1
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
$x^2-18 x+56=0$
2
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
3
JEE Main 2023 (Online) 30th January Morning Shift
+4
-1

Let the system of linear equations

$$x+y+kz=2$$

$$2x+3y-z=1$$

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

have infinitely many solutions. Then the system

$$(k+1)x+(2k-1)y=7$$

$$(2k+1)x+(k+5)y=10$$

has :

A
unique solution satisfying $$x-y=1$$
B
infinitely many solutions
C
no solution
D
unique solution satisfying $$x+y=1$$
4
JEE Main 2023 (Online) 30th January Morning Shift
+4
-1
Out of Syllabus

Let $$A=\left(\begin{array}{cc}\mathrm{m} & \mathrm{n} \\ \mathrm{p} & \mathrm{q}\end{array}\right), \mathrm{d}=|\mathrm{A}| \neq 0$$ and $$\mathrm{|A-d(A d j A)|=0}$$. Then

A
$$1+\mathrm{d}^{2}=\mathrm{m}^{2}+\mathrm{q}^{2}$$
B
$$1+d^{2}=(m+q)^{2}$$
C
$$(1+d)^{2}=m^{2}+q^{2}$$
D
$$(1+d)^{2}=(m+q)^{2}$$
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