If a , b , c , d , e , f are in G.P ., then value of $left| {begin{array}{*{20}{c}} {{a^2}}&a

English

Gujarati

Hindi

3 and 4 .Determinants and Matrices

normal

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

$x, y$

$x, z$

$y, z$

None

Solution

$\mathrm{b}=\mathrm{ar}, \mathrm{c}=\mathrm{ar}^{2}, \mathrm{d}=\mathrm{ar}^{3}, \mathrm{e}=\mathrm{ar}^{4}, \mathrm{f}=\mathrm{ar}^{6}$

$\therefore\left|\begin{array}{ccc}{a^{2}} & {a^{2} r^{6}} & {x} \\ {a^{2} r^{2}} & {a^{2} r^{3}} & {y} \\ {a^{2} r^{6}} & {a^{2} r^{10}} & {z}\end{array}\right|$

$=a^{2} \times a^{2} r^{6}\left|\begin{array}{ccc}{1} & {1} & {x} \\ {r^{2}} & {r^{2}} & {y} \\ {r^{4}} & {r^{4}} & {z}\end{array}\right|=0$

Standard 12

Mathematics

The value of the determinant given below $\left| {{\rm{ }}\begin{array}{*{20}{c}}1&2&3\\3&5&7\\8&{14}&{20}\end{array}} \right|$ is

normal

$\left| {\begin{array}{*{20}{c}}0&a&{ – b}\\{ – a}&0&c\\b&{ – c}&0\end{array}} \right| = $

easy

The system of equations ${x_1} – {x_2} + {x_3} = 2,$ $\,3{x_1} – {x_2} + 2{x_3} = – 6$ and $3{x_1} + {x_2} + {x_3} = – 18$ has

hard

If the system of linear equations $x + y + z = 5$ ; $x = 2y + 2z = 6$ ; $x + 3y + \lambda z = u (\lambda \, \mu \in R)$, has infinitely many solutions then the value of $\lambda + \mu $ is

hard

(JEE MAIN-2019)

Let $S$ be the set of all values of $\theta \in[-\pi, \pi]$ for which the system of linear equations

$x+y+\sqrt{3} z=0$

$-x+(\tan \theta) y+\sqrt{7} z=0$

$x+y+(\tan \theta) z=0$

has non-trivial solution. Then $\frac{120}{\pi} \sum_{\theta \in s} \theta$ is equal to

hard

(JEE MAIN-2023)

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

Solution

Similar Questions

The value of the determinant given below $\left| {{\rm{ }}\begin{array}{*{20}{c}}1&2&3\\3&5&7\\8&{14}&{20}\end{array}} \right|$ is

$\left| {\begin{array}{*{20}{c}}0&a&{ – b}\\{ – a}&0&c\\b&{ – c}&0\end{array}} \right| = $

The system of equations ${x_1} – {x_2} + {x_3} = 2,$ $\,3{x_1} – {x_2} + 2{x_3} = – 6$ and $3{x_1} + {x_2} + {x_3} = – 18$ has

If the system of linear equations $x + y + z = 5$ ; $x = 2y + 2z = 6$ ; $x + 3y + \lambda z = u (\lambda \, \mu \in R)$, has infinitely many solutions then the value of $\lambda + \mu $ is

Let $S$ be the set of all values of $\theta \in[-\pi, \pi]$ for which the system of linear equations $x+y+\sqrt{3} z=0$ $-x+(\tan \theta) y+\sqrt{7} z=0$ $x+y+(\tan \theta) z=0$ has non-trivial solution. Then $\frac{120}{\pi} \sum_{\theta \in s} \theta$ is equal to

Start a Free Trial Now

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

Let $S$ be the set of all values of $\theta \in[-\pi, \pi]$ for which the system of linear equations

$x+y+\sqrt{3} z=0$

$-x+(\tan \theta) y+\sqrt{7} z=0$

$x+y+(\tan \theta) z=0$

has non-trivial solution. Then $\frac{120}{\pi} \sum_{\theta \in s} \theta$ is equal to