જો શ્રેણિક A =left[begin{array}{cc}1 & -α α & βend{array}right], મા

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3 and 4 .Determinants and Matrices

medium

જો શ્રેણિક $A =\left[\begin{array}{cc}1 & -\alpha \\ \alpha & \beta\end{array}\right],$ માટે, $AA ^{ T }= I _{2}$હોય, તો $\alpha^{4}+\beta^{4}$ નું મૂલ્ય ....... થાય.

$4$

$2$

$3$

$1$

(JEE MAIN-2021)

Solution

$A =\left[\begin{array}{cc}1 & -\alpha \\ \alpha & \beta\end{array}\right] \quad AA ^{ T }= I _{2}$

$\Rightarrow\left[\begin{array}{cc}1 & -\alpha \\ \alpha & \beta\end{array}\right]\left[\begin{array}{cc}1 & \alpha \\ -\alpha & \beta\end{array}\right]=\left[\begin{array}{cc}1 & 0 \\ 0 & 1\end{array}\right]$

$\Rightarrow\left[\begin{array}{cc}1+\alpha^{2} & \alpha-\alpha \beta \\ \alpha-\alpha \beta & \alpha^{2}+\beta^{2}\end{array}\right]=\left[\begin{array}{cc}1 & 0 \\ 0 & 1\end{array}\right]$

$\Rightarrow \alpha^{2}=0 \;and\; \beta^{2}=1$

$\therefore \alpha^{4}+\beta^{4}=1$

Standard 12

Mathematics

જો $A = \left[ {\begin{array}{{20}{c}}1&{ – 2}&1\\2&1&3\end{array}} \right]$ અને $B = \left[ {\begin{array}{{20}{c}}2&1\\3&2\\1&1\end{array}} \right]$, તો ${(AB)^T} = $

easy

જો શ્રેણિક $\left( {A – \frac{I}{2}} \right)$ અને ${A + \frac{I}{2}}$ એ લંબચ્છેદિ હોય તો

normal

જો $P = \left[ {\begin{array}{{20}{c}}
{\frac{{\sqrt 3 }}{2}}&{\frac{1}{2}}\\
{ – \frac{1}{2}}&{\frac{{\sqrt 3 }}{2}}
\end{array}} \right],\,A = \,\left[ {\begin{array}{{20}{c}}
1&1\\
0&1
\end{array}} \right]$ અને $Q=PAP^T,$ તો $P^T$ $Q^{2015}$ $P$ = . . . .

medium

(JEE MAIN-2016)

શ્રેણિક

$A = \left[ {\begin{array}{*{20}{c}}
{{{10}^{30}} + 5}&{{{10}^{20}} + 4}&{{{10}^{20}} + 6}\\
{{{10}^4} + 2}&{{{10}^8} + 7}&{{{10}^{10}} + 2n}\\
{{{10}^4} + 8}&{{{10}^6} + 4}&{{{10}^{15}} + 9}
\end{array}} \right]$ , $n \in N$, હોય તો . ..

normal

ધારો કે $A=\left[\begin{array}{cc}0 & -2 \\ 2 & 0\end{array}\right]$. જો બે શ્રેણિકો $M$ અને $N$ એ $M =\sum \limits_{ k =1}^{10} A ^{2 k }$ અને $N =\sum \limits_{ k =1}^{10} A ^{2 k -1}$ પ્રમાણે આપેલા હોય, તો $MN ^{2}$ એ……..

medium

(JEE MAIN-2022)

જો શ્રેણિક $A =\left[\begin{array}{cc}1 & -\alpha \\ \alpha & \beta\end{array}\right],$ માટે, $AA ^{ T }= I _{2}$હોય, તો $\alpha^{4}+\beta^{4}$ નું મૂલ્ય ....... થાય.

Solution

Similar Questions

જો $A = \left[ {\begin{array}{*{20}{c}}1&{ – 2}&1\\2&1&3\end{array}} \right]$ અને $B = \left[ {\begin{array}{*{20}{c}}2&1\\3&2\\1&1\end{array}} \right]$, તો ${(AB)^T} = $

જો શ્રેણિક $\left( {A – \frac{I}{2}} \right)$ અને ${A + \frac{I}{2}}$ એ લંબચ્છેદિ હોય તો

જો $P = \left[ {\begin{array}{*{20}{c}} {\frac{{\sqrt 3 }}{2}}&{\frac{1}{2}}\\ { – \frac{1}{2}}&{\frac{{\sqrt 3 }}{2}} \end{array}} \right],\,A = \,\left[ {\begin{array}{*{20}{c}} 1&1\\ 0&1 \end{array}} \right]$ અને $Q=PAP^T,$ તો $P^T$ $Q^{2015}$ $P$ = . . . .

શ્રેણિક $A = \left[ {\begin{array}{*{20}{c}} {{{10}^{30}} + 5}&{{{10}^{20}} + 4}&{{{10}^{20}} + 6}\\ {{{10}^4} + 2}&{{{10}^8} + 7}&{{{10}^{10}} + 2n}\\ {{{10}^4} + 8}&{{{10}^6} + 4}&{{{10}^{15}} + 9} \end{array}} \right]$ , $n \in N$, હોય તો . ..

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જો $A = \left[ {\begin{array}{{20}{c}}1&{ – 2}&1\\2&1&3\end{array}} \right]$ અને $B = \left[ {\begin{array}{{20}{c}}2&1\\3&2\\1&1\end{array}} \right]$, તો ${(AB)^T} = $

જો $P = \left[ {\begin{array}{{20}{c}}
{\frac{{\sqrt 3 }}{2}}&{\frac{1}{2}}\\
{ – \frac{1}{2}}&{\frac{{\sqrt 3 }}{2}}
\end{array}} \right],\,A = \,\left[ {\begin{array}{{20}{c}}
1&1\\
0&1
\end{array}} \right]$ અને $Q=PAP^T,$ તો $P^T$ $Q^{2015}$ $P$ = . . . .

શ્રેણિક

$A = \left[ {\begin{array}{*{20}{c}}
{{{10}^{30}} + 5}&{{{10}^{20}} + 4}&{{{10}^{20}} + 6}\\
{{{10}^4} + 2}&{{{10}^8} + 7}&{{{10}^{10}} + 2n}\\
{{{10}^4} + 8}&{{{10}^6} + 4}&{{{10}^{15}} + 9}
\end{array}} \right]$ , $n \in N$, હોય તો . ..