1
JEE Main 2023 (Online) 25th January Morning Shift
MCQ (Single Correct Answer)
+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
$$2^4$$
2
JEE Main 2023 (Online) 25th January Morning Shift
MCQ (Single Correct Answer)
+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
3
JEE Main 2023 (Online) 24th January Evening Shift
MCQ (Single Correct Answer)
+4
-1
Out of Syllabus

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
$$\sqrt6$$
4
JEE Main 2023 (Online) 24th January Evening Shift
MCQ (Single Correct Answer)
+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
$$\left( {{{72} \over 5}, - {{21} \over 5}} \right)$$
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