Statement-1: `cos theta= x+1/x` is possible for some real value of `x`.
Statement:2: `sin1^(@) lt sin 1`.
Statement-3: The minimum value of `5cos x-12 sinx +13` is `0`

Options

A. TTT
B. FTT
C. FFT
D. FTF

Answer

To solve the problem, we need to analyze each statement one by one.

### Statement 1:
**Statement:** \( \cos \theta = x + \frac{1}{x} \) is possible for some real value of \( x \).

**Solution Steps:**

1. **Understanding the Expression:** We need to find the range of the expression \( x + \frac{1}{x} \) for real values of \( x \).

2. **Applying AM-GM Inequality:** By the Arithmetic Mean-Geometric Mean inequality (AM-GM), we know that:
\[ \frac{x + \frac{1}{x}}{2} \geq \sqrt{x \cdot \frac{1}{x}} = 1 \]
This implies:
\[ x + \frac{1}{x} \geq 2 \quad \text{for } x > 0 \]

3. **Considering Negative Values of \( x \):** If \( x < 0 \), we can rewrite \( x + \frac{1}{x} \) as:
\[ -\left(-x - \frac{1}{-x}\right) = -\left(-x + \frac{1}{-x}\right) \]
By applying AM-GM again, we find:
\[ -\left(-x + \frac{1}{-x}\right) \leq -2 \quad \text{for } x < 0 \]

4. **Conclusion on Range:** Therefore, the minimum value of \( x + \frac{1}{x} \) is \( 2 \) (for \( x > 0 \)) and the maximum value is \( -2 \) (for \( x < 0 \)). Thus, the expression \( x + \frac{1}{x} \) can take values outside the range of \( \cos \theta \), which is \([-1, 1]\).

5. **Final Verdict:** Since \( x + \frac{1}{x} \) cannot equal \( \cos \theta \) for any real \( x \), **Statement 1 is false.**

### Statement 2:
**Statement:** \( \sin 1^\circ < \sin 1 \).

**Solution Steps:**

1. **Understanding Degrees and Radians:** We need to convert \( 1^\circ \) to radians. We know:
\[ 1^\circ = \frac{\pi}{180} \text{ radians} \]

2. **Comparing Values:** We know that the sine function is increasing in the interval \( [0, \frac{\pi}{2}] \). Since \( 1^\circ \) (in radians) is approximately \( 0.01745 \) radians, and \( 1 \) radian is approximately \( 57.2958^\circ \), we have:
\[ 1^\circ < 1 \text{ radian} \]
Thus, \( \sin(1^\circ) < \sin(1) \).

3. **Final Verdict:** Therefore, **Statement 2 is true.**

### Statement 3:
**Statement:** The minimum value of \( 5\cos x - 12\sin x + 13 \) is \( 0 \).

**Solution Steps:**

1. **Identifying the Function:** We need to find the minimum value of the function:
\[ f(x) = 5\cos x - 12\sin x + 13 \]

2. **Using the Minimum Value Formula:** The minimum value of \( a\cos x + b\sin x + c \) can be calculated using the formula:
\[ \text{Minimum value} = c - \sqrt{a^2 + b^2} \]
Here, \( a = 5 \), \( b = -12 \), and \( c = 13 \).

3. **Calculating \( \sqrt{a^2 + b^2} \):**
\[ \sqrt{5^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \]

4. **Finding the Minimum Value:**
\[ \text{Minimum value} = 13 - 13 = 0 \]

5. **Final Verdict:** Therefore, **Statement 3 is true.**

### Summary of Statements:
- Statement 1: False
- Statement 2: True
- Statement 3: True

Thus, the correct answer is that Statement 1 is false, while Statements 2 and 3 are true.