The number of integral value(s) of 'p' for which the equation `99 cos 2theta - 20 sin 2theta = 20p + 35`, will have a solution is

Options

A. 8
B. 9
C. 10
D. 11

Answer

To solve the equation \( 99 \cos 2\theta - 20 \sin 2\theta = 20p + 35 \) for the number of integral values of \( p \), we can follow these steps:

### Step 1: Determine the range of \( 99 \cos 2\theta - 20 \sin 2\theta \)

We can express the left-hand side in the form \( a \sin x + b \cos x \) to find its maximum and minimum values. Here, \( a = -20 \) and \( b = 99 \).

The maximum and minimum values of \( a \sin x + b \cos x \) can be found using the formula:
\[ \text{Maximum} = \sqrt{a^2 + b^2}, \quad \text{Minimum} = -\sqrt{a^2 + b^2} \]

Calculating \( a^2 + b^2 \):
\[ a^2 + b^2 = (-20)^2 + (99)^2 = 400 + 9801 = 10201 \]
Thus,
\[ \text{Maximum} = \sqrt{10201} = 101, \quad \text{Minimum} = -\sqrt{10201} = -101 \]

### Step 2: Set up the inequality

From the range we found, we have:
\[ -101 \leq 99 \cos 2\theta - 20 \sin 2\theta \leq 101 \]

### Step 3: Substitute into the equation

Now substituting \( 20p + 35 \) into the inequality:
\[ -101 \leq 20p + 35 \leq 101 \]

### Step 4: Solve the inequalities

**First inequality:**
\[ -101 \leq 20p + 35 \]
Subtracting 35 from both sides:
\[ -101 - 35 \leq 20p \implies -136 \leq 20p \]
Dividing by 20:
\[ -\frac{136}{20} \leq p \implies -6.8 \leq p \]

**Second inequality:**
\[ 20p + 35 \leq 101 \]
Subtracting 35 from both sides:
\[ 20p \leq 101 - 35 \implies 20p \leq 66 \]
Dividing by 20:
\[ p \leq \frac{66}{20} \implies p \leq 3.3 \]

### Step 5: Combine the results

Now we have:
\[ -6.8 \leq p \leq 3.3 \]

### Step 6: Determine the integral values of \( p \)

The integral values of \( p \) that satisfy this inequality are:
\[ p = -6, -5, -4, -3, -2, -1, 0, 1, 2, 3 \]
Counting these values, we find there are 10 integral values.

### Conclusion

The number of integral values of \( p \) for which the equation has a solution is **10**.

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