The number of integral values of `k` for which the equation `7cos x +5 sinx=2k+1` has a solution is (1) `4` (2) `8` (3) `10` (4) `12`

Options

A. 4
B. 8
C. 10
D. 112

Explanation

To find the number of integral values of \( k \) for which the equation \( 7\cos x + 5\sin x = 2k + 1 \) has a solution, we can follow these steps:

### Step 1: Determine the Range of \( 7\cos x + 5\sin x \)

We know that any expression of the form \( a\cos x + b\sin x \) can be expressed in terms of its maximum and minimum values. The maximum value is given by \( \sqrt{a^2 + b^2} \) and the minimum value is \( -\sqrt{a^2 + b^2} \).

Here, \( a = 7 \) and \( b = 5 \).

\[ \text{Maximum value} = \sqrt{7^2 + 5^2} = \sqrt{49 + 25} = \sqrt{74} \]
\[ \text{Minimum value} = -\sqrt{74} \]

### Step 2: Set Up the Inequality

The expression \( 7\cos x + 5\sin x \) must satisfy the inequality:

\[ -\sqrt{74} \leq 2k + 1 \leq \sqrt{74} \]

### Step 3: Solve the Inequalities

1. **For the upper bound**:
\[ 2k + 1 \leq \sqrt{74} \]
Subtracting 1 from both sides:
\[ 2k \leq \sqrt{74} - 1 \]
Dividing by 2:
\[ k \leq \frac{\sqrt{74} - 1}{2} \]

2. **For the lower bound**:
\[ -\sqrt{74} \leq 2k + 1 \]
Subtracting 1 from both sides:
\[ -\sqrt{74} - 1 \leq 2k \]
Dividing by 2:
\[ k \geq \frac{-\sqrt{74} - 1}{2} \]

### Step 4: Calculate the Numerical Values

Now we need to compute the numerical values of \( \frac{\sqrt{74} - 1}{2} \) and \( \frac{-\sqrt{74} - 1}{2} \).

Calculating \( \sqrt{74} \):
\[ \sqrt{74} \approx 8.6 \]

1. **Upper bound**:
\[ k \leq \frac{8.6 - 1}{2} = \frac{7.6}{2} = 3.8 \]

2. **Lower bound**:
\[ k \geq \frac{-8.6 - 1}{2} = \frac{-9.6}{2} = -4.8 \]

### Step 5: Determine the Integral Values of \( k \)

From the inequalities:
\[ -4.8 \leq k \leq 3.8 \]

The integral values of \( k \) that satisfy this inequality are:
\[ k = -4, -3, -2, -1, 0, 1, 2, 3 \]

### Step 6: Count the Integral Values

The integral values of \( k \) are:
-4, -3, -2, -1, 0, 1, 2, 3 (total of 8 values).

### Conclusion

Thus, the number of integral values of \( k \) for which the equation \( 7\cos x + 5\sin x = 2k + 1 \) has a solution is **8**.