1
JEE Main 2024 (Online) 31st January Evening Shift
Numerical
+4
-1

Let A be a $$3 \times 3$$ matrix and $$\operatorname{det}(A)=2$$. If $$n=\operatorname{det}(\underbrace{\operatorname{adj}(\operatorname{adj}(\ldots . .(\operatorname{adj} A))}_{2024-\text { times }}))$$, then the remainder when $$n$$ is divided by 9 is equal to __________.

2
JEE Main 2024 (Online) 29th January Evening Shift
Numerical
+4
-1

Let for any three distinct consecutive terms $$a, b, c$$ of an A.P, the lines $$a x+b y+c=0$$ be concurrent at the point $$P$$ and $$Q(\alpha, \beta)$$ be a point such that the system of equations

\begin{aligned} & x+y+z=6, \\ & 2 x+5 y+\alpha z=\beta \text { and } \end{aligned}

$$x+2 y+3 z=4$$, has infinitely many solutions. Then $$(P Q)^2$$ is equal to _________.

3
JEE Main 2024 (Online) 27th January Evening Shift
Numerical
+4
-1

Let $$A$$ be a $$2 \times 2$$ real matrix and $$I$$ be the identity matrix of order 2. If the roots of the equation $$|\mathrm{A}-x \mathrm{I}|=0$$ be $$-1$$ and 3, then the sum of the diagonal elements of the matrix $$\mathrm{A}^2$$ is

4
JEE Main 2024 (Online) 27th January Morning Shift
Numerical
+4
-1
Let $A=\left[\begin{array}{lll}2 & 0 & 1 \\ 1 & 1 & 0 \\ 1 & 0 & 1\end{array}\right], B=\left[B_1, B_2, B_3\right]$, where $B_1, B_2, B_3$ are column matrics, and

$$\mathrm{AB}_1=\left[\begin{array}{l} 1 \\ 0 \\ 0 \end{array}\right], \mathrm{AB}_2=\left[\begin{array}{l} 2 \\ 3 \\ 0 \end{array}\right], \quad \mathrm{AB}_3=\left[\begin{array}{l} 3 \\ 2 \\ 1 \end{array}\right]$$

If $\alpha=|B|$ and $\beta$ is the sum of all the diagonal elements of $B$, then $\alpha^3+\beta^3$ is equal to ____________.