Left| {,begin{array}{*{20}{c}}1&a&{{a^2}}1&b&{{b^2}}1&c&{{c^2}}end{array},}

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3 and 4 .Determinants and Matrices

medium

$\left| {\,\begin{array}{*{20}{c}}1&a&{{a^2}}\\1&b&{{b^2}}\\1&c&{{c^2}}\end{array}\,} \right| = $

${a^2} + {b^2} + {c^2}$

$(a + b)\,(b + c)\,(c + a)$

$(a - b)(b - c)(c - a)$

इनमें से कोई नहीं

Solution

(c) $\left| {\,\begin{array}{*{20}{c}}1&a&{{a^2}}\\1&b&{{b^2}}\\1&c&{{c^2}}\end{array}\,} \right|\, = \left| {\,\begin{array}{*{20}{c}}0&{a – b}&{{a^2} – {b^2}}\\0&{b – c}&{{b^2} – {c^2}}\\1&c&{{c^2}}\end{array}\,} \right|,$ by $\begin{array}{l}{R_1} \to {R_1} – {R_2}\\{R_2} \to {R_2} – {R_3}\end{array}$

= $(a – b)\,(b – c)\,\left| {\,\begin{array}{*{20}{c}}0&1&{a + b}\\0&1&{b + c}\\1&c&{{c^2}}\end{array}\,} \right|$

= $(a – b)\,\,(b – c)\,\left| {\,\begin{array}{*{20}{c}}0&0&{a – c}\\0&1&{b + c}\\1&c&{{c^2}}\end{array}\,} \right|$, by ${R_1} \to {R_1} – {R_2}$

= $(a – b)\,(b – c)\,(a – c)\,\left| {\,\begin{array}{*{20}{c}}0&0&1\\0&1&{b + c}\\1&c&{{c^2}}\end{array}\,} \right|$

= $(a – b)\,(b – c)\,(a – c)\,.\,( – 1) = (a – b)\,(b – c)\,(c – a)$.

Standard 12

Mathematics

$\left| {\,\begin{array}{*{20}{c}}{{b^2} – ab}&{b – c}&{bc – ac}\\{ab – {a^2}}&{a – b}&{{b^2} – ab}\\{bc – ac}&{c – a}&{ab – {a^2}}\end{array}\,} \right| = $

easy

$\left| {\,\begin{array}{*{20}{c}}1&1&1\\{\cos (nx)}&{\cos (n + 1)x}&{\cos (n + 2)x}\\{\sin (nx)}&{\sin (n + 1)x}&{\sin (n + 2)x}\end{array}\,} \right|$ निर्भर नहीं करता है

medium

यदि $ab + bc + ca = 0$ और $\left| {\,\begin{array}{*{20}{c}}{a – x}&c&b\\c&{b – x}&a\\b&a&{c – x}\end{array}\,} \right| = 0$, तो $x$ का एक मान होगा

hard

$\left| {\,\begin{array}{*{20}{c}}1&1&1\\{bc}&{ca}&{ab}\\{b + c}&{c + a}&{a + b}\end{array}\,} \right|$

medium

यदि $a,b,c$ धनात्मक पूर्णांक हैं, तो सारणिक $\Delta = \left| {\,\begin{array}{*{20}{c}}{{a^2} + x}&{ab}&{ac}\\{ab}&{{b^2} + x}&{bc}\\{ac}&{bc}&{{c^2} + x}\end{array}\,} \right|$ विभाज्य है

medium

$\left| {\,\begin{array}{*{20}{c}}1&a&{{a^2}}\\1&b&{{b^2}}\\1&c&{{c^2}}\end{array}\,} \right| = $

Solution

Similar Questions

$\left| {\,\begin{array}{*{20}{c}}{{b^2} – ab}&{b – c}&{bc – ac}\\{ab – {a^2}}&{a – b}&{{b^2} – ab}\\{bc – ac}&{c – a}&{ab – {a^2}}\end{array}\,} \right| = $

$\left| {\,\begin{array}{*{20}{c}}1&1&1\\{\cos (nx)}&{\cos (n + 1)x}&{\cos (n + 2)x}\\{\sin (nx)}&{\sin (n + 1)x}&{\sin (n + 2)x}\end{array}\,} \right|$ निर्भर नहीं करता है

यदि $ab + bc + ca = 0$ और $\left| {\,\begin{array}{*{20}{c}}{a – x}&c&b\\c&{b – x}&a\\b&a&{c – x}\end{array}\,} \right| = 0$, तो $x$ का एक मान होगा

$\left| {\,\begin{array}{*{20}{c}}1&1&1\\{bc}&{ca}&{ab}\\{b + c}&{c + a}&{a + b}\end{array}\,} \right|$

यदि $a,b,c$ धनात्मक पूर्णांक हैं, तो सारणिक $\Delta = \left| {\,\begin{array}{*{20}{c}}{{a^2} + x}&{ab}&{ac}\\{ab}&{{b^2} + x}&{bc}\\{ac}&{bc}&{{c^2} + x}\end{array}\,} \right|$ विभाज्य है

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