If `2 cos theta + sin theta =1 then `7 cos theta + 6 sin theta` is equal to

Options

A. 1 or 2
B. 2 or 3
C. 2 or 4
D. 2 or 6

Answer

To solve the problem, we need to find the value of \( 7 \cos \theta + 6 \sin \theta \) given that \( 2 \cos \theta + \sin \theta = 1 \).

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ 2 \cos \theta + \sin \theta = 1 \]

2. **Rearrange the equation:**
\[ 2 \cos \theta = 1 - \sin \theta \]

3. **Square both sides:**
\[ (2 \cos \theta)^2 = (1 - \sin \theta)^2 \]
This gives:
\[ 4 \cos^2 \theta = 1 - 2 \sin \theta + \sin^2 \theta \]

4. **Use the Pythagorean identity \( \cos^2 \theta = 1 - \sin^2 \theta \):**
\[ 4(1 - \sin^2 \theta) = 1 - 2 \sin \theta + \sin^2 \theta \]
Expanding this gives:
\[ 4 - 4 \sin^2 \theta = 1 - 2 \sin \theta + \sin^2 \theta \]

5. **Rearranging the equation:**
\[ 4 - 1 = 4 \sin^2 \theta + \sin^2 \theta - 2 \sin \theta \]
This simplifies to:
\[ 3 = 5 \sin^2 \theta - 2 \sin \theta \]
Rearranging gives:
\[ 5 \sin^2 \theta - 2 \sin \theta - 3 = 0 \]

6. **Factor the quadratic equation:**
\[ (5 \sin \theta + 3)(\sin \theta - 1) = 0 \]
This gives us two possible solutions:
\[ \sin \theta = 1 \quad \text{or} \quad 5 \sin \theta + 3 = 0 \Rightarrow \sin \theta = -\frac{3}{5} \]

7. **Find \( \cos \theta \) for both cases:**
- For \( \sin \theta = 1 \):
\[ \cos \theta = 0 \]
- For \( \sin \theta = -\frac{3}{5} \):
\[ \sin^2 \theta + \cos^2 \theta = 1 \Rightarrow \left(-\frac{3}{5}\right)^2 + \cos^2 \theta = 1 \]
\[ \frac{9}{25} + \cos^2 \theta = 1 \Rightarrow \cos^2 \theta = 1 - \frac{9}{25} = \frac{16}{25} \Rightarrow \cos \theta = \frac{4}{5} \text{ or } -\frac{4}{5} \]

8. **Calculate \( 7 \cos \theta + 6 \sin \theta \) for both cases:**
- For \( \sin \theta = 1 \) and \( \cos \theta = 0 \):
\[ 7 \cos \theta + 6 \sin \theta = 7(0) + 6(1) = 6 \]
- For \( \sin \theta = -\frac{3}{5} \) and \( \cos \theta = \frac{4}{5} \):
\[ 7 \cos \theta + 6 \sin \theta = 7\left(\frac{4}{5}\right) + 6\left(-\frac{3}{5}\right) = \frac{28}{5} - \frac{18}{5} = \frac{10}{5} = 2 \]

### Final Answer:
Thus, the values of \( 7 \cos \theta + 6 \sin \theta \) can be \( 6 \) or \( 2 \). The answer is \( 2 \).