If a,b,c are non-coplanar vectors and `lamda` is a real number, then the vectors `a+2b+3c,lamdab+4c and (2lamda-1)c` are non-coplanar for

Options

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

Explanation

To determine the conditions under which the vectors \( \mathbf{a} + 2\mathbf{b} + 3\mathbf{c} \), \( \lambda \mathbf{b} + 4\mathbf{c} \), and \( (2\lambda - 1)\mathbf{c} \) are non-coplanar, we can use the determinant method. Vectors are non-coplanar if the determinant of their coefficients is non-zero.

### Step-by-step Solution:

1. **Identify the Vectors**:
We have three vectors:
- \( \mathbf{v_1} = \mathbf{a} + 2\mathbf{b} + 3\mathbf{c} \)
- \( \mathbf{v_2} = \lambda \mathbf{b} + 4\mathbf{c} \)
- \( \mathbf{v_3} = (2\lambda - 1)\mathbf{c} \)

2. **Write the Coefficients**:
The coefficients of \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) in these vectors can be arranged into a matrix:
\[ \begin{bmatrix} 1 & 2 & 3 \\ 0 & \lambda & 4 \\ 0 & 0 & 2\lambda - 1 \end{bmatrix} \]

3. **Calculate the Determinant**:
We need to compute the determinant of this matrix:
\[ \text{Det} = \begin{vmatrix} 1 & 2 & 3 \\ 0 & \lambda & 4 \\ 0 & 0 & 2\lambda - 1 \end{vmatrix} \]
Since the matrix is upper triangular, the determinant is the product of the diagonal elements:
\[ \text{Det} = 1 \cdot \lambda \cdot (2\lambda - 1) = \lambda(2\lambda - 1) \]

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

5. **Find the Values of \( \lambda \)**:
The determinant is zero when:
- \( \lambda = 0 \)
- \( 2\lambda - 1 = 0 \) which gives \( \lambda = \frac{1}{2} \)

6. **Conclusion**:
Therefore, the vectors are non-coplanar for all values of \( \lambda \) except \( \lambda = 0 \) and \( \lambda = \frac{1}{2} \).

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