Question

Assertion :The quadratic equation $10x^2-28x+17=0$ has atleast one root in $[1,2]$ Reason: $f\left(x\right)=e^{10x}(x-1)(x-2)$ satisfies all the conditions for Rolles theorem in $[1,2]$

• 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 but Reason is correct

Answer 1

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

$y={10x}^2-28x+17$
$y\left(1\right)=10-28+17=-1$
$y\left(2\right)=40-56+17=1$
As y is +ve for $x=2$ and y is -ve for $x=1$ y has atleast one root in $(1,2).$

$\because$ $f\left(x\right)$ is continuous,
derivable & $f\left(1\right)=f\left(2\right)=0$
$\Rightarrow f\left(x\right)=0$ has atleast one root in $(1,2).$
$\Rightarrow e^{10x}(2x-3)+10e^{10x}(x^2-3x+2)=0$
has atleast one root in $(1,2)$. $\Rightarrow 10x^2-28x+17=0$ has at least one root in $(1,2).$

Thus the assertion and reason are correct and the reason explains the assertion.