Question

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

• A. 2
• B. 3
• C. 1

Answer 1

The correct option is B 3

$x^2+px+q=0$ has roots $tan{30}^{\circ }tan{15}^{\circ }$.
$\therefore tan{30}^{\circ }+tan{15}^{\circ }=-p.....\left(1\right)$
and $tan{30}^{\circ }tan{15}^{\circ }=q......\left(2\right)$
Now $tan{45}^{\circ }=tan({30}^{\circ }+{15}^{\circ })$
$\Rightarrow 1=\frac{tan{30}^{\circ }+tan{15}^{\circ }}{1-tan{30}^{\circ }tan{15}^{\circ }}$
$\Rightarrow 1=\frac{-p}{1-q}$ [Using(1) and (2)]
$\Rightarrow 1-q=-p\Rightarrow q-p=1$
$\Rightarrow 2+q-p=3$