If `veca, vecb, vecc` are non coplanar vectors and `lamda` is a real number, then the vectors `veca+2vecb+3vecc, lamdavecb+4vecc` and `(2lamda-1)vecc` are non coplanar for

Options

A. no value of `lamda`
B. all except one value of `lamda`
C. all except two values of `lamda`
D. all values of `lamda`

Explanation

To determine the values of \(\lambda\) for which the vectors \(\vec{a} + 2\vec{b} + 3\vec{c}\), \(\lambda\vec{b} + 4\vec{c}\), and \((2\lambda - 1)\vec{c}\) are non-coplanar, we can use the concept of the determinant of a matrix formed by these vectors. The vectors are non-coplanar if the determinant of the matrix formed by their coefficients is non-zero.

### Step-by-step Solution:

1. **Identify the Vectors**:
- Let \(\vec{x} = \vec{a} + 2\vec{b} + 3\vec{c}\)
- Let \(\vec{y} = \lambda\vec{b} + 4\vec{c}\)
- Let \(\vec{z} = (2\lambda - 1)\vec{c}\)

2. **Set Up the Determinant**:
We need to form a matrix with the coefficients of \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\):
\[ \begin{vmatrix} 1 & 2 & 3 \\ 0 & \lambda & 4 \\ 0 & 0 & 2\lambda - 1 \end{vmatrix} \]

3. **Calculate the Determinant**:
The determinant can be calculated as follows:
\[ D = 1 \cdot \begin{vmatrix} \lambda & 4 \\ 0 & 2\lambda - 1 \end{vmatrix} - 2 \cdot \begin{vmatrix} 0 & 4 \\ 0 & 2\lambda - 1 \end{vmatrix} + 3 \cdot \begin{vmatrix} 0 & \lambda \\ 0 & 0 \end{vmatrix} \]

The second and third determinants are zero since they have rows of zeros. Thus, we only need to compute the first determinant:
\[ D = 1 \cdot (\lambda(2\lambda - 1) - 0) = \lambda(2\lambda - 1) \]

4. **Set the Determinant Not Equal to Zero**:
For the vectors to be non-coplanar, we require:
\[ \lambda(2\lambda - 1) \neq 0 \]

5. **Solve the Inequality**:
This gives us two conditions:
- \(\lambda \neq 0\)
- \(2\lambda - 1 \neq 0 \Rightarrow \lambda \neq \frac{1}{2}\)

6. **Conclusion**:
The vectors \(\vec{a} + 2\vec{b} + 3\vec{c}\), \(\lambda\vec{b} + 4\vec{c}\), and \((2\lambda - 1)\vec{c}\) are non-coplanar for all values of \(\lambda\) except \(\lambda = 0\) and \(\lambda = \frac{1}{2}\).