Question

The number of integral values of ${}^{\prime }{a}^{\prime }$ for which $a(a-4)x^2+3ax+2>0$ for all $x$ belongs to real numbers is

• A. $30$
• B. $31$
• C. $32$
• D. $33$

Answer 1

The correct option is A $30$

$\begin{array}{c}a\left(a-4\right)x^2+3ax+2>0\\ \Rightarrow {\left(3a\right)}^2-4\times 2a\left(a-4\right)>0\\ \Rightarrow 9a^2-8a^2+32a>0\\ \Rightarrow a^2=32a>0\\ \Rightarrow a\left(a+32\right)>0\end{array}$

Hence, number of integral -32

Value of $a$ is (30) $\left(-\propto ,-32\right)\cup \left(0,\propto \right)$