Question

If $\Delta(x)$ = $\left| \begin{matrix} x & 1 + x^{2} & x^{3} \\ \log\left( 1 + x^{2} \right) & e^{x} & \sin x \\ \cos x & \tan x & \sin^{2}x \end{matrix} \right|$, then

• A. $\Delta(x)$ is divisible by x
• B. $\Delta(x) = 0$
• C. $\Delta'(x) = 0$
• D. None of these.

Answer 1

Answer: (A)

$\Delta(x)$ is divisible by x

Answer 2

Let ∆ $(x) = A + Bx + Cx^{2} + Dx^{3} + .........$

∴ ∆ $(0) = \left| \begin{matrix} 0 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{matrix} \right| = 0 \Rightarrow A = 0$

∆' $(0) = \left| \begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix} \right| + \left| \begin{matrix} 0 & 1 & 0 \\ 0 & 1 & 1 \\ 1 & 0 & 0 \end{matrix} \right| + \left| \begin{matrix} 0 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{matrix} \right| = 1$

∴ B= 1

⇒ then ∆ $(x)$ = $x + Cx^{2} + Dx^{3} + .......$

∴ ∆ x is divisible by x.