Question

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

• 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. Assertion is incorrect, Reason is correct.

Answer 1

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

Here, the coefficient of $x^2=2>0$,

Then, the conditions are $D\ge 0,f\left(1\right)>0$ and $f\left(2\right)>0$

$\Rightarrow 1-8a\ge 0,2-1+a>0$ and $8-2+a>0$

$\Rightarrow a\le \frac{1}{8},a>-1$ and $a>-6$

$\Rightarrow -1<a\le \frac{1}{8}$.