Question

If ${\Delta }_1=\left|\begin{array}{cc}{e}^x& 1\\ e^x& 3\end{array}\right|$ and ${\Delta }_2=\left|\begin{array}{cc}{x}^2& -1\\ 2x& 1\end{array}\right|$ are given determinants, then which of the following is correct

• A. $\frac{d{\Delta }_1}{dx}=2e^x$
• B. $\frac{d{\Delta }_2}{dx}=x^2+2$
• C. $\frac{d{\Delta }_2}{dx}=2x+2$
• D. $\frac{d{\Delta }_1}{dx}=\frac{d{\Delta }_2}{dx}$ has no solution

Answer 1

Answer: (A), (C)

$\frac{d{\Delta }_2}{dx}=2x+2$

${\Delta }_1=3e^x-e^x=2e^x$
$\therefore \frac{d{\Delta }_1}{dx}=2e^x$

${\Delta }_2=x^2-(-2x)=x^2+2x$
$\therefore \frac{d{\Delta }_2}{dx}=2x+2$

For $\frac{d{\Delta }_1}{dx}=\frac{d{\Delta }_2}{dx}$
$\Rightarrow 2e^x=2x+2\Rightarrow e^x=x+1$
$\therefore x=0$ is solution of equation.