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 ${R_1} \to {R_1} - {R_2}$
= $(a - b)\,(b - c)\,(a - c)\,\left| {\,\begin{array}{*{20}{c}}0&0&1\\0&1&{b + c}\\1&c&{{c^2}}\end{array}\,} \right|$
= $(a - b)\,(b - c)\,(a - c)\,.\,( - 1) = (a - b)\,(b - c)\,(c - a)$.
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
$2 x+y \leq 10, x+3 y \leq 15, x, y \geq 0$ are $(0,0),(5,0),(3,4)$ and $(0,5) .$ Let $Z =p x+q y,$ where $p, q\,>\,0 .$ Condition on $p$ and $q$ so that the maximum of $Z$ occurs at both $(3,4)$ and $(0,5)$ is $....$
$(A)$ There exist $r , s \in R$, where $r < s$, such that $f$ is one-one on the open interval $( r , s )$
$(B)$ There exists $x 0 \in(-4,0)$ such that $\left| f ^{\prime}\left( x _0\right)\right| \leq 1$
$(C)$ $\lim _{x \rightarrow \infty} f(x)=1$
$(D)$ There exists a $\in(-4,4)$ such that $f(a)+f^{\prime \prime}(a)=0$ and $f^{\prime}(a) \neq 0$