1
JEE Main 2023 (Online) 25th January Evening Shift
+4
-1

Let A, B, C be 3 $$\times$$ 3 matrices such that A is symmetric and B and C are skew-symmetric. Consider the statements

(S1) A$$^{13}$$ B$$^{26}$$ $$-$$ B$$^{26}$$ A$$^{13}$$ is symmetric

(S2) A$$^{26}$$ C$$^{13}$$ $$-$$ C$$^{13}$$ A$$^{26}$$ is symmetric

Then,

A
Only S2 is true
B
Only S1 is true
C
Both S1 and S2 are false
D
Both S1 and S2 are true
2
JEE Main 2023 (Online) 25th January Evening Shift
+4
-1

Let $$A = \left[ {\matrix{ {{1 \over {\sqrt {10} }}} & {{3 \over {\sqrt {10} }}} \cr {{{ - 3} \over {\sqrt {10} }}} & {{1 \over {\sqrt {10} }}} \cr } } \right]$$ and $$B = \left[ {\matrix{ 1 & { - i} \cr 0 & 1 \cr } } \right]$$, where $$i = \sqrt { - 1}$$. If $$\mathrm{M=A^T B A}$$, then the inverse of the matrix $$\mathrm{AM^{2023}A^T}$$ is

A
$$\left[ {\matrix{ 1 & { - 2023i} \cr 0 & 1 \cr } } \right]$$
B
$$\left[ {\matrix{ 1 & 0 \cr {2023i} & 1 \cr } } \right]$$
C
$$\left[ {\matrix{ 1 & {2023i} \cr 0 & 1 \cr } } \right]$$
D
$$\left[ {\matrix{ 1 & 0 \cr { - 2023i} & 1 \cr } } \right]$$
3
JEE Main 2023 (Online) 25th January Morning Shift
+4
-1
Out of Syllabus

Let $$x,y,z > 1$$ and $$A = \left[ {\matrix{ 1 & {{{\log }_x}y} & {{{\log }_x}z} \cr {{{\log }_y}x} & 2 & {{{\log }_y}z} \cr {{{\log }_z}x} & {{{\log }_z}y} & 3 \cr } } \right]$$. Then $$\mathrm{|adj~(adj~A^2)|}$$ is equal to

A
$$6^4$$
B
$$2^8$$
C
$$4^8$$
D
$$2^4$$
4
JEE Main 2023 (Online) 25th January Morning Shift
+4
-1

Let S$$_1$$ and S$$_2$$ be respectively the sets of all $$a \in \mathbb{R} - \{ 0\}$$ for which the system of linear equations

$$ax + 2ay - 3az = 1$$

$$(2a + 1)x + (2a + 3)y + (a + 1)z = 2$$

$$(3a + 5)x + (a + 5)y + (a + 2)z = 3$$

has unique solution and infinitely many solutions. Then

A
$$\mathrm{n({S_1}) = 2}$$ and S$$_2$$ is an infinite set
B
$$\mathrm{{S_1} = \Phi}$$ and $$\mathrm{{S_2} = \mathbb{R} - \{ 0\}}$$
C
$$\mathrm{{S_1} = \mathbb{R} - \{ 0\}}$$ and $$\mathrm{{S_2} = \Phi}$$
D
S$$_1$$ is an infinite set and n(S$$_2$$) = 2
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