Question

${\Delta }_1=\left|\begin{array}{c}x\\ a\\ a\end{array}\begin{array}{c}b\\ x\\ a\end{array}\begin{array}{c}b\\ b\\ x\end{array}\right|$ and ${\Delta }_2=\left|\begin{array}{cc}x& b\\ a& x\end{array}\right|$ are the given determinants, then

• A. ${\Delta }_1={3\left({\Delta }_2\right)}^2$
• B. $\frac{d}{dx}\left({\Delta }_1\right)={\Delta }_2$
• C. $\frac{d}{dx}\left({\Delta }_1\right)={3\left({\Delta }_2\right)}^2$
• D. ${\Delta }_1={3\left({\Delta }_2\right)}^{3/2}$

Answer 1

Answer: (B)

$\frac{d}{dx}\left({\Delta }_1\right)={\Delta }_2$

${△}_1=\left|\begin{array}{ccc}x& b& b\\ a& x& b\\ a& a& x\end{array}\right|{△}_2=\left[\begin{array}{cc}x& b\\ a& x\end{array}\right]$=$x^2-ab$

$\frac{d}{dx}\left({△}_1\right)=\left|\begin{array}{ccc}1& b& b\\ 0& x& b\\ 0& a& x\end{array}\right|$

$=x^2-ab$

$\frac{d}{dx}\left({△}_1\right)={△}_2$