Question

If the curve $x^3-x^2(\sin \theta \cos \theta +\sin \theta +1)+x({\sin }^2\theta \cos \theta +\sin \theta \cos \theta +\sin \theta )-{\sin }^2\theta \cos \theta$ has roots $x_1,x_2\&x_3$ , then -

• A. $x_1^2+\left(\frac{x_1^2}{x_2^2}\right)+x_3^2=2\forall \theta \in R$only for $3$ set of value of $x_1,x_2\&x_3$
• B. The number of values of $\theta \in [-\pi ,\pi ]$ for which $2$ different roots exists is $4$
• C. There are two different roots of the equation for $\theta \in [-\pi ,\pi ]$
• D. The greatest possible difference between two roots for $\theta \in [-\pi ,\pi ]$ is $2$

Answer 1

Answer: (B), (D)

The correct options are
B The number of values of $\theta \in [-\pi ,\pi ]$ for which $2$ different roots exists is $4$
D The greatest possible difference between two roots for $\theta \in [-\pi ,\pi ]$ is $2$

$f\left(x\right)=x^3-x^2(\sin \theta \cos \theta +\sin \theta +1)+x({\sin }^2\theta \cos \theta +\sin \theta \cos \theta +\sin \theta )-{\sin }^2\theta \cos \theta$
$f\left(1\right)=0$
$\Rightarrow \frac{x^3-x^2(\sin \theta \cos \theta +\sin \theta +1)+x({\sin }^2\theta \cos \theta +\sin \theta \cos \theta +\sin \theta )-{\sin }^2\theta \cos \theta }{(x-1)}=0\Rightarrow (x-1)(x-\sin \theta \cos \theta )(x-\sin \theta )=0$
The roots are
$x=1,\sin \theta \cos \theta ,\sin \theta$
For the given condition,
$x_1^2+\left(\frac{x_2^2}{x_1^2}\right)+x_3^2=2$
When
$x_1=\sin \theta ,x_2=\sin \theta \cos \theta \&x_3=1$
For $\theta \in R$ there exist infinite triplets of $x_1,x_2,x_3$

For two different roots to exist -
$\sin \theta =\sin \theta \cos \theta \ne 1\Rightarrow \theta =0,-\pi ,\pi$

$\sin \theta \cos \theta =1\Rightarrow \sin 2\theta =2(\text{not possible}$

$\sin \theta =1\Rightarrow \theta =\frac{\pi }{2}$
$4$ values of $\theta \in [-\pi ,\pi ]$ exists for which $2$ different roots exists.

For
$\theta \in (-\pi ,\pi )-\{0,\frac{\pi }{2}\}$ the equation has $3$ different roots.
The greatest possible difference between two roots if $\theta \in [-\pi ,\pi ]$ is
$|1-\sin \theta |\le 2|1-\frac{\sin 2\theta }{2}|\le 2|\frac{\sin 2\theta }{2}-\sin \theta |\le 2$