Question

Consider the cubic equation $x^3-(1+\cos \theta +\sin \theta )x^2+(\cos \theta \sin \theta +\cos \theta +\sin \theta )x-\sin \theta \cos \theta =0$
whose roots are $x_1,x_2,x_3$
The number of values of $\theta$ in $[0,2\pi ]$ for which at least two roots are equal

• A. $3$
• B. $4$
• C. $5$
• D. $6$

Answer 1

The correct option is C $5$

Roots are $\sin \theta ,\cos \theta$ and $1$
When $\sin \theta =1$, 2 equal roots are $\sin \theta$ and $1$
When $\cos \theta =1$. 2 equal roots are $\cos \theta$ and $1$
When $\sin \theta =\cos \theta$, 2 equal roots are $\sin \theta$ and $\cos \theta$
Hence $\theta =0,\frac{\pi }{2},2\pi ,\frac{\pi }{4},\frac{5\pi }{4}$