1
JEE Main 2023 (Online) 12th April Morning Shift
+4
-1

Let $$A=\left[\begin{array}{cc}1 & \frac{1}{51} \\ 0 & 1\end{array}\right]$$. If $$\mathrm{B}=\left[\begin{array}{cc}1 & 2 \\ -1 & -1\end{array}\right] A\left[\begin{array}{cc}-1 & -2 \\ 1 & 1\end{array}\right]$$, then the sum of all the elements of the matrix $$\sum_\limits{n=1}^{50} B^{n}$$ is equal to

A
50
B
75
C
100
D
125
2
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
3
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
$$4 x^{2}+24 x+27=0$$
4
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
$$\frac{5}{2}$$
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