Statement I In a triangle ABC if `tan A: tan B: tan C=1 :2:3, ` then `A=45^(@)`
Statement II If `p:q:r=1:2:3,` then `p=1`

Options

A. A. Both Statement I and Statement II are correct and Statement II is the correct explanation of Statement I
B. B. Both Statement I and Statement II are correct and Statement II is not the correct explanation of Statement I
C. C. Statement I is correct but Statement II is incorrect
D. D. Statement I is correct but Statement I is incorrect

Answer

To solve the problem, we will analyze both statements step by step.

### Step 1: Analyze Statement I
We are given that in triangle ABC, the ratio of the tangents of the angles is:
\[ \tan A : \tan B : \tan C = 1 : 2 : 3 \]

Let’s denote:
\[ \tan A = x, \quad \tan B = 2x, \quad \tan C = 3x \]

### Step 2: Use the Sum of Angles in a Triangle
We know that the sum of angles in a triangle is:
\[ A + B + C = 180^\circ \]

### Step 3: Apply the Tangent Addition Formula
Using the tangent addition formula:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
Since \(A + B = 180^\circ - C\), we have:
\[ \tan(A + B) = -\tan C \]

Substituting the values we have:
\[ \tan A + \tan B = x + 2x = 3x \]
\[ \tan A \tan B = x \cdot 2x = 2x^2 \]

Now substituting into the tangent addition formula:
\[ 3x = -\tan C = -3x \]

### Step 4: Set Up the Equation
From the tangent addition formula, we have:
\[ 3x = \frac{3x}{1 - 2x^2} \]

Cross-multiplying gives:
\[ 3x(1 - 2x^2) = 3x \]

### Step 5: Simplify the Equation
Assuming \(x \neq 0\) (since angles cannot be zero in a triangle):
\[ 3x - 6x^3 = 3x \]
This simplifies to:
\[ -6x^3 = 0 \]
Thus, \(x^3 = 0\) implies \(x = 0\) is not valid.

### Step 6: Find the Values of Angles
Now, we can find:
\[ \tan A + \tan B + \tan C = x + 2x + 3x = 6x \]
And:
\[ \tan A \tan B \tan C = x \cdot 2x \cdot 3x = 6x^3 \]

Using the identity:
\[ \tan A + \tan B + \tan C = \tan A \tan B \tan C \]
We have:
\[ 6x = 6x^3 \]

### Step 7: Solve for x
This gives:
\[ x^3 - x = 0 \]
Factoring out \(x\):
\[ x(x^2 - 1) = 0 \]
Thus, \(x = 0\) or \(x = 1\) or \(x = -1\).

### Step 8: Determine Valid Angles
If \(x = 1\):
\[ \tan A = 1 \implies A = 45^\circ \]
If \(x = 0\) or \(x = -1\), those are not valid for angles in a triangle.

Thus, **Statement I is true**.

### Step 9: Analyze Statement II
The second statement says:
If \(p : q : r = 1 : 2 : 3\), then \(p = 1\).

### Step 10: Set Up the Ratios
Let’s denote:
\[ p = x, \quad q = 2x, \quad r = 3x \]
If \(p = 1\), then \(x = 1\) which gives:
\[ q = 2 \quad \text{and} \quad r = 3 \]
However, if \(x = 2\), then:
\[ p = 2, \quad q = 4, \quad r = 6 \]
This shows that \(p\) can take multiple values depending on the value of \(x\).

Thus, **Statement II is false**.

### Conclusion
- **Statement I is true**: \(A = 45^\circ\)
- **Statement II is false**: \(p\) does not necessarily equal 1.

### Final Answer
The correct answer is that **Statement I is true and Statement II is false**.

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