Question

If the point $(\alpha ,0)$ lies inside the quadrilateral formed by lines $2x+5y=15,5x-4y=21,3x+5y+17=0$ and $y=x+3$, then which of the following is $TRUE$ ?

• A. number of prime value(s) of $\alpha$ is $4$.
• B. number of integral value(s) of $\alpha$ is $7$.
• C. minimum integral value of $\alpha$ is $-3$.
• D. maximum integral value of $\alpha$ is $4$.

Answer 1

Answer: (B)

number of integral value(s) of $\alpha$ is $7$.

Since $(\alpha ,0)$ lies inside

So, $\frac{0-\alpha -3}{1}<0\Rightarrow \alpha >-3$

$\frac{2\alpha -15}{5}<0\Rightarrow \alpha <15/2$

$\frac{3\alpha +17}{5}>0\Rightarrow \alpha >-17/3$

$\frac{5\alpha -21}{-4}>0\Rightarrow \alpha <21/5$

So, overall $\alpha ϵ(-3,\frac{21}{5})$

So, no. of prime values of $\alpha$ is $2$

No. of integral values of $\alpha$ is $7$