1
JEE Main 2023 (Online) 11th April Morning Shift
MCQ (Single Correct Answer)
+4
-1

Let $$\mathrm{A}$$ be a $$2 \times 2$$ matrix with real entries such that $$\mathrm{A}'=\alpha \mathrm{A}+\mathrm{I}$$, where $$\alpha \in \mathbb{R}-\{-1,1\}$$. If $$\operatorname{det}\left(A^{2}-A\right)=4$$, then the sum of all possible values of $$\alpha$$ is equal to :

A
2
B
$$\frac{3}{2}$$
C
0
D
$$\frac{5}{2}$$
2
JEE Main 2023 (Online) 10th April Evening Shift
MCQ (Single Correct Answer)
+4
-1
Out of Syllabus

If $$\mathrm{A}=\frac{1}{5 ! 6 ! 7 !}\left[\begin{array}{ccc}5 ! & 6 ! & 7 ! \\ 6 ! & 7 ! & 8 ! \\ 7 ! & 8 ! & 9 !\end{array}\right]$$, then $$|\operatorname{adj}(\operatorname{adj}(2 \mathrm{~A}))|$$ is equal to :

A
$$2^{12}$$
B
$$2^{20}$$
C
$$2^{8}$$
D
$$2^{16}$$
3
JEE Main 2023 (Online) 10th April Morning Shift
MCQ (Single Correct Answer)
+4
-1
Out of Syllabus

If A is a 3 $$\times$$ 3 matrix and $$|A| = 2$$, then $$|3\,adj\,(|3A|{A^2})|$$ is equal to :

A
$${3^{12}}\,.\,{6^{10}}$$
B
$${3^{11}}\,.\,{6^{10}}$$
C
$${3^{12}}\,.\,{6^{11}}$$
D
$${3^{10}}\,.\,{6^{11}}$$
4
JEE Main 2023 (Online) 10th April Morning Shift
MCQ (Single Correct Answer)
+4
-1

For the system of linear equations

$$2x - y + 3z = 5$$

$$3x + 2y - z = 7$$

$$4x + 5y + \alpha z = \beta$$,

which of the following is NOT correct?

A
The system has infinitely many solutions for $$\alpha=-6$$ and $$\beta=9$$
B
The system has a unique solution for $$\alpha$$ $$\ne$$ $$-5$$ and $$\beta=8$$
C
The system is inconsistent for $$\alpha=-5$$ and $$\beta=8$$
D
The system has infinitely many solutions for $$\alpha=-5$$ and $$\beta=9$$
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