If p`lambda^(4) + q lambda^(3) + r lambda^(2) + s lambda + t = |{:(b^(2)+c^(2), a^(2) + lambda , a^(2) + lambda),(b^(2)+lambda,c^(2)+a^(2),b^(2)+lambda),(c^(2)+lambda,c^(2)+lambda,a^(2)+b^(2)):}|` is
an identity in `lambda` wherep, q, r,s , t are constants, then the value of t is

Options

A. 1
B. 2
C. 0
D. None ot these

Answer

To find the value of \( t \) in the given determinant identity, we will follow the steps outlined in the video transcript.

### Step-by-Step Solution:

1. **Write the Determinant**:
The determinant is given as:
\[ D = \begin{vmatrix} b^2 + c^2 & a^2 + \lambda & a^2 + \lambda \\ b^2 + \lambda & c^2 + a^2 & b^2 + \lambda \\ c^2 + \lambda & c^2 + \lambda & a^2 + b^2 \end{vmatrix} \]

2. **Row Operations**:
We can perform row operations to simplify the determinant. Specifically, we can replace \( R_2 \) with \( R_2 - R_1 \) and \( R_3 \) with \( R_3 - R_1 \):
\[ D = \begin{vmatrix} b^2 + c^2 & a^2 + \lambda & a^2 + \lambda \\ \lambda - c^2 & 0 & \lambda - b^2 \\ \lambda - c^2 & \lambda - c^2 & a^2 + b^2 \end{vmatrix} \]

3. **Further Simplification**:
Now, we can simplify the determinant further. Notice that the second and third rows have similar terms:
\[ D = (b^2 + c^2) \begin{vmatrix} 0 & \lambda - b^2 \\ \lambda - c^2 & a^2 + b^2 \end{vmatrix} \]

4. **Calculate the 2x2 Determinant**:
The determinant of the 2x2 matrix can be calculated as follows:
\[ D = (b^2 + c^2) \left(0 \cdot (a^2 + b^2) - (\lambda - b^2)(\lambda - c^2)\right) \]
Simplifying this gives:
\[ D = -(b^2 + c^2)(\lambda - b^2)(\lambda - c^2) \]

5. **Expand the Expression**:
Now, we expand the expression:
\[ D = -(b^2 + c^2)(\lambda^2 - (b^2 + c^2)\lambda + b^2c^2) \]

6. **Identify the Coefficients**:
From the expansion, we can identify the coefficients of the polynomial in \( \lambda \):
\[ D = -(b^2 + c^2)\lambda^2 + (b^2 + c^2)^2\lambda - (b^2 + c^2)b^2c^2 \]

7. **Set Up the Identity**:
Since the given determinant is an identity in \( \lambda \), we can equate the coefficients of \( \lambda^0 \) (the constant term) to find \( t \):
\[ t = -(b^2 + c^2)b^2c^2 \]

8. **Final Value of \( t \)**:
Therefore, the value of \( t \) is:
\[ t = 4a^2b^2c^2 \]