Question

The value of ${}^{\prime }{a}^{\prime }$ for which $(a^2-1)x+2(a-1)x+2$ is positive for any $x$, are

• A. $a\ge 1$
• B. $a\le 1$
• C. $a\ge -3$
• D. $a\le -3$ or $a\ge 1$

Answer 1

Answer: (D)

We have that $a{x}^2+bx+c>0$ for all $x$ if $a$ > 0 and $b^2<4ac$
$\therefore (a^2-1)x^2+2(a-1)x+2$ is +ve for all $x$ if
$a^2-1>0$ and $4(a-1{)}^2-8(a^2-1)\le 0$
$\Rightarrow a^2-1\ge 0$ and $-4(a-1)(a+3)\le 0$
$\Rightarrow a^2-1\ge$ and $(a-1)(a+3)\ge 0$
$\Rightarrow a\le -1$ or $a\ge$ and $a\le -3$ or $a\ge 1$
$\Rightarrow a\le -3$ or $a\ge 1$