Question

Consider the two sets:
$A=\{m\in R:\text{both the roots of}{x}^2-(m+1)x+m+4=0\text{are real}\}$ and $B=[-3,5).$
Which of the following is not true ?

• A. $A-B=(-\infty ,-3)\cup (5,\infty )$
• B. $A\cap B=\{-3\}$
• C. $B-A=(-3,5)$
• D. $A\cup B=R$

Answer 1

The correct option is A $A-B=(-\infty ,-3)\cup (5,\infty )$

$A:D\ge 0$
$\Rightarrow {(m+1)}^2-4(m+4)\ge 0$
$\Rightarrow m^2-2m-15\ge 0$
$\Rightarrow (m-5)(m+3)\ge 0$
$\Rightarrow m\in (-\infty ,-3]\cup [5,\infty )$
$\therefore A=(-\infty ,-3]\cup [5,\infty )$
$B=[-3,5)$

$A-B=(-\infty ,-3)\cup [5,\infty )$
$A\cap B=\{-3\}$
$B-A=(-3,5)$
$A\cup B=R$