JEE Main 2023 (Online) 30th January Evening Shift | Matrices and Determinants Question 70 | Mathematics

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 :

$x^2+18 x+56=0$

$x^2-10 x+16=0$

$x^2+14 x+24=0$

$x^2-18 x+56=0$

JEE Main 2023 (Online) 30th January Evening Shift

MCQ (Single Correct Answer)

-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 :

$|A d j P|=1$

$|A d j P|>1$

$|A d j P|=\frac{1}{2}$

$P$ is a singular matrix

JEE Main 2023 (Online) 30th January Morning Shift

MCQ (Single Correct Answer)

-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 :

unique solution satisfying $$x-y=1$$

infinitely many solutions

no solution

unique solution satisfying $$x+y=1$$

JEE Main 2023 (Online) 30th January Morning Shift

MCQ (Single Correct Answer)

-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

$$1+\mathrm{d}^{2}=\mathrm{m}^{2}+\mathrm{q}^{2}$$

$$1+d^{2}=(m+q)^{2}$$

$$(1+d)^{2}=m^{2}+q^{2}$$

$$(1+d)^{2}=(m+q)^{2}$$

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