If (a, a2) falls inside the angle made by the lines y=(x/2), x 0 and y=3x, x 0, then a belongs to:

Question

If (a, a2) falls inside the angle made by the lines y=(x/2), x> 0 and y=3x, x> 0, then a belongs to: (A) (3, ∞) (B) ((1/2), 3) (C) (-3, -(1/2)) (D) (0, (1/2))

Tardigrade Admin · Accepted Answer

The graph of equations x-2y = 0 and 3x - y = 0 is as shown in the figure. Since given point (a, a2) lies in the shaded region. Then a-2a2 < 0 and 3a-a2 > 0 ⇒ a ∈ (-∞, 0) ∪((1/2), ∞) and a ∈ (0,3) ⇒ a ∈ ((1/2), 3)

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

$(3, \infty)$

$(\frac{1}{2}, 3)$

$(- 3, - \frac{1}{2})$

$(0, \frac{1}{2})$

The graph of equations $x - 2 y = 0$ and $3 x - y = 0$ is as shown in the figure. Since given point $(a, a^{2})$ lies in the shaded region.
Then $a - 2 a^{2} < 0$ and $3 a - a^{2} > 0$
$\Rightarrow a \in (- \infty, 0) \cup (\frac{1}{2}, \infty)$
and $a \in (0, 3) \Rightarrow a \in (\frac{1}{2}, 3)$

Q. If (a,a2) falls inside the angle made by the lines y=2x​,x>0 and y=3x,x>0, then a belongs to:

(3,∞)

(21​,3)

(−3,−21​)

(0,21​)