Question

If $ac>b^2$, then the sum of the coefficients in the expansion of ${(a{\alpha }^2{x}^2+2b\alpha x+c)}^n$ is ,where $a,b,c$ $\alpha \in R$ and $n\in N$

• A. positive if $a>0$
• B. positive if $c>0$
• C. negative if $a<0$ and $n$ is odd
• D. positive if $c<0$ and $n$ is even

Answer 1

The correct option is A positive if $a>0$

We have

${\left(a{\alpha }^2+2b\alpha x+c\right)}^n$

Sum of coefficients $={\left(a{\alpha }^2+2b\alpha +c\right)}^n$

$a{\alpha }^2+2b\alpha +c$

$D<0$

if $a<0$ $n$ odd

$a{\alpha }^2+2b\alpha +c>0$

${\left(a{\alpha }^2+2b\alpha +c\right)}^n>0$

$ifa<0nodd$

$a{\alpha }^2+2b\alpha +c<0$

${\left(a{\alpha }^2+2b\alpha +c\right)}^n<0$

Hence,

Option $\left(A\right)$ is correct answer.