The determinant
`Delta = |(b,c,b alpha +c),(c,d,c alpha + d),(b alpha + c,c alpha + d,a a^(3) - c alpha)|`
is equal to zero, if

Options

A. b, c, d are in A.P
B. b, c, d are in G.P
C. b, c, d are in H.P
D. `alpha` is a root of `ax^(3) - cx - d = 0`

Answer

To solve the given determinant problem, we need to analyze the determinant and find the conditions under which it equals zero. The determinant is given as:

\[ \Delta = \begin{vmatrix} b & c & b\alpha + c \\ c & d & c\alpha + d \\ b\alpha + c & c\alpha + d & a\alpha^3 - c\alpha \end{vmatrix} \]

### Step 1: Apply Column Operations
We will perform the column operation \( C_3 \to C_3 - \alpha C_1 - C_2 \). This operation will help simplify the determinant.

\[ C_3 = (b\alpha + c) - \alpha(b) - (c) = 0 \]

\[ C_3 = (c\alpha + d) - \alpha(c) - (d) = 0 \]

After performing these operations, the determinant simplifies to:

\[ \Delta = \begin{vmatrix} b & c & 0 \\ c & d & 0 \\ 0 & 0 & a\alpha^3 - 3c\alpha - b\alpha^2 - d \end{vmatrix} \]

### Step 2: Calculate the Determinant
The determinant can now be calculated as follows:

\[ \Delta = 0 \cdot \begin{vmatrix} c & 0 \\ 0 & a\alpha^3 - 3c\alpha - b\alpha^2 - d \end{vmatrix} - 0 + b \cdot \begin{vmatrix} c & 0 \\ 0 & a\alpha^3 - 3c\alpha - b\alpha^2 - d \end{vmatrix} \]

Since the first column has been reduced to zeros, we only need to consider the last term:

\[ \Delta = b \cdot (c(a\alpha^3 - 3c\alpha - b\alpha^2 - d)) \]

### Step 3: Set the Determinant to Zero
For the determinant to equal zero, we need:

\[ b \cdot c(a\alpha^3 - 3c\alpha - b\alpha^2 - d) = 0 \]

This means either \( b = 0 \), \( c = 0 \), or:

\[ a\alpha^3 - 3c\alpha - b\alpha^2 - d = 0 \]

### Step 4: Analyze the Conditions
1. If \( b = 0 \) or \( c = 0 \), the determinant will be zero.
2. The polynomial \( a\alpha^3 - 3c\alpha - b\alpha^2 - d = 0 \) gives us a cubic equation in \( \alpha \).

### Conclusion
The determinant \( \Delta \) is equal to zero if either \( b \) or \( c \) is zero, or \( \alpha \) is a root of the cubic equation \( a\alpha^3 - b\alpha^2 - 3c\alpha - d = 0 \).

Thus, the correct condition is that \( b, c, d \) are in geometric progression (GP) when \( bd = c^2 \).