If p=`tan(-(11pi)/(6)).q=tan((21pi)/(4))` and r=cot`((383pi)/(6))`, then which of the following is/are correct?
1.The value of `pxxr` is 2
2.p.q and r are in G.P
Select the correct answer using the code given below:

Options

A. Only 1
B. Only 2
C. Both 1 and 2
D. Neither 1 nor 2

Answer

To solve the problem, we need to find the values of \( p \), \( q \), and \( r \) based on the given expressions and check the conditions provided in the question.

### Step 1: Calculate \( p = \tan\left(-\frac{11\pi}{6}\right) \)

Using the property of tangent, we have:
\[ \tan(-\theta) = -\tan(\theta) \]
Thus,
\[ p = \tan\left(-\frac{11\pi}{6}\right) = -\tan\left(\frac{11\pi}{6}\right) \]

Next, we can express \( \frac{11\pi}{6} \) in terms of a reference angle:
\[ \frac{11\pi}{6} = 2\pi - \frac{\pi}{6} \]
Since \( \frac{11\pi}{6} \) is in the fourth quadrant, where tangent is negative:
\[ \tan\left(\frac{11\pi}{6}\right) = -\tan\left(\frac{\pi}{6}\right) = -\frac{1}{\sqrt{3}} \]
Therefore,
\[ p = -\left(-\frac{1}{\sqrt{3}}\right) = \frac{1}{\sqrt{3}} \]

### Step 2: Calculate \( q = \tan\left(\frac{21\pi}{4}\right) \)

First, simplify \( \frac{21\pi}{4} \):
\[ \frac{21\pi}{4} = 5\pi + \frac{\pi}{4} = 2\pi + 3\pi + \frac{\pi}{4} \]
Since \( \tan \) has a period of \( \pi \):
\[ \tan\left(\frac{21\pi}{4}\right) = \tan\left(\frac{\pi}{4}\right) = 1 \]
Thus,
\[ q = 1 \]

### Step 3: Calculate \( r = \cot\left(\frac{383\pi}{6}\right) \)

First, simplify \( \frac{383\pi}{6} \):
\[ \frac{383\pi}{6} = 63\pi + \frac{5\pi}{6} \quad (\text{since } 63 = 10 \times 6 + 3) \]
Using the periodicity of \( \cot \):
\[ \cot\left(\frac{383\pi}{6}\right) = \cot\left(\frac{5\pi}{6}\right) \]
In the second quadrant, where cotangent is negative:
\[ \cot\left(\frac{5\pi}{6}\right) = -\cot\left(\frac{\pi}{6}\right) = -\sqrt{3} \]
Thus,
\[ r = -\sqrt{3} \]

### Step 4: Check the conditions

1. **Calculate \( p \times r \)**:
\[ p \times r = \left(\frac{1}{\sqrt{3}}\right) \times (-\sqrt{3}) = -1 \]
Since \( -1 \neq 2 \), the first statement is **false**.

2. **Check if \( p, q, r \) are in Geometric Progression (G.P.)**:
For \( p, q, r \) to be in G.P., the condition \( q^2 = p \cdot r \) must hold:
\[ q^2 = 1^2 = 1 \]
Calculate \( p \cdot r \):
\[ p \cdot r = \left(\frac{1}{\sqrt{3}}\right) \cdot (-\sqrt{3}) = -1 \]
Since \( 1 \neq -1 \), they are **not** in G.P. Thus, the second statement is also **false**.

### Final Answer
Both statements are false.