Question

Assertion :If The equation $x^2+2(k+1)x+9k-5=0$ has only negative roots, then $k\le 6$ Reason: The equation $f\left(x\right)=0$ will have both roots negative if and only if

(i) Discriminant $\ge 0,$

(ii) Sum of roots $<0,$

(iii) Product of roots $>0.$

• 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 is incorrect and Reason are correct.

Answer 1

The correct option is D Both Assertion is incorrect and Reason are correct.

Reason is true as,

Let $f\left(x\right)=x^2+2(k+1)x+9k-5.$

Let $\alpha ,\beta$ be the roots of $f\left(x\right)=0$.

The equation $f\left(x\right)=0$ will have both negative roots if and only if

(1) Disc. $\ge 0$

(2) $\alpha +\beta <0$

(3) $f\left(0\right)>0$

Now, (1) $\Rightarrow$ discriminant $\ge 0$

$\Rightarrow 4{(k+1)}^2-36k+20\ge 0\Rightarrow k^2-7k+6\ge 0$

$\Rightarrow (k-1)(k-6)\ge 0$$\Rightarrow k\le 1$ or $k\ge 6$ ...(a)

(2) $\Rightarrow (\alpha +\beta )<0\Rightarrow -2(k+1)<0$

$\Rightarrow k+1>0\Rightarrow k>-1$ ...(b)

and, (3) $\Rightarrow$ $f\left(0\right)>0\Rightarrow 9k-5>0\Rightarrow k>\frac{5}{9}$ ...(c)

From (a),(b) and (c)

$k\ge 6$ hence assertion is incorrect .