1
JEE Main 2023 (Online) 11th April Evening Shift
+4
-1

If the system of linear equations

\begin{aligned} & 7 x+11 y+\alpha z=13 \\\\ & 5 x+4 y+7 z=\beta \\\\ & 175 x+194 y+57 z=361 \end{aligned}

has infinitely many solutions, then $$\alpha+\beta+2$$ is equal to :

A
6
B
4
C
5
D
3
2
JEE Main 2023 (Online) 11th April Evening Shift
+4
-1

$$\left|\begin{array}{ccc}x+1 & x & x \\ x & x+\lambda & x \\ x & x & x+\lambda^{2}\end{array}\right|=\frac{9}{8}(103 x+81)$$, then $$\lambda, \frac{\lambda}{3}$$ are the roots of the equation :

A
$$4 x^{2}+24 x-27=0$$
B
$$4 x^{2}-24 x+27=0$$
C
$$4 x^{2}-24 x-27=0$$
D
$$4 x^{2}+24 x+27=0$$
3
JEE Main 2023 (Online) 11th April Morning Shift
+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
$$\frac{5}{2}$$
4
JEE Main 2023 (Online) 10th April Evening Shift
+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
$$2^{16}$$
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