A) \[\left( \frac{4}{3},2 \right)\]
B) \[(2,\infty )\]
C) \[\left( -1,-\frac{4}{5} \right)\]
D) \[(-\infty ,-1)\]
Correct Answer: D
Solution :
Idea For a quadratic equation \[a{{x}^{2}}+bx+c=0\]if roots are real then, \[{{b}^{2}}-4ac\ge 0\]We have given that \[4{{x}^{2}}-20\lambda x+25{{\lambda }^{2}}+15\lambda -66=0\] \[{{(2x-5\lambda )}^{2}}=66-15\lambda \] The roots are\[x=\frac{5\lambda \pm \sqrt{66-15\lambda }}{2}\] Discriminate \[\ge 0\] \[\therefore \] \[66-15\lambda \ge 0\Rightarrow \lambda \le \frac{22}{5}\] ?(i) \[\frac{5\lambda +\sqrt{66-15\lambda }}{2}<2\] \[\Rightarrow \] \[5\lambda +\sqrt{66-15\lambda }<4\] \[\Rightarrow \] \[\sqrt{66-15\lambda }<4-5\lambda \] \[\Rightarrow \] \[66-15\lambda <16-40\lambda +25{{\lambda }^{2}}\] \[\Rightarrow \] \[{{\lambda }^{2}}-\lambda -2>0\] \[\Rightarrow \] \[(\lambda +1)(\lambda -2)>0\] ?(ii) \[\Rightarrow \] \[\lambda \in (-\infty .-1)\cup (2,\infty )\] ?(iii) From above equations \[-\infty <\lambda <-1\] \[\therefore \] \[\lambda \in (-\infty ,-1)\] TEST Edge Relation between the roots and nature of roots. Maximum and minimum values of \[a{{x}^{2}}+bx+c=0\] related questions are asked. To solve these questions, students learn the formulae of above concept and acquainted yourself for wary curve method.
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