Question

The number of real roots of the equation $\csc \theta +\sec \theta -\sqrt{15}=0$ lying in $[0,\pi ]$ is

• A. $3$
• B. $8$
• C. $4$
• D. $0$

Answer 1

The correct option is A $3$

We have,

$f\left(\theta \right)=\csc \theta +\sec \theta -\sqrt{15}$

Now,

$f^{\prime }\left(\theta \right)=0$

$\Rightarrow \csc \theta +\sec \theta =\sqrt{15}$

Let

$g\left(\theta \right)=csc\theta +sec\theta$

So, the number of soution $f\left(\theta \right)=0$ is same as number of points of intersection of the curve

$y=sec\theta +csc\theta$

$andy=\sqrt{15}$

It is horizontal line.

Now,

$g^{\prime }\left(\theta \right)=-csc\theta cot\theta +sec\theta tan\theta$

$=\frac{-\cos \theta }{{\sin }^2\theta }+\frac{\sin \theta }{{\cos }^2\theta }$

$=\frac{{\sin }^3\theta -{\cos }^3\theta }{{\sin }^2\theta {\cos }^2\theta }$

So,

$If,\theta \in (0,\frac{\pi }{4},)\Rightarrow g^{\prime }\left(\theta \right)<0$

$If,\theta \in (\frac{\pi }{4},\frac{\pi }{2})\Rightarrow g^{\prime }\left(\theta \right)>0$

$If,\theta \in (\frac{\pi }{2},\pi )\Rightarrow g^{\prime }\left(\theta \right)>0$

Draw the horizontal line $y=\sqrt{15}(\simeq 3.9)$

So, there are three solutions $(0,\pi )$.

Hence, this is the answer.