Jee Main 2024 27 01 2024 Shift 2 - Question9
Question 9
Let $R$ be the interior region between the lines $3 x-y+1=0$ and $x+2 y-5=0$ containing the origin. The set of all values of ’ $a$ ’ for which the points $\left(a^{2}, a+1\right)$ lies in $R$ is
(1) $(-\infty,-1) \cup(3, \infty)$
(2) $(-3,0) \cup\left(\frac{1}{3}, 1\right)$
(3) $(-\infty,-1) \cup\left(0, \frac{1}{3}\right)$
(4) $(-\infty,-2) \cup\left(0, \frac{1}{3}\right)$
Show Answer
Answer: (2)
Solution:
$R$ is the shaded region where $\left(a^{2}, a+1\right)$ should lie.
For line $L _1$,
$$ \begin{align*} & \therefore \quad a^{2}+2(a+1)-5<0 \\ & a^{2}+2 a-3<0 \\ &(a+3)(a-1)<0 \\ & \Rightarrow a \in(-3,1) \tag{1} \end{align*} $$
Also, for line $L _2$
$3 a^{2}-a-1+1>0$
$3 a^{2}-a>0$
$a(3 a-1)>0$
$a \in(-\infty, 0) \cup\left(\frac{1}{3}, \infty\right)$
$\therefore(1) \cap(2)$
$a \in(-3,0) \cup\left(\frac{1}{3}, 1\right)$
BETA
