Question

If the roots of 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. 3
• B. 0
• C. 1
• D. 2

Answer 1

Answer: (C)

1

Let $\tan {30}^{\circ }$ and $\tan {15}^{\circ }$ be the roots of $x^2+px+q=0$

Sum of the roots$=\tan {30}^{\circ }+\tan {15}^{\circ }=-p$

Product of the roots$=\tan {30}^{\circ }\tan {15}^{\circ }=q$

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

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

$\Rightarrow 1-q+p=0$

$\Rightarrow 1+1+p-q=1$

$\therefore p-q+2=1$