Let `f (x) =x^(2)+ lamda x+ mu cos x, lamda` being an integer and `mu` is a real number. The number of ordered pairs `(lamda, mu)` for which the equation `f(x) =0 and f (f (x))=0` have the same (non empty) set of real roots is:

Options

A. 2
B. 1
C. 4
D. 6

Answer

To solve the problem, we need to analyze the function \( f(x) = x^2 + \lambda x + \mu \cos x \) and find the ordered pairs \( (\lambda, \mu) \) such that the equations \( f(x) = 0 \) and \( f(f(x)) = 0 \) have the same non-empty set of real roots.

### Step 1: Understand the function \( f(x) \)
The function is given by:
\[ f(x) = x^2 + \lambda x + \mu \cos x \]
where \( \lambda \) is an integer and \( \mu \) is a real number.

### Step 2: Set up the equation \( f(x) = 0 \)
To find the roots of \( f(x) = 0 \):
\[ x^2 + \lambda x + \mu \cos x = 0 \]
This is a quadratic equation in \( x \).

### Step 3: Analyze \( f(f(x)) = 0 \)
Next, we need to compute \( f(f(x)) \):
\[ f(f(x)) = f(x^2 + \lambda x + \mu \cos x) \]
Substituting \( f(x) \) into itself gives:
\[ f(f(x)) = (f(x))^2 + \lambda f(x) + \mu \cos(f(x)) \]

### Step 4: Find conditions for \( f(x) = 0 \) and \( f(f(x)) = 0 \) to have the same roots
For \( f(x) = 0 \) and \( f(f(x)) = 0 \) to have the same set of real roots, we need to analyze the conditions under which this occurs.

### Step 5: Substitute \( f(x) = 0 \) into \( f(f(x)) \)
If \( f(x) = 0 \), then:
\[ f(f(x)) = f(0) = 0^2 + \lambda(0) + \mu \cos(0) = \mu \]
Thus, for \( f(f(x)) = 0 \), we require:
\[ \mu = 0 \]

### Step 6: Substitute \( \mu = 0 \) back into \( f(x) \)
Substituting \( \mu = 0 \) into the original function gives:
\[ f(x) = x^2 + \lambda x \]
Factoring this, we get:
\[ f(x) = x(x + \lambda) = 0 \]
This gives the roots \( x = 0 \) and \( x = -\lambda \).

### Step 7: Determine the number of ordered pairs \( (\lambda, \mu) \)
Since \( \lambda \) is an integer and \( \mu \) must be \( 0 \), the ordered pairs \( (\lambda, \mu) \) can be expressed as \( (\lambda, 0) \) where \( \lambda \) can take any integer value.

### Conclusion
The number of ordered pairs \( (\lambda, \mu) \) for which the equations \( f(x) = 0 \) and \( f(f(x)) = 0 \) have the same non-empty set of real roots is infinite, as \( \lambda \) can be any integer.