1
JEE Main 2023 (Online) 24th January Evening Shift
+4
-1 Let A be a 3 $$\times$$ 3 matrix such that $$\mathrm{|adj(adj(adj~A))|=12^4}$$. Then $$\mathrm{|A^{-1}~adj~A|}$$ is equal to

A
12
B
2$$\sqrt3$$
C
1
D
$$\sqrt6$$
2
JEE Main 2023 (Online) 24th January Evening Shift
+4
-1 If the system of equations

$$x+2y+3z=3$$

$$4x+3y-4z=4$$

$$8x+4y-\lambda z=9+\mu$$

has infinitely many solutions, then the ordered pair ($$\lambda,\mu$$) is equal to :

A
$$\left( {{{72} \over 5},{{21} \over 5}} \right)$$
B
$$\left( { - {{72} \over 5}, - {{21} \over 5}} \right)$$
C
$$\left( { - {{72} \over 5},{{21} \over 5}} \right)$$
D
$$\left( {{{72} \over 5}, - {{21} \over 5}} \right)$$
3
JEE Main 2023 (Online) 24th January Morning Shift
+4
-1 If A and B are two non-zero n $$\times$$ n matrices such that $$\mathrm{A^2+B=A^2B}$$, then :

A
$$\mathrm{A^2B=I}$$
B
$$\mathrm{A^2=I}$$ or $$\mathrm{B=I}$$
C
$$\mathrm{A^2B=BA^2}$$
D
$$\mathrm{AB=I}$$
4
JEE Main 2023 (Online) 24th January Morning Shift
+4
-1 Let $$\alpha$$ be a root of the equation $$(a - c){x^2} + (b - a)x + (c - b) = 0$$ where a, b, c are distinct real numbers such that the matrix $$\left[ {\matrix{ {{\alpha ^2}} & \alpha & 1 \cr 1 & 1 & 1 \cr a & b & c \cr } } \right]$$ is singular. Then, the value of $${{{{(a - c)}^2}} \over {(b - a)(c - b)}} + {{{{(b - a)}^2}} \over {(a - c)(c - b)}} + {{{{(c - b)}^2}} \over {(a - c)(b - a)}}$$ is

A
3
B
6
C
12
D
9
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