Question

Let $\left|\begin{array}{ccc}3& 3x& 3x^2+2a^2\\ 3x& 3x^2+2a^2& 3x^3+6a^{2}x\\ 3x^2+2a^2& 3x^3+6a^{2}x& 3x^4+12a^2{x}^2+2a^4\end{array}\right|$
then:

• A. $\Delta {}^{\prime }\left(x\right)=0$
• B. $\Delta \left(x\right)$ is independent of $x$
• C. ${\int }_0^1\Delta \left(x\right)dx=4a^6$
• D. $y=\Delta \left(x\right)$ is a straight line

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A $\Delta {}^{\prime }\left(x\right)=0$
B ${\int }_0^1\Delta \left(x\right)dx=4a^6$
C $\Delta \left(x\right)$ is independent of $x$
D $y=\Delta \left(x\right)$ is a straight line

Applying $C_3\to C_3-x{C}_2,C_2\to C_2-x{C}_1$
we obtain
$\Delta \left(x\right)=\left|\begin{array}{ccc}3& 0& 2a^2\\ 3x& 2a^2& 4a^{2}x\\ 3x^2+2a^2& 4a^{2}x& 6a^2{x}^2+2a^4\end{array}\right|$
$\Delta \left(x\right)=4a^4\left|\begin{array}{ccc}3& 0& 1\\ 3x& 1& 2x\\ 3x^2+2a^2& 2x& 3x^2+a^2\end{array}\right|=4a^4\times a^2=4a^6$ ... .is independent of $x$.
${\Delta }^{\prime }\left(x\right)=0$

${\int }_0^1\Delta \left(x\right)={\int }_0^{1}4a^6=[4a^{6}x{]}_0^1=4a^6$

$y=4a^6$ is a straight line parallel to $x$ axis.

This is independent of $x$.