3 statements are given below each of which is either True of false. Indicate the correct order sequence
`I (log_(3) 169) (log_(13) 243)=10" "II cos (cos pi)=cos (cos 0^(@))`
`III cos x + 1/(cos x)=3/2`

Options

A. TFF
B. TFT
C. TTT
D. TTF

Answer

To solve the problem, we need to evaluate the truth of three statements and determine their order based on whether they are true or false. Let's analyze each statement step by step.

### Statement I: \( \log_{3}(169) \cdot \log_{13}(243) = 10 \)

1. **Rewrite the logarithms**:
- \( 169 = 13^2 \)
- \( 243 = 3^5 \)

2. **Apply the logarithm property**:
- \( \log_{3}(169) = \log_{3}(13^2) = 2 \cdot \log_{3}(13) \)
- \( \log_{13}(243) = \log_{13}(3^5) = 5 \cdot \log_{13}(3) \)

3. **Substitute back into the equation**:
\[ \log_{3}(169) \cdot \log_{13}(243) = (2 \cdot \log_{3}(13)) \cdot (5 \cdot \log_{13}(3)) = 10 \cdot \log_{3}(13) \cdot \log_{13}(3) \]

4. **Use the change of base formula**:
\[ \log_{3}(13) \cdot \log_{13}(3) = 1 \]
Therefore,
\[ 10 \cdot \log_{3}(13) \cdot \log_{13}(3) = 10 \cdot 1 = 10 \]

5. **Conclusion**: The statement is **True**.

### Statement II: \( \cos(\cos \pi) = \cos(\cos 0) \)

1. **Evaluate \( \cos \pi \)**:
\[ \cos \pi = -1 \]

2. **Evaluate \( \cos 0 \)**:
\[ \cos 0 = 1 \]

3. **Substitute back into the equation**:
\[ \cos(-1) \quad \text{and} \quad \cos(1) \]

4. **Check if they are equal**:
- \( \cos(-1) \) is not equal to \( \cos(1) \) (since \( \cos \) is not equal at these points).

5. **Conclusion**: The statement is **False**.

### Statement III: \( \cos x + \frac{1}{\cos x} = \frac{3}{2} \)

1. **Let \( t = \cos x \)**, then rewrite the equation:
\[ t + \frac{1}{t} = \frac{3}{2} \]

2. **Multiply through by \( t \)** (assuming \( t \neq 0 \)):
\[ t^2 - \frac{3}{2}t + 1 = 0 \]

3. **Use the quadratic formula**:
\[ t = \frac{\frac{3}{2} \pm \sqrt{(\frac{3}{2})^2 - 4 \cdot 1}}{2} \]
\[ = \frac{\frac{3}{2} \pm \sqrt{\frac{9}{4} - 4}}{2} = \frac{\frac{3}{2} \pm \sqrt{\frac{9 - 16}{4}}}{2} = \frac{\frac{3}{2} \pm \sqrt{-\frac{7}{4}}}{2} \]

4. **Conclusion**: Since the discriminant is negative, there are no real solutions. Thus, the statement is **False**.

### Final Evaluation of Statements:
- Statement I: True
- Statement II: False
- Statement III: False

### Correct Order Sequence:
The correct order of the statements is **True, False, False** (TTF).