Question

If $\tan A,\tan B,\tan C$ are the roots of the equation $2x^3-5x^2-2x+4=0,$ then the value of $\tan (A+B+C)$ is

• A. $\frac{7}{4}$
• B. $\frac{1}{4}$
• C. $\frac{9}{4}$
• D. undefined

Answer 1

The correct option is C $\frac{9}{4}$

Given, $\tan A,\tan B,\tan C$ are the roots of the equation $2x^3-5x^2-2x+4=0$
So,
$\sum \tan A=\tan A+\tan B+\tan C=\frac{5}{2}$
$\sum \tan A\tan B=\tan A\tan B+\tan B\tan C+\tan C\tan A=-1$
$\tan A\tan B\tan C=-2$

Now,
$\tan (A+B+C)=\frac{\sum \tan A-\tan A\tan B\tan C}{1-\sum \tan A\tan B}=\frac{\frac{5}{2}-(-2)}{1+1}=\frac{9}{4}$