| Assertion [A]: One root of \[{{x}^{3}}-2{{x}^{2}}-1=0\] and lies between 2 and 3. |
| Reason [R]: If \[f\left( x \right)\] is continuous function and f [a], f[b] have opposite signs then at least one or odd number of roots of \[f\left( x \right)=0\] lies between a and b. |
| [A] Statement-1 is true, Statement-2 is true; statement-I 2 is a correct explanation of statement-1. |
| [B] Statement-1 is true, Statement-2 is true, statement- 2 is not correct explanation of Statement-1 |
| [C] Statement-1 is true, Statement-2 is false |
| [D] Statement-1 is false, Statement-2 is True. |
A) Both A and R are individually true and R is the correct explanation of A.
B) Both A and R are individually 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
E) Both A and R are false.
Correct Answer: A
Solution :
Given \[f(x)={{x}^{3}}-2{{x}^{2}}-1=0\] Here, \[f(2)={{(2)}^{3}}-2{{(2)}^{2}}-1=8-8-1=-1\] and \[f(3)={{(3)}^{3}}-2{{(3)}^{2}}-1=27-18-1=8\] \[\therefore f(2)\,\,\,f(3)=(-1)\,\,8=-8<0\] \[\Rightarrow \] One root of \[f\left( x \right)\]lies between 2 and 3 \[\therefore \] Given Assertion is true Also Reason R is true and valid reason \[\therefore \] Both A and R are correct and R is correct explanation of A.
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