Question

Let $a,b,c,a_1,b_1,c_1,\in R$ and $a{x}^2+bx+c>0\forall x\in R$ and $a_1{x}^2+b_{1}x+c_1>0\forall x\in R.$ Then

• A. $a{a}_1{x}^2+b{b}_{1}x+c{c}_1>0\forall x\in R$
• B. $a{a}_1{x}^2+b{b}_{1}x+c{c}_1<0\forall x\in R$
• C. $a{a}_1{x}^2+b{b}_{1}x+c{c}_1=0$, will have real roots.
• D. Nothing can be said in general about the nature of roots of, $a{a}_1{x}^2+b{b}_{1}x+c{c}_1=0$

Answer 1

The correct option is D Nothing can be said in general about the nature of roots of, $a{a}_1{x}^2+b{b}_{1}x+c{c}_1=0$

Solution : We have $a{x}^2+bx+c>0\forall x\in R$
$a_1{x}^2{b}_{1}x+c_1>0\forall x\in R$
So, $b^2-4ac<0,a>0,c>0$
and $b_1^2-4a_1{c}_1<0,a_1>0,c_1>0$
$b^2<4ac$ and $b_1^2<4a_1{c}_1$
$b^2{b}_1^2<16a_1{c}_{1}ac$
$(b{b}_1{)}^2-4a_{1}ac{c}_1<12a_{1}ac.c_1$ (positive)
$b^2{b}_1^2-4a{a}_{1}c{c}_1$ is discriminate of $a{a}_1{x}^2+b{b}_{1}x+c{c}_1=0$
If may be negative, zero, or positive that way we can't say nature of the root.