Question

Assertion (A) : One root of $x^3-2x^2-1=0$ lies between 2 and 3.
Reason (R) : If $f\left(x\right)$ is continuous function and $f\left(a\right),f\left(b\right)$ have opposite sings then atleast one root of $f\left(x\right)=0$ lies between $a$ and $b$.

• A. Both A and R are true and R is the correct explanation of A
• B. Both A and R are true and R is not the correct explanation of A
• C. A is true but R is false
• D. A is false but R is true

Answer 1

The correct option is A Both A and R are true and R is the correct explanation of A

From above figure, if $f\left(a\right)$ and $f\left(b\right)$ are of opp. signs then clearly one root lies between a and b

$f\left(x\right)=x^3-2x^2-1=0$

$f\left(2\right)={\left(2\right)}^3-2\times {\left(2\right)}^2-1=-1$

$f\left(3\right)={\left(3\right)}^3-2\times {\left(3\right)}^2-1=8$

$f\left(2\right)$ and $f\left(3\right)$ are opposite signs, thus there is a root between $2$ and $3$ of $f\left(3\right)$

Thus, Assertion is true and Reason is true and Reason is the correct explanation of Assertion.