જો A =frac{1}{2}left[begin{array}{cc}1 & sqrt{3} -sqrt{3} & 1end{array}right], તો:

Q: જો $A =\frac{1}{2}\left[\begin{array}{cc}1 & \sqrt{3} \\ -\sqrt{3} & 1\end{array}\right],$ તો:

$ A =\frac{1}{2}\left[\begin{array}{cc}1 & \sqrt{3} \\ -\sqrt{3} & 1\end{array}\right] $ $ A =\left[\begin{array}{cc}\cos 60^{\circ} & \sin 60^{\circ} \\ -\sin 60^{\circ} & \cos 60^{\circ}\end{array}\right]$ $\text { If } A =\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right]$ Here $ \alpha=\frac{\pi}{3} ~$ $ A ^2=\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right]$ $\left[\begin{array}{cc}\cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha\end{array}\right] ~$ $ =\left[\begin{array}{cc}\cos 2 \alpha & \sin 2 \alpha \\ -\sin 2 \alpha & \cos 2 \alpha\end{array}\right]$ $A ^{30}=\left[\begin{array}{cc}\cos 30 \alpha & \sin 30 \alpha \\ -\sin 30 \alpha & \cos 30 \alpha\end{array}\right] ~\\ A ^{30}=\left[\begin{array}{cc}1 & 0 \\ 0 & 1\end{array}\right]= I$ $A^{25}=\left[\begin{array}{cc}\cos 25 \alpha & \sin 25 \alpha \\ -\sin 25 \alpha & \cos 25 \alpha\end{array}\right]$ $=\left[\begin{array}{cc}\frac{1}{2} & \frac{\sqrt{3}}{2} \\ \frac{-\sqrt{3}}{2} & \frac{1}{2}\end{array}\right]$ $A^{25}=A$ $A^{25}-A=0$

MCQ

ગુજરાતી માધ્યમ

જો $A =\frac{1}{2}\left[\begin{array}{cc}1 & \sqrt{3} \\ -\sqrt{3} & 1\end{array}\right],$ તો:

A$A ^{30}- A ^{25}=2 I$
B$A ^{30}+ A ^{25}+ A = I$
C$A ^{30}+ A ^{25}- A = I$
D$A ^{30}= A ^{25}$

JEE MAIN 2023, Difficult

ધોરણ 12 સાયન્સ
ગણિત
3 and 4 . determinant and metrices
JEE

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