1
JEE Main 2023 (Online) 13th April Morning Shift
+4
-1

For the system of linear equations

$$2 x+4 y+2 a z=b$$

$$x+2 y+3 z=4$$

$$2 x-5 y+2 z=8$$

which of the following is NOT correct?

A
It has infinitely many solutions if $$a=3, b=8$$
B
It has infinitely many solutions if $$a=3, b=6$$
C
It has unique solution if $$a=b=8$$
D
It has unique solution if $$a=b=6$$
2
JEE Main 2023 (Online) 13th April Morning Shift
+4
-1

Let $$B=\left[\begin{array}{lll}1 & 3 & \alpha \\ 1 & 2 & 3 \\ \alpha & \alpha & 4\end{array}\right], \alpha > 2$$ be the adjoint of a matrix $$A$$ and $$|A|=2$$. Then $$\left[\begin{array}{ccc}\alpha & -2 \alpha & \alpha\end{array}\right] B\left[\begin{array}{c}\alpha \\ -2 \alpha \\ \alpha\end{array}\right]$$ is equal to :

A
32
B
$$-$$16
C
0
D
16
3
JEE Main 2023 (Online) 13th April Morning Shift
+4
-1

The number of symmetric matrices of order 3, with all the entries from the set $$\{0,1,2,3,4,5,6,7,8,9\}$$ is :

A
$$10^{9}$$
B
$$9^{10}$$
C
$$10^{6}$$
D
$$6^{10}$$
4
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
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