If $2 \sin x - \cos 2x = 1$,then $(3 - 2 \sin^2 x) = $

Let $PQRS$ be a quadrilateral in a plane,where $QR = 1$,$\angle PQR = \angle QRS = 70^{\circ}$,$\angle PQS = 15^{\circ}$ and $\angle PRS = 40^{\circ}$. If $\angle RPS = \theta^{\circ}$,$PQ = \alpha$ and $PS = \beta$,then the interval$(s)$ that contain$(s)$ the value of $4 \alpha \beta \sin \theta^{\circ}$ is/are
$(A)$ $(0, \sqrt{2})$
$(B)$ $(1, 2)$
$(C)$ $(\sqrt{2}, 3)$
$(D)$ $(2 \sqrt{2}, 3 \sqrt{2})$

For a $\Delta ABC$,if $a \cos^2 \frac{C}{2} + c \cos^2 \frac{A}{2} = \frac{3b}{2}$,then the sides $a, b, c$ are in:

In a $\triangle ABC$,if $r: R: r_2 = 1: 3: 7$,then $\sin(A+C) + \sin B = $

In $\Delta ABC$,$(a - b)^2 \cos^2 \frac{C}{2} + (a + b)^2 \sin^2 \frac{C}{2} = $

The sides $a, b, c$ of a triangle satisfy the relations $c^2=2ab$ and $a^2+c^2=3b^2$. Then the measure of $\angle BAC$,in degrees,is