Given `a_(1)cos alpha_(1)+alpha_(2)cos alpha_(2)+…+a_(n)cos alpha_(n)=0` and `a_(1)cos(alpha_(1)+theta)+a_(2)cos(alpha_(2)+theta)+…+a_(n)cos(alpha_(n)+theta)=0 (theta ne k pi)`, then the value of `a_(1)cos(alpha_(1)+lambda)+a_(2)cos(alpha_(2)+lambda)+…+a_(n)cos(alpha_(n)+lambda)` is

Options

A. `theta-lambda`
B. `theta+lambda`
C. `lambda`
D. 0

Explanation

To solve the problem step by step, we need to analyze the given equations and apply trigonometric identities.

### Step 1: Understand the Given Equations
We have two equations:
1. \( a_1 \cos \alpha_1 + a_2 \cos \alpha_2 + \ldots + a_n \cos \alpha_n = 0 \)
2. \( a_1 \cos(\alpha_1 + \theta) + a_2 \cos(\alpha_2 + \theta) + \ldots + a_n \cos(\alpha_n + \theta) = 0 \) (where \( \theta \neq k\pi \))

### Step 2: Apply the Cosine Addition Formula
Using the cosine addition formula:
\[ \cos(x + y) = \cos x \cos y - \sin x \sin y \]
we can rewrite the second equation:
\[ a_1 (\cos \alpha_1 \cos \theta - \sin \alpha_1 \sin \theta) + a_2 (\cos \alpha_2 \cos \theta - \sin \alpha_2 \sin \theta) + \ldots + a_n (\cos \alpha_n \cos \theta - \sin \alpha_n \sin \theta) = 0 \]

### Step 3: Factor Out Common Terms
Factoring out the common terms:
\[ \cos \theta (a_1 \cos \alpha_1 + a_2 \cos \alpha_2 + \ldots + a_n \cos \alpha_n) - \sin \theta (a_1 \sin \alpha_1 + a_2 \sin \alpha_2 + \ldots + a_n \sin \alpha_n) = 0 \]
Since \( a_1 \cos \alpha_1 + a_2 \cos \alpha_2 + \ldots + a_n \cos \alpha_n = 0 \), we can substitute this into the equation:
\[ \cos \theta \cdot 0 - \sin \theta (a_1 \sin \alpha_1 + a_2 \sin \alpha_2 + \ldots + a_n \sin \alpha_n) = 0 \]
This simplifies to:
\[ -\sin \theta (a_1 \sin \alpha_1 + a_2 \sin \alpha_2 + \ldots + a_n \sin \alpha_n) = 0 \]

### Step 4: Analyze the Result
Since \( \theta \neq k\pi \) implies \( \sin \theta \neq 0 \), we conclude:
\[ a_1 \sin \alpha_1 + a_2 \sin \alpha_2 + \ldots + a_n \sin \alpha_n = 0 \]

### Step 5: Evaluate the Expression for \( \lambda \)
Now we need to evaluate:
\[ a_1 \cos(\alpha_1 + \lambda) + a_2 \cos(\alpha_2 + \lambda) + \ldots + a_n \cos(\alpha_n + \lambda) \]
Using the cosine addition formula again:
\[ = a_1 (\cos \alpha_1 \cos \lambda - \sin \alpha_1 \sin \lambda) + a_2 (\cos \alpha_2 \cos \lambda - \sin \alpha_2 \sin \lambda) + \ldots + a_n (\cos \alpha_n \cos \lambda - \sin \alpha_n \sin \lambda) \]

### Step 6: Factor Out Common Terms Again
Factoring out the common terms gives:
\[ \cos \lambda (a_1 \cos \alpha_1 + a_2 \cos \alpha_2 + \ldots + a_n \cos \alpha_n) - \sin \lambda (a_1 \sin \alpha_1 + a_2 \sin \alpha_2 + \ldots + a_n \sin \alpha_n) \]
Substituting the known results:
\[ = \cos \lambda \cdot 0 - \sin \lambda \cdot 0 = 0 \]

### Conclusion
Thus, the value of \( a_1 \cos(\alpha_1 + \lambda) + a_2 \cos(\alpha_2 + \lambda) + \ldots + a_n \cos(\alpha_n + \lambda) \) is:
\[ \boxed{0} \]