Question

If the points $(\alpha ,{\alpha }^2)$ and (1, 2) lie in the same angular region between the lines $3x+4y-1=0$ and $2x+y-3=0$ then

• A. $\alpha <-3or\alpha >1$
• B. $-3\le a\le 1$
• C. $\alpha <\frac{1}{2}or\alpha >1$
• D. none of these

Answer 1

The correct option is A $\alpha <-3or\alpha >1$

Given points $(\alpha ,{\alpha }^2)$ lies between the lines

$3x+4y-1=0----\left(1\right)$

$2x+y-3=0----\left(2\right)$

Hence

$3\alpha +4{\alpha }^2-1>0$

$4{\alpha }^2+4\alpha -\alpha -1>0$

$\alpha >-1,\frac{1}{4}$

Hence $\alpha >1$

$2\alpha +{\alpha }^2-3<0$

${\alpha }^2+3\alpha -\alpha -3>0$

$\alpha >-3$

Hence $\alpha <-3$

So $\alpha <-3$ or $\alpha >1$