JEE 2022 — Mathematics PYQ

1. Find $M^2$:

2. Analyze the sum $N$:

$N$ is a geometric series of matrices:

Since $M^2=-{\alpha }^{2}I$, then $M^{2k}=(-{\alpha }^2{)}^{k}I$.

3. Use the given equation $(I-M^2)N=-2I$:

The expression $(I-M^2)N$ is the telescoping sum/product of a geometric series:

Substitute the known values:

4. Solve for $\alpha$:

Equating the coefficients:

Let $x={\alpha }^2$. Then $x^{50}+x-2=0$.

By inspection, $x=1$ is a solution ($1^{50}+1-2=0$).

Since $x={\alpha }^2$, we have ${\alpha }^2=1$, which means $\alpha =\pm 1$.

The question asks for the positive integral value.

Final Answer:

The value of $\alpha$ is 1.