Question

Assertion :If both roots of the equation $x^2+2(a-1)x+a+5=0$ $\forall a\in R$ lie in interval $(1,3)$, then $\frac{-8}{7}<a\le -1$. Reason: If $f\left(x\right)=x^2+2(a-1)x+a+5$ then, $D\ge 0,$ $f\left(1\right)>0,$ $f\left(3\right)>0$ gives $\frac{-8}{7}<a\le -1$.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Both Assertion and Reason are incorrect

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

As both roots lies in the interval $(1,3)$ then the following conditions must hold simultaneously
(i) $D\ge 0$ where $D$ is discriminant of $f\left(x\right)=0$
(ii) $f\left(1\right)>0$
(iii) $f\left(3\right)>0$
Hence, Reason (R) is true.
Now, $D\ge 0$
$\Rightarrow 4(a-1{)}^2-4(a+5)\ge 0$
$\Rightarrow a^2-3a-4\ge 0$
$\Rightarrow (a-4)(a+1)\ge 0$
$\Rightarrow a\in (-\infty ,-1]\cup [4,\infty )$ .....(A)
Again $f\left(1\right)>0$
$\Rightarrow 1+2(a-1)+a+a>0$
$4+3a>0$
i.e $a>-\frac{4}{3}$ ....(B)
and $f\left(3\right)>0$
$\Rightarrow 9+6(a-1)+a+5>0$
$\Rightarrow 7a+8>0\Rightarrow a>-\frac{8}{7}$ .....(C)
From (A), (B) & (C) we have $a\in (-\frac{8}{7},-1]$.