1
JEE Main 2023 (Online) 31st January Morning Shift
+4
-1

For the system of linear equations

$$x+y+z=6$$

$$\alpha x+\beta y+7 z=3$$

$$x+2 y+3 z=14$$

which of the following is NOT true ?

A
If $$\alpha=\beta=7$$, then the system has no solution
B
For every point $$(\alpha, \beta) \neq(7,7)$$ on the line $$x-2 y+7=0$$, the system has infinitely many solutions
C
There is a unique point $$(\alpha, \beta)$$ on the line $$x+2 y+18=0$$ for which the system has infinitely many solutions
D
If $$\alpha=\beta$$ and $$\alpha \neq 7$$, then the system has a unique solution
2
JEE Main 2023 (Online) 31st January Morning Shift
+4
-1

Let $$A = \left( {\matrix{ 1 & 0 & 0 \cr 0 & 4 & { - 1} \cr 0 & {12} & { - 3} \cr } } \right)$$. Then the sum of the diagonal elements of the matrix $${(A + I)^{11}}$$ is equal to :

A
4094
B
2050
C
6144
D
4097
3
JEE Main 2023 (Online) 30th January Evening Shift
+4
-1
For $\alpha, \beta \in \mathbb{R}$, suppose the system of linear equations

\begin{aligned} & x-y+z=5 \\ & 2 x+2 y+\alpha z=8 \\ & 3 x-y+4 z=\beta \end{aligned}

has infinitely many solutions. Then $\alpha$ and $\beta$ are the roots of :
A
$x^2+18 x+56=0$
B
$x^2-10 x+16=0$
C
$x^2+14 x+24=0$
D
$x^2-18 x+56=0$
4
JEE Main 2023 (Online) 30th January Evening Shift
+4
-1
Out of Syllabus
If $P$ is a $3 \times 3$ real matrix such that $P^T=a P+(a-1) I$, where $a>1$, then :
A
$|A d j P|=1$
B
$|A d j P|>1$
C
$|A d j P|=\frac{1}{2}$
D
$P$ is a singular matrix
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