Write the number of solutions of the equation`2sinx-3cosx=7.`

Options

A. Two
B. Infinite
C. Three
D. No solution

Answer

To find the number of solutions for the equation \(2\sin x - 3\cos x = 7\), we can follow these steps:

### Step-by-Step Solution

1. **Rearrange the Equation**:
Start by isolating \(2\sin x\):
\[ 2\sin x = 3\cos x + 7 \]

2. **Determine the Range of \(2\sin x\)**:
The sine function, \(\sin x\), has a range of \([-1, 1]\). Therefore, multiplying by 2 gives:
\[ 2\sin x \text{ has a range of } [-2, 2]. \]

3. **Determine the Range of \(3\cos x + 7\)**:
The cosine function, \(\cos x\), also has a range of \([-1, 1]\). Thus, multiplying by 3 gives:
\[ 3\cos x \text{ has a range of } [-3, 3]. \]
Adding 7 to this range results in:
\[ 3\cos x + 7 \text{ has a range of } [4, 10]. \]

4. **Compare the Ranges**:
Now we compare the ranges of both sides:
- The left-hand side (LHS) \(2\sin x\) ranges from \([-2, 2]\).
- The right-hand side (RHS) \(3\cos x + 7\) ranges from \([4, 10]\).

5. **Conclusion**:
Since there is no overlap between the ranges of the LHS and RHS, we conclude that there are no values of \(x\) for which \(2\sin x - 3\cos x = 7\) holds true. Therefore, the equation has:
\[ \text{No solutions.} \]