Question

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. $(\frac{1}{2},3)$
• C. $(-3,-\frac{1}{2})$
• D. $(0,\frac{1}{2})$

Answer 1

The correct option is B $(\frac{1}{2},3)$

Since, the given point $(a,a^2)$ lies inside the angle between two lines.

Then, $a^2-\frac{a}{2}>0$ and $a^2-3a<0$

$\Rightarrow a(a-\frac{1}{2})>0$ and $a(a-3)<0$

$\Rightarrow a>0\text{and}a>\frac{1}{2}$ or $a<0\text{and}a<\frac{1}{2}$

and $a<0,a>3$ or $a>0,a<3$

$\Rightarrow a>\frac{1}{2}$ or $a<0$ and $0<a<3$

$\Rightarrow a\in (-\infty ,0)\cup (\frac{1}{2},\infty )$ and $a\in (0,3)$

$\Rightarrow a\in (\frac{1}{2},3)$