Let the vectors `vec(PQ),vec(QR),vec(RS), vec(ST), vec(TU)` and `vec(UP)` represent the sides of a regular hexagon.
Statement I:`vec(PQ) xx (vec(RS) + vec(ST)) ne vec0`
Statement II: `vec(PQ) xx vec(RS) = vec0` and `vec(PQ) xx vec(RS) = vec0` and `vec(PQ) xx vec(ST) ne vec0`
For the following question, choose the correct answer from the codes (A), (B) , (C) and (D) defined as follows:

Options

A. Statement I is true , statement II is also true, Statement II is the correct explanation of statement I.
B. statement I is true, statement II is also true, statement II is not the corret explanation of statement I.
C. Statement I is false, statement II is false.
D. Statement I is false, statement II is true.

Answer

To solve the problem, we need to analyze the two statements regarding the vectors of a regular hexagon.

### Step 1: Understand the structure of a regular hexagon
A regular hexagon has six equal sides and can be represented by the vertices \( P, Q, R, S, T, U \). The vectors representing the sides are:
- \( \vec{PQ} \)
- \( \vec{QR} \)
- \( \vec{RS} \)
- \( \vec{ST} \)
- \( \vec{TU} \)
- \( \vec{UP} \)

### Step 2: Analyze Statement I
**Statement I:** \( \vec{PQ} \times (\vec{RS} + \vec{ST}) \neq \vec{0} \)

To evaluate this statement, we need to find \( \vec{RS} + \vec{ST} \):
- By the triangle rule, \( \vec{RS} + \vec{ST} = \vec{RT} \).

Now, we check if \( \vec{PQ} \) is parallel to \( \vec{RT} \):
- In a regular hexagon, \( \vec{PQ} \) and \( \vec{RT} \) are not parallel because they are not along the same line.
- Therefore, \( \vec{PQ} \times \vec{RT} \neq \vec{0} \).

Thus, **Statement I is true**.

### Step 3: Analyze Statement II
**Statement II:** \( \vec{PQ} \times \vec{RS} = \vec{0} \) and \( \vec{PQ} \times \vec{ST} \neq \vec{0} \)

1. **Check \( \vec{PQ} \times \vec{RS} \)**:
- Since \( \vec{PQ} \) and \( \vec{RS} \) are not parallel (they are adjacent sides of the hexagon), their cross product is not zero.
- Therefore, \( \vec{PQ} \times \vec{RS} \neq \vec{0} \).

2. **Check \( \vec{PQ} \times \vec{ST} \)**:
- \( \vec{PQ} \) and \( \vec{ST} \) are also not parallel (they are not adjacent sides), so their cross product is also not zero.
- Thus, \( \vec{PQ} \times \vec{ST} \neq \vec{0} \).

Since both conditions in Statement II are not satisfied, **Statement II is false**.

### Conclusion
- **Statement I is true**: \( \vec{PQ} \times (\vec{RS} + \vec{ST}) \neq \vec{0} \)
- **Statement II is false**: \( \vec{PQ} \times \vec{RS} \neq \vec{0} \) and \( \vec{PQ} \times \vec{ST} \neq \vec{0} \)

### Final Answer
The correct option is (C), as both statements are evaluated, and we found that Statement I is true and Statement II is false.