| Assertion (A): If two zeroes of the polynomial \[f(x)={{x}^{3}}-2{{x}^{2}}-3x+6\] are \[\sqrt{3}\] and \[-\sqrt{3},\] then its third zero is 4. |
| Reason (R): If \[\alpha ,\beta \] and \[\gamma \] be the zeroes of the polynomial \[f(x)=a{{x}^{3}}+b{{x}^{2}}+cx+d\]. Then, |
| Sum of the zeroes \[=-\frac{Coefficient\,\,of\,\,{{x}^{2}}}{Coefficient\,\,of\,\,{{x}^{2}}}\] |
A) Both assertion (A) and reason (R) are true and reason (R) is the correct explanation of assertion (A).
B) Both assertion (A) and reason (R) are true but reason (R) is not the correct explanation of assertion (A).
C) Assertion (A) is true but reason (R) is false.
D) Assertion (A) is false but reason (R) is true.
Correct Answer: D
Solution :
| [d] Clearly, reason is true. |
| Let \[\alpha =\sqrt{3},\] \[\beta =-\sqrt{3}\] be the \[\alpha \] zeroes of the given polynomial |
| \[f(x)={{x}^{3}}-2{{x}^{2}}-3x+6\] and \[\gamma \] be the third zero. |
| Then, \[\alpha +\beta +\gamma =-\left( \frac{-2}{1} \right)\] |
| \[\Rightarrow \,\,\,\,\,\,\,\,\,\,\,\,\sqrt{3}-\sqrt{3}+\gamma =2\,\,\,\,\,\,\,\,\,\,\Rightarrow \,\,\,\gamma =2\] |
| \[\therefore \]Assertion is false. |
| \[\therefore \] Assertion: False; Reason: True. |
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