If the range of `f(x) =cos^(-1)[5x]` is {a, b,c} and `a+b+c = (lambda pi)/(2)`, then `lambda` is equal to ([.] denotes G.I.F.)

Answer

To find the value of \( \lambda \) in the problem, we need to analyze the function \( f(x) = \cos^{-1}(5x) \) and determine its range.

### Step 1: Determine the range of \( 5x \)
The function \( \cos^{-1}(x) \) is defined for \( x \) in the interval \([-1, 1]\). Therefore, for \( f(x) = \cos^{-1}(5x) \) to be defined, \( 5x \) must also lie within this interval:
\[ -1 \leq 5x \leq 1 \]
Dividing the entire inequality by 5 gives:
\[ -\frac{1}{5} \leq x \leq \frac{1}{5} \]

### Step 2: Identify the values of \( 5x \)
The greatest integer function (GIF) applied to \( 5x \) will yield integer values within the range of \( 5x \). As \( x \) varies from \(-\frac{1}{5}\) to \(\frac{1}{5}\), \( 5x \) will vary from \(-1\) to \(1\). The possible integer values that \( 5x \) can take are:
\[ \{-1, 0, 1\} \]

### Step 3: Calculate \( f(x) \) for these values
Next, we calculate \( f(x) \) for each of these integer values:
- For \( 5x = -1 \):
\[ f(x) = \cos^{-1}(-1) = \pi \]
- For \( 5x = 0 \):
\[ f(x) = \cos^{-1}(0) = \frac{\pi}{2} \]
- For \( 5x = 1 \):
\[ f(x) = \cos^{-1}(1) = 0 \]

### Step 4: Identify the range \( \{a, b, c\} \)
Thus, the range of \( f(x) \) is:
\[ \{0, \frac{\pi}{2}, \pi\} \]
We can assign:
- \( a = 0 \)
- \( b = \frac{\pi}{2} \)
- \( c = \pi \)

### Step 5: Calculate \( a + b + c \)
Now, we compute:
\[ a + b + c = 0 + \frac{\pi}{2} + \pi = \frac{\pi}{2} + \frac{2\pi}{2} = \frac{3\pi}{2} \]

### Step 6: Relate to \( \lambda \)
According to the problem, we have:
\[ a + b + c = \frac{\lambda \pi}{2} \]
Setting this equal to our previous result:
\[ \frac{3\pi}{2} = \frac{\lambda \pi}{2} \]
Dividing both sides by \( \frac{\pi}{2} \):
\[ 3 = \lambda \]

### Final Answer
Thus, the value of \( \lambda \) is:
\[ \boxed{3} \]