Question

Let $A=\{x\in R:x^2-|x|-2=0\}$ and $B=\{\alpha +\beta ,\alpha \beta \}$ where $\alpha ,\beta$ are real roots of the quadratic equation $x^2+|x|-2=0.$ If $(a,b)\in A\times B,$ then the quadratic equation whose roots are $a,b$ is

• A. $x^2-2x=0$
• B. $x^2-x-2=0$
• C. $x^2+2x=0$
• D. $x^2+3x+2=0$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A $x^2-2x=0$
B $x^2-x-2=0$
C $x^2+2x=0$
D $x^2+3x+2=0$

$x^2-|x|-2=0$
$\Rightarrow |x{|}^2-|x|-2=0$
$\Rightarrow (|x|-2)(|x|+1)=0$
$\Rightarrow |x|=2$
$\therefore A=\{2,-2\}$

$x^2+|x|-2=0$
$\Rightarrow (|x|-1)(|x|+2)=0$
$\Rightarrow x=\pm 1$
$\therefore B=\{-1,0\}$
Hence, $A\times B=\left\{\right(2,-1),(2,0),(-2,-1),(-2,0\left)\right\}$