MCQ
$\left| {\,\begin{array}{*{20}{c}}1&a&{{a^2}}\\1&b&{{b^2}}\\1&c&{{c^2}}\end{array}\,} \right| = $
- A${a^2} + {b^2} + {c^2}$
- B$(a + b)\,(b + c)\,(c + a)$
- ✓$(a - b)(b - c)(c - a)$
- DNone of these
= $(a - b)\,(b - c)\,\left| {\,\begin{array}{*{20}{c}}0&1&{a + b}\\0&1&{b + c}\\1&c&{{c^2}}\end{array}\,} \right|$
= $(a - b)\,\,(b - c)\,\left| {\,\begin{array}{*{20}{c}}0&0&{a - c}\\0&1&{b + c}\\1&c&{{c^2}}\end{array}\,} \right|$, by ${R_1} \to {R_1} - {R_2}$
= $(a - b)\,(b - c)\,(a - c)\,\left| {\,\begin{array}{*{20}{c}}0&0&1\\0&1&{b + c}\\1&c&{{c^2}}\end{array}\,} \right|$
= $(a - b)\,(b - c)\,(a - c)\,.\,( - 1) = (a - b)\,(b - c)\,(c - a)$.
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$f\left( x \right)\left\{ \begin{array}{l}
\frac{{2{x^2}}}{a}\,\,\,\,\,\,\,\,\,\,\,\,,\,\,\,\,\,0 \le x < 1\,\,\,\\
a\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,,\,\,\,\,\,1 \le x < \sqrt 2 \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\
\frac{{2{b^2} - 4b}}{{{x^3}}}\,\,\,,\,\,\,\,\,\sqrt 2 \le x < \infty
\end{array} \right.\,\,\,\,$
is continuous in the interval $\left[ {0,\infty } \right)$ , then an ordered pair $(a, b)$ is