Number of integral values of k for which the equation `4 cos^(-1)(-|x|)=k` has exactly two solutions, is: (a) 4 (b) 5 (c) 6 (d) 7

Options

A. 4
B. 5
C. 6
D. 7

Explanation

To solve the problem of finding the number of integral values of \( k \) for which the equation \( 4 \cos^{-1}(-|x|) = k \) has exactly two solutions, we can follow these steps:

### Step 1: Understand the Range of the Function
The function \( \cos^{-1}(-|x|) \) is defined for \( -1 \leq -|x| \leq 1 \), which implies \( 0 \leq |x| \leq 1 \). Therefore, \( x \) can take values in the interval \( [-1, 1] \).

### Step 2: Determine the Range of \( 4 \cos^{-1}(-|x|) \)
The function \( \cos^{-1}(y) \) has a range of \( [0, \pi] \) for \( y \in [-1, 1] \). Thus, \( 4 \cos^{-1}(-|x|) \) will have a range of:
\[ 4 \cdot 0 = 0 \quad \text{to} \quad 4 \cdot \pi \approx 12.56 \]
So, the range of \( 4 \cos^{-1}(-|x|) \) is \( [0, 4\pi] \).

### Step 3: Analyze the Function for Solutions
The equation \( 4 \cos^{-1}(-|x|) = k \) will have solutions based on the value of \( k \). We need to find conditions under which this equation has exactly two solutions.

### Step 4: Determine the Values of \( k \)
The function \( 4 \cos^{-1}(-|x|) \) is decreasing in the interval \( [-1, 1] \). It will take each value in its range twice (once for \( x \) and once for \( -x \)) except at the endpoints where it takes the value at \( x = 0 \) only once.

1. At \( k = 0 \), there is 1 solution (at \( x = 0 \)).
2. For \( 0 < k < 4\pi \), there are 2 solutions.
3. At \( k = 4\pi \), there is 1 solution (at \( x = -1 \) and \( x = 1 \)).

### Step 5: Identify Integral Values of \( k \)
The range of \( k \) is from \( 0 \) to \( 4\pi \). We can approximate \( 4\pi \) as \( 12.56 \). Therefore, the integral values of \( k \) that can be considered are:
\[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 \]
This gives us a total of 12 integral values. However, we are interested in the values of \( k \) that yield exactly two solutions.

### Step 6: Count the Valid Integral Values
From the analysis:
- The values \( k = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 \) will yield exactly two solutions.
- The values \( k = 0 \) and \( k = 12 \) yield only one solution.

Thus, the integral values of \( k \) for which the equation has exactly two solutions are \( 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 \), which gives us a total of **11 values**.

### Conclusion
The number of integral values of \( k \) for which the equation \( 4 \cos^{-1}(-|x|) = k \) has exactly two solutions is **11**.