Question

$\text{Let}f\left(\alpha \right)={\alpha }^2-a\alpha +1,f\left(\beta \right)=b\beta -a{\beta }^2-\frac{3}{a},\text{where}a,b\in I^{-}\text{and}|f\left(\alpha \right)+f\left(\beta \right)|=|f\left(\alpha \right)|+|f\left(\beta \right)|\text{is satisfied}\forall \alpha ,\beta \in R,\text{then which of the following is/are true?}$

• A. Sum of all values of $a$ is $-3$
• B. Sum of all values of $b$ is $-6$
• C. Sum of all values of $b$ is $-3$
• D. Sum of all values of $a$ is $-2$

Answer 1

Answer: (A), (B)

Sum of all values of $b$ is $-6$

$\because |f\left(\alpha \right)+f\left(\beta \right)|=|f\left(\alpha \right)|+|f\left(\beta \right)|$
$\therefore f\left(\alpha \right).f\left(\beta \right)\ge 0$

So $f\left(\alpha \right)\ge 0\&f\left(\beta \right)\ge 0$
$\begin{array}{c}{\alpha }^2-a\alpha +1\ge 0\forall \alpha \in R\\ \Rightarrow D\le 0\\ \Rightarrow a^2-4\le 0\\ -2\le a\le 2\\ a=-2,-1\end{array}|\begin{array}{c}\frac{abp-a^2{p}^2-3}{a}\ge 0\\ (a^2{p}^2-b(ap)+3)\ge 0\\ \Rightarrow D\le 0\\ \Rightarrow b^2-12\le 0\\ -2\sqrt{3}\le b\le 2\sqrt{3}\\ b=-3,-2,-1\end{array}$