The number of solutions of $\sin ^2 x + (2 + 2x - x^2) \sin x - 3(x - 1)^2 = 0$,where $-\pi \leq x \leq \pi$,is....................

In a triangle $ABC$,with usual notations,$\frac{\cos B+\cos C}{b+c}+\frac{\cos A}{a}$ has the value

In a triangle $ABC$ with a fixed base $BC$,the vertex $A$ moves such that $\cos B + \cos C = 4 \sin^2 \frac{A}{2}$. If $a, b,$ and $c$ denote the lengths of the sides of the triangle opposite to the angles $A, B,$ and $C$,respectively,then:
$(A) b+c=4a$
$(B) b+c=2a$
$(C) \text{locus of point } A \text{ is an ellipse}$
$(D) \text{locus of point } A \text{ is a pair of straight lines}$

If in a $\Delta ABC$,$\cos A \cos B + \sin A \sin B \sin^2 C = 1$,then the statement which is incorrect is:

If in a triangle $ABC$,$\cos A \cos B + \sin A \sin B \sin C = 1$,then $a : b : c =$

In a right-angled triangle,if the difference between the two acute angles is $60^{\circ}$,then the ratio of the length of the hypotenuse to the length of the perpendicular drawn to the hypotenuse from its opposite vertex is: (in $: 1$)