Question

For the equation $|x^2-2x-3|=k,$ consider the following statements :
$\left(1\right)$ For $k=-1,$ there is no solution.
$\left(2\right)$ For $k=2,$ there are four solutions.
$\left(3\right)$ For $k=4,$ there are two solutions.
$\left(4\right)$ For $k=0,$ there are two solutions.

the CORRECT statements are

• A. $\left(1\right),$ $\left(2\right)$ and $\left(3\right)$ only
• B. $\left(1\right)$ and $\left(2\right)$ only
• C. $\left(1\right),$ $\left(2\right)$ and $\left(4\right)$ only
• D. $\left(2\right)$ and $\left(4\right)$ only

Answer 1

The correct option is C $\left(1\right),$ $\left(2\right)$ and $\left(4\right)$ only

$|x^2-2x-3|$
$=|(x-3)(x+1)|$
For $x^2-2x-3,$
$\frac{-D}{4a}=\frac{-(4+12)}{4}=-4$


For $k<0,$ there is no solution.
For $k=0,$ there are two solutions.
For $k\in (0,4),$ there are four solutions.
For $k=4,$ there are three solutions.
For $k>4,$ there are two solutions.