Question

For the quadratic equation $x^2+2(a+1)x+9a-5=0$ which of the following is/are true?

• A. If $2<a<5$ , then roots are of opposite sign.
• B. If $a<0$ , then roots are of opposite sign.
• C. If $a>7$ , then both roots are negative.
• D. If $2\le a\le 5$, then roots are unreal.

Answer 1

Answer: (B), (C), (D)

The correct options are
B If $a<0$ , then roots are of opposite sign.
C If $a>7$ , then both roots are negative.
D If $2\le a\le 5$, then roots are unreal.

$x^2+2(a+1)x+9a-5=0$
If the roots are of opposite sign, then the product of the roots will be less than zero.
$\Rightarrow 9a-5<0\Rightarrow a<\frac{9}{5}$
If both roots are negative then sum of roots will be less than zero.
$\Rightarrow -2(a+1)<0\Rightarrow a>-1$

if both roots are unreal
then,

$D=4(a+1{)}^2-4(9a-5)=4(a-1\left)\right(a-6)$

$\therefore D\ge 0\Rightarrow a\le 1$ or $a\ge 6\Rightarrow$ roots are real
$\therefore D\le 0\Rightarrow 1\le a\le 6\Rightarrow$ roots are unreal

Ans: B,C,D