding Values of \( x \)**:
We need to find the values of \( x \) in the interval \([0, 2\pi]\) where \( \cos x \) takes integer values:
- \( \cos x = 1 \) at \( x = 0 \) and \( x = 2\pi \).
- \( \cos x = -1 \) at \( x = \pi \).
- \( \cos x = 0 \) at \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \).

4. **Identifying Points**:
The integer values of \( \cos x \) in the interval \([0, 2\pi]\) are:
- \( \cos x = 1 \) at \( x = 0 \) and \( x = 2\pi \).
- \( \cos x = -1 \) at \( x = \pi \).
- \( \cos x = 0 \) at \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \).

5. **Counting the Points**:
The points where \( f(x) \neq |\cos x| \) occur at:
- \( x = 0 \)
- \( x = \pi \)
- \( x = 2\pi \)
- \( x = \frac{\pi}{2} \)
- \( x = \frac{3\pi}{2} \)

Thus, the values of \( x \) for which \( f(x) \neq |\cos x| \) are \( 0, \pi, 2\pi, \frac{\pi}{2}, \frac{3\pi}{2} \).

6. **Conclusion**:
The total number of elements in the set \( S \) is 5.

### Final Answer:
The number of elements in the set \( S \) is **5**.