question_answer

If \[(a,{{a}^{2}})\] falls inside the angle made by the lines \[y=\frac{x}{2},x>0\]and\[y=3x,x>0,\], then a belongs to

A) \[(3,\infty )\]                    

B) \[\left( \frac{1}{2},3 \right)\]

C) \[\left( -3,-\frac{1}{2} \right)\]                  

D) \[\left( 0,\frac{1}{2} \right)\]

Correct Answer: B

Solution :

Since, the given point \[(a,{{a}^{2}})\]  lies inside the  angle between two lines. Then, \[{{a}^{2}}-\frac{a}{2}>0\] and \[{{a}^{2}}-3a<0\] \[\Rightarrow \]               \[a\in (-\infty ,0)\cup \left( \frac{1}{2},\infty  \right)\]and \[a\in (0,3)\]  \[\Rightarrow \]              \[a\in \left( \frac{1}{2},3 \right)\]