Question

lf the roots of quadratic equation $x^2+px+q=0$ are $\tan {30}^0$ and $\tan {15}^0$ respectively, then the value of $2+q-p$ is:

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is D $3$

Given quadratic equation is $x^2+px+q=0$

$\therefore \tan {30}^0+\tan {15}^0=\frac{-p}{1}=-p$

$\Rightarrow \tan {30}^0\tan {15}^0=\frac{q}{1}=q$

$\Rightarrow \frac{\tan {30}^0+\tan {15}^0}{1-\tan {30}^0\tan {15}^0}=\tan ({30}^0+{15}^0)$

$=\tan {45}^0$

$\Rightarrow \frac{-p}{1-q}=1$

$\Rightarrow -p=1-q$

$\Rightarrow q-p=1$

$\therefore 2+q-p=3$