Question

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

• A. $1$
• B. $2$
• C. $3$
• D. $0$

Answer 1

The correct option is C $3$

Roots of $x^2+px+q=0$ are $\tan {30}^o$ and $\tan {15}^o$

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

Product of roots $=q=\tan {30}^o.\tan {15}^o$

$\tan ({30}^o+{15}^o)=\frac{\tan {30}^o+\tan {15}^o}{1-\tan {30}^o\tan {15}^o}=\frac{-p}{1-q}$

$\tan \left({45}^o\right)=1=\frac{-p}{1-q}$

$\Rightarrow 1-q+p=0$

$\Rightarrow -p+q-1=0$

Adding $3$ on both sides

$\Rightarrow -p+q-1+3=3$

$\Rightarrow 2+q-p=3$