JEE 2023 — Mathematics PYQ

Step 1: Calculate $A^4$ in terms of $A^2$

Given the equation $A^2=3A+\alpha I$, we square both sides to find $A^4$:

Expanding the bracket:

Since $I^2=I$ and $AI=A$:

Step 2: Substitute $A^2$ back into the equation

Now, replace $A^2$ with the original expression $3A+\alpha I$:

Grouping the terms of $A$ and $I$:

Step 3: Compare with the given $A^4$ expression

The problem states that $A^4=21A+\beta I$. By comparing the coefficients of $A$ and $I$, we get two equations:

1. $27+6\alpha =21$

2. $9\alpha +{\alpha }^2=\beta$

Step 4: Solve for $\alpha$ and $\beta$

From equation (1):

Now, substitute $\alpha =-1$ into equation (2):

Final Answer

The value of $\beta$ is $-8$. Therefore, option (1) is correct.