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3 and 4 .Determinants and Matrices
hard
माना $M =\left[\begin{array}{cc}0 & -\alpha \\ \alpha & 0\end{array}\right]$ है, जहाँ $\alpha$ अशून्य वास्तविक तथा $N =\sum \limits_{ k =1}^{49} M ^{2 k }$ है। यदि $\left( I - M ^2\right) N =-2 I$ है, तो $\alpha$ का धनात्मक पूर्णांक मान है।
A
$4$
B
$3$
C
$2$
D
$1$
(JEE MAIN-2022)
Solution
$M =\left[\begin{array}{cc}0 & -\alpha \\ \alpha & 0\end{array}\right] ; M ^{2}=\left[\begin{array}{cc}-\alpha^{2} & 0 \\ 0 & -\alpha^{2}\end{array}\right]=-\alpha^{2} I$
$N = M ^{2}+ M ^{4}+\ldots \ldots+ M ^{98}=\left[-\alpha^{2}+\alpha^{4}-\alpha^{6}+\ldots .\right] I$
$=-\alpha^{2} \frac{\left(1-\left(-\alpha^{2}\right)^{49}\right)}{1+\alpha^{2}} . I$
$I – M ^{2}=\left(1+\alpha^{2}\right) I$
$\left( I – M ^{2}\right) N =-\alpha^{2}\left(\alpha^{98}+1\right)=-2$
$\alpha=1$
Standard 12
Mathematics
