If `13 cos theta=12k-5` where `0lethetale90^(@)` and k is an integer then number of possible value of k is

Options

A. `0`
B. `1`
C. `2`
D. more than `2`

Answer

To solve the equation \( 13 \cos \theta = 12k - 5 \) with the constraints \( 0 \leq \theta \leq 90^\circ \) and \( k \) being an integer, we will follow these steps:

### Step 1: Understand the range of \( \cos \theta \)
Since \( \theta \) is between \( 0 \) and \( 90^\circ \), the value of \( \cos \theta \) will range from \( 1 \) (when \( \theta = 0^\circ \)) to \( 0 \) (when \( \theta = 90^\circ \)). Therefore, we have:
\[ 0 \leq \cos \theta \leq 1 \]

### Step 2: Rewrite the equation
We can rewrite the equation \( 13 \cos \theta = 12k - 5 \) as:
\[ \cos \theta = \frac{12k - 5}{13} \]

### Step 3: Set up inequalities based on the range of \( \cos \theta \)
Since \( \cos \theta \) must be between \( 0 \) and \( 1 \), we can set up the following inequalities:
1. \( 0 \leq \frac{12k - 5}{13} \)
2. \( \frac{12k - 5}{13} \leq 1 \)

### Step 4: Solve the first inequality
From the first inequality:
\[ 0 \leq 12k - 5 \]
Adding \( 5 \) to both sides gives:
\[ 12k \geq 5 \]
Dividing by \( 12 \):
\[ k \geq \frac{5}{12} \]
Since \( k \) is an integer, the smallest integer satisfying this inequality is \( k \geq 1 \).

### Step 5: Solve the second inequality
From the second inequality:
\[ 12k - 5 \leq 13 \]
Adding \( 5 \) to both sides gives:
\[ 12k \leq 18 \]
Dividing by \( 12 \):
\[ k \leq \frac{18}{12} = 1.5 \]
Since \( k \) is an integer, the largest integer satisfying this inequality is \( k \leq 1 \).

### Step 6: Combine the results
From the inequalities, we have:
\[ 1 \leq k \leq 1 \]
This means the only possible integer value for \( k \) is \( k = 1 \).

### Conclusion
Thus, the number of possible values of \( k \) is **1**.