જો Delta = left| {,begin{array}{*{20}{c}}x&y&zp&q&ra&b&cend{array},} rig

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3 and 4 .Determinants and Matrices

easy

જો $\Delta = \left| {\,\begin{array}{{20}{c}}x&y&z\\p&q&r\\a&b&c\end{array}\,} \right|,$ તો $\left| {\,\begin{array}{{20}{c}}x&{2y}&z\\{2p}&{4q}&{2r}\\a&{2b}&c\end{array}\,} \right|$ = . . .

${\Delta ^2}$

$4\Delta $

$3\Delta $

એકપણ નહી.

Solution

(b) Let ${\Delta _1} = \left| {\,\begin{array}{*{20}{c}}x&{2y}&z\\{2p}&{4q}&{2r}\\a&{2b}&c\end{array}\,} \right|$=$4\,\left| {\,\begin{array}{*{20}{c}}x&y&z\\p&q&r\\a&b&c\end{array}\,} \right| = 4\Delta $.

(Taking common $'2'$ from $II^{nd}$ row and $'2'$ from $II^{nd}$ column).

Standard 12

Mathematics

જો સમીકરણ સંહતીઓ $2 x+\lambda y+3 z=5$, $3 x+2 y-z=7$, $4 x+5 y+\mu z=9$ ને અનંત ઉકેલ હોય તો $\left(\lambda^2+\mu^2\right)$ ની કિમંત મેળવો.

hard

(JEE MAIN-2025)

સમીકરણ સંહતિ ${x_2} – {x_3} = 1,\,\, – {x_1} + 2{x_3} = – 2,$ ${x_1} – 2{x_2} = 3$ ના ઉકેલની સંખ્યા મેળવો.

easy

જો સમીકરણની સંહતિ ${(\alpha + 1)^3}x + {(\alpha + 2)^3}y – {(\alpha + 3)^3} = 0$ અને $(\alpha + 1)x + (\alpha + 2)y – (\alpha + 3) = 0,x + y – 1 = 0$ એ અચળ હોય તો $\alpha $ ની કિમત મેળવો.

hard

સમીકરણ સંહતિ $2x + y – z = 7,\,\,x – 3y + 2z = 1,\,x + 4y – 3z = 5$ ના ઉકેલની સંખ્યા મેળવો.

medium

$\left| {\begin{array}{*{20}{c}}
{\sin \alpha }&{\cos \alpha }&{\sin \left( {\alpha + \gamma } \right)}\\
{\sin \beta }&{\cos \beta }&{\sin \left( {\beta + \gamma } \right)}\\
{\sin \delta }&{\cos \delta }&{\sin \left( {\gamma + \delta } \right)}
\end{array}} \right|$ મેળવો.

normal

જો $\Delta = \left| {\,\begin{array}{*{20}{c}}x&y&z\\p&q&r\\a&b&c\end{array}\,} \right|,$ તો $\left| {\,\begin{array}{*{20}{c}}x&{2y}&z\\{2p}&{4q}&{2r}\\a&{2b}&c\end{array}\,} \right|$ = . . .