| Assertion (A): If \[\alpha ,\beta ,\gamma \] are the zeroes of \[{{x}^{3}}-2{{x}^{2}}+qx-r\]and \[\alpha +\beta =0,\] then \[2q=r\]. |
| Reason (R): If a, p, y are the zeroes of \[a{{x}^{3}}+b{{x}^{2}}+cx+d,\]then |
| \[\alpha +\beta +\gamma =-\frac{b}{a}\] |
| \[\alpha \beta +\beta \gamma +\gamma \alpha =\frac{c}{a}\] |
| \[\alpha \beta \gamma =-\frac{d}{a}\] |
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: A
Solution :
| [a] Clearly, reason is true. (Standard result) |
| \[\alpha +\beta +\gamma =-(2)=2\] |
| \[0+\gamma =2\,\,\,\,\Rightarrow \,\,\,\,\gamma =2\] |
| \[\alpha \beta \gamma =-(-r)=r\] |
| \[\alpha \beta (2)=r\] |
| \[\alpha \beta =\frac{r}{2}\] |
| \[\alpha \beta +\beta \gamma +\gamma \alpha =q\] |
| \[\frac{r}{2}+\gamma (0)=q\] |
| \[r=2q\] |
| Assertion is true. Since, reason gives assertion. |
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