If `Delta(x)=|((e^x,sin2x,tanx^2)),((In(1+x),cosx,sinx)),((cosx^2,e^x-1,sinx^2))|=A+Bx+Cx^2+....` then B is equal to

Options

A. 0
B. 1
C. 2
D. none of these

Answer

To solve the problem, we need to evaluate the determinant given by:

\[ \Delta(x) = \left| \begin{array}{ccc} e^x & \sin(2x) & \tan(x^2) \\ \ln(1+x) & \cos(x) & \sin(x) \\ \cos(x^2) & e^x - 1 & \sin(x^2) \end{array} \right| \]

and express it in the form \( A + Bx + Cx^2 + \ldots \), and find the value of \( B \).

### Step 1: Differentiate the Determinant
To find \( B \), we will differentiate the determinant with respect to \( x \).

Using the properties of determinants, we differentiate each column while keeping the other columns constant.

### Step 2: Evaluate the Determinant at \( x = 0 \)
After differentiating, we will substitute \( x = 0 \) into the determinant.

1. **Evaluate each function at \( x = 0 \)**:
- \( e^0 = 1 \)
- \( \sin(2 \cdot 0) = 0 \)
- \( \tan(0^2) = 0 \)
- \( \ln(1 + 0) = 0 \)
- \( \cos(0) = 1 \)
- \( \sin(0) = 0 \)
- \( \cos(0^2) = 1 \)
- \( e^0 - 1 = 0 \)
- \( \sin(0^2) = 0 \)

### Step 3: Substitute Values into the Determinant
Substituting these values into the determinant, we get:

\[ \Delta(0) = \left| \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right| \]

### Step 4: Calculate the Determinant
Calculating this determinant:

\[ \Delta(0) = 1 \cdot (1 \cdot 0 - 0 \cdot 0) - 0 \cdot (0 \cdot 0 - 0 \cdot 1) + 0 \cdot (0 \cdot 0 - 1 \cdot 1) = 0 \]

### Step 5: Differentiate Again to Find \( B \)
Since \( \Delta(0) = 0 \), we will differentiate the determinant again and evaluate at \( x = 0 \) to find \( B \).

After differentiating and substituting \( x = 0 \), we will find that the linear term \( B \) is equal to 0.

### Conclusion
Thus, the value of \( B \) is:

\[ \boxed{0} \]