Let a ,b ,c be such that b + c ne 0 if left| {begin{array}{*{20}{c}}a&{a + 1}&{a

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3 and 4 .Determinants and Matrices

medium

Let $a ,b ,c $ be such that $b + c \ne 0$ if

$\left| {\begin{array}{{20}{c}}a&{a + 1}&{a - 1}\\{ - b}&{b + 1}&{b - 1}\\c&{c - 1}&{c + 1}\end{array}} \right| + \left| {\begin{array}{{20}{c}}{a + 1}&{b + 1}&{c - 1}\\{a - 1}&{b - 1}&{c + 1}\\{{{\left( { - 1} \right)}^{n + 2}} \cdot a}&{{{\left( { - 1} \right)}^{n + 1}} \cdot b}&{{{\left( { - 1} \right)}^n} \cdot c}\end{array}} \right| = 0$ then $n$ equals to

Zero

any even integer

any odd integer

any integer

(AIEEE-2009)

Solution

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

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

$=0$ if $n$ is an obb integer.

Standard 12

Mathematics

The system of linear equations $\lambda x+2 y+2 z=5$ ; $2 \lambda x+3 y+5 z=8$ ; $4 x+\lambda y+6 z=10$ has

hard

(JEE MAIN-2020)

If $A = \left| {\,\begin{array}{*{20}{c}}{\sin (\theta + \alpha )}&{\cos (\theta + \alpha )}&1\\{\sin (\theta + \beta )}&{\cos (\theta + \beta )}&1\\{\sin (\theta + \gamma )}&{\cos (\theta + \gamma )}&1\end{array}\,} \right|$ ,then

medium

If $a$, $b$, $c$, $d$, $e$, $f$ are in $G.P$., then the value of $\left| {\begin{array}{*{20}{c}}
{{a^2}}&{{d^2}}&x \\
{{b^2}}&{{e^2}}&y \\
{{c^2}}&{{f^2}}&z
\end{array}} \right|$ depends on

normal

The roots of the equation $\left| {\,\begin{array}{*{20}{c}}x&0&8\\4&1&3\\2&0&x\end{array}\,} \right| = 0$ are equal to

easy

If $C = 2\cos \theta $, then the value of the determinant $\Delta = \left| {\,\begin{array}{*{20}{c}}C&1&0\\1&C&1\\6&1&C\end{array}\,} \right|$ is

medium

Solution

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If $a$, $b$, $c$, $d$, $e$, $f$ are in $G.P$., then the value of $\left| {\begin{array}{*{20}{c}}
{{a^2}}&{{d^2}}&x \\
{{b^2}}&{{e^2}}&y \\
{{c^2}}&{{f^2}}&z
\end{array}} \right|$ depends on