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. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

Answer: (D)

$3$

Since,$\tan {30}^{\circ }$ and $\tan {15}^{\circ }$ are the roots of equation

$x^2+px+q=0$

$x^2+px+q=0$

$\therefore \tan {30}^{\circ }+\tan {15}^{\circ }=-p$

and $\tan {30}^{\circ }\tan {15}^{\circ }=q$

Now,$2+q-p=2+\tan {30}^{\circ }\tan {15}^{\circ }+\tan {30}^{\circ }+\tan {15}^{\circ }$

We know that $\tan {45}^{\circ }=\frac{\tan {30}^{\circ }+\tan {15}^{\circ }}{1-\tan {15}^{\circ }\tan {30}^{\circ }}$

$\Rightarrow 2+q-p=2+\tan {30}^{\circ }\tan {15}^{\circ }+1-\tan {30}^{\circ }\tan {15}^{\circ }$

$\Rightarrow 2+q-p=3$