Question

If $\Delta_{1} = \left| \begin{matrix} x & b & b \\ a & x & b \\ a & a & x \end{matrix} \right|$ and $\Delta_{2} = \left| \begin{matrix} x & b \\ a & x \end{matrix} \right|$are the given

determinants, then

• A. $\Delta_{1} = 3(\Delta_{2})^{2}$
• B. $\frac{d}{dx}(\Delta_{1}) = 3\Delta_{2}$
• C. $\frac{d}{dx}(\Delta_{1}) = 2(\Delta_{2})^{2}$
• D. $\Delta_{1} = 3\Delta_{2}^{3/2}$

Answer 1

Answer: (B)

$\frac{d}{dx}(\Delta_{1}) = 3\Delta_{2}$

Answer 2

$\Delta_{1} = \left| \begin{matrix} x & b & b \\ a & x & b \\ a & a & x \end{matrix} \right| = x^{3} - 3abx \Rightarrow \frac{d}{dx}(\Delta_{1}) = 3(x^{2} - ab)$ and

x & b \\ a & x \end{matrix} \right| = x^{2} - ab$$ $$\therefore\frac{d}{dx}(\Delta_{1}) = 3(x^{2} - ab) = 3\Delta_{2}$$