Question

If a is any real number,the number of roots of cot x =a in the first quadrant is (are).

• A. 2
• B. 0
• C. 1
• D. none of these

Answer 1

The correct option is C

1

$Given:cotx-tanx=a\Rightarrow \frac{1}{tanx}-tanx=a\Rightarrow 1-ta{n}^{2}x=atanx\Rightarrow ta{n}^{2}x+atanx-1=0\text{It is a quadratic equation.}Iftanx=z,thentheequationbecomes{z}^2+az-1=0\Rightarrow z=\frac{-a\pm \sqrt{a^2+4}}{2}\Rightarrow tanx=\frac{-a\pm \sqrt{a^2+4}}{2}\Rightarrow x=ta{n}^{-1}\left(\frac{-a\pm \sqrt{a^2+4}}{2}\right)\text{There are two roots of the given equation,but we need to find the number of roots in the first quadrant.}\text{that is exactly one root of the equation,}thatis,x=ta{n}^{-1}\left(\frac{-a\pm \sqrt{a^2+4}}{2}\right)$