Question

Coefficient of \(x\) in \(f(x) = \begin{vmatrix} x & (1 + \sin x)^3 & \cos x \\ 1 & \log(1 + x) & 2 \\ x^2 & (1 + x)^2 & 0 \end{vmatrix}\) is

• A. 0
• B. 1
• C. -2
• D. Cannot be determined

Hint:

Expand determinant and collect linear terms, or use derivative.

Answer 1

The Correct Option is C

Step 1: Formula / Definition}
\[ \text{Coefficient of } x \text{ in } f(x) = f'(0) \]
Step 2: Calculation / Simplification}
Expand functions around \(x=0\):
\(\sin x = x - \frac{x^3}{6} + \dots \Rightarrow (1+\sin x)^3 = 1 + 3x + \dots\)
\(\cos x = 1 - \frac{x^2}{2} + \dots\)
\(\log(1+x) = x - \frac{x^2}{2} + \dots\)
\((1+x)^2 = 1 + 2x + \dots\)
\(f(x) = \begin{vmatrix} x & 1+3x & 1 \\ 1 & x & 2 \\ x^2 & 1 & 0 \end{vmatrix} + \text{higher terms}\)
Coefficient of \(x\) = \(x(0-2) - 1(0-2x^2) + 1(1-x^3)\) evaluated at linear term
= \(-2x + \dots \Rightarrow\) coefficient is \(-2\)
Step 3: Final Answer
\[ -2 \]