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| = $

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

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

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

None of these

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

If $a^2 + b^2 + c^2 = – 2$ and $f (x) = $ $\left| {\,\begin{array}{*{20}{c}}{1 + {a^2}x}&{(1 + {b^2})x}&{(1 + {c^2})x}\\{(1 + {a^2})x}&{1 + {b^2}x}&{(1 + {c^2})x}\\ {(1 + {a^2})x}&{(1 + {b^2})x}&{1 + {c^2}x}\end{array}\,} \right|$ then $f (x)$ is a polynomial of degree

normal

Solve the equations $\left|\begin{array}{ccc}x+a & x & x \\ x & x+a & x \\ x & x & x+a\end{array}\right|=0, a \neq 0$

medium

Let the numbers $2, b, c$ be in an $A.P$ and $A = \left[ {\begin{array}{*{20}{c}}
1&1&1 \\
2&b&c \\
4&{{b^2}}&{{c^2}}
\end{array}} \right]$. If $det(A) \in [2,16]$ then $c$ lies in the interval

hard

(JEE MAIN-2019)

Suppose $a_1, a_2, …….$ real numbers, with $a_1 \ne 0$. If $a_1, a_2, a_3, ……….$ are in $A.P$. then

normal

If $p, q, r, s$ are in $A.P.$ and $f (x) =$ $\left| {\,\begin{array}{*{20}{c}} {p\,\, + \,\,\sin \,x}&{q\,\, + \,\,\sin \,x}&{p\,\, – \,\,r\,\, + \,\,\sin \,x}\\ {q\,\, + \,\,\sin \,x}&{r\,\, + \,\,\sin \,x}&{ – \,1\,\, + \,\,\sin \,x}\\ {r\,\, + \,\,\sin \,x}&{s\,\, + \,\,\sin \,x}&{s\,\, – \,\,q\,\, + \,\,\sin \,x} \end{array}\,} \right|$ such that $f (x)dx = – 4$ then the common difference of the $A.P.$ can be :

normal

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

Solution

Similar Questions

If $a^2 + b^2 + c^2 = – 2$ and $f (x) = $ $\left| {\,\begin{array}{*{20}{c}}{1 + {a^2}x}&{(1 + {b^2})x}&{(1 + {c^2})x}\\{(1 + {a^2})x}&{1 + {b^2}x}&{(1 + {c^2})x}\\ {(1 + {a^2})x}&{(1 + {b^2})x}&{1 + {c^2}x}\end{array}\,} \right|$ then $f (x)$ is a polynomial of degree

Solve the equations $\left|\begin{array}{ccc}x+a & x & x \\ x & x+a & x \\ x & x & x+a\end{array}\right|=0, a \neq 0$

Let the numbers $2, b, c$ be in an $A.P$ and $A = \left[ {\begin{array}{*{20}{c}} 1&1&1 \\ 2&b&c \\ 4&{{b^2}}&{{c^2}} \end{array}} \right]$. If $det(A) \in [2,16]$ then $c$ lies in the interval

Suppose $a_1, a_2, …….$ real numbers, with $a_1 \ne 0$. If $a_1, a_2, a_3, ……….$ are in $A.P$. then

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Let the numbers $2, b, c$ be in an $A.P$ and $A = \left[ {\begin{array}{*{20}{c}}
1&1&1 \\
2&b&c \\
4&{{b^2}}&{{c^2}}
\end{array}} \right]$. If $det(A) \in [2,16]$ then $c$ lies in the interval