A) \[(\alpha -\gamma )(\beta -\gamma )(\alpha +\delta )(\beta +\delta )\]
B) \[(\alpha +\gamma )(\beta +\gamma )(\alpha -\delta )(\beta +\delta )\]
C) \[(\alpha +\gamma )(\beta +\gamma )(\alpha +\delta )(\beta +\delta )\]
D) None of these
Correct Answer: A
Solution :
As given, \[\alpha +\beta =-p,\,\,\alpha \beta =1,\gamma +\delta =-q\]and \[\gamma \delta =1\] Now, \[(\alpha -\gamma )(\beta -\gamma )(\alpha +\delta )(\beta +\delta )\] = \[\{\alpha \beta -\gamma (\alpha +\beta )+{{\gamma }^{2}}\}\{\alpha \beta +\delta (\alpha +\beta )+{{\delta }^{2}}\}\] = \[(1+p\gamma +{{\gamma }^{2}})(1-p\delta +{{\delta }^{2}})\]\[=(p\gamma -q\gamma )(-p\delta -q\delta )\](Since \[\gamma \] is a root of \[{{x}^{2}}+qx+1=0\]) Þ \[{{\gamma }^{2}}+q\gamma +1=0\,\,\,\,\Rightarrow {{\gamma }^{2}}+1=-q\gamma \] and similarly \[{{\delta }^{2}}+1=-q\delta \]\[=-\gamma \delta (p-q)(p+q)={{q}^{2}}-{{p}^{2}}\].
You need to login to perform this action.
You will be redirected in
3 sec
