In $\triangle ABC$,with usual notations,if $a, b, c$ are in $A.P.$,then $a \cos^2\left(\frac{C}{2}\right) + c \cos^2\left(\frac{A}{2}\right) = $

The set of values of $a$ for which the equation $\cos\, 2x + a\, \sin\, x = 2a - 7$ possesses a solution is:

Observe the following statements:
$(I)$ In $\triangle ABC$,$b \cos^2 \frac{C}{2} + c \cos^2 \frac{B}{2} = s$
$(II)$ In $\triangle ABC$,$\cot \frac{A}{2} = \frac{b+c}{a} \implies B = 90^{\circ}$
Which of the following is correct?

If $A$ is the solution set of the equation $\cos ^2 x = \cos ^2 \frac{\pi}{6}$ and $B$ is the solution set of the equation $\cos ^2 x = \log _{16} P$ where $P + \frac{16}{P} = 10$,then $B - A =$

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

In a $\triangle ABC$,with usual notation,match the items in List-$I$ with the items in List-$II$ and choose the correct option.

List-$I$	List-$II$
$(A) \ r_1 r_2 \sqrt{\frac{4R-r_1-r_2}{r_1+r_2}}$	$1. \ b$
$(B) \ \frac{r_2(r_3+r_1)}{\sqrt{r_1r_2+r_2r_3+r_3r_1}}$	$2. \ a^2, b^2, c^2 \text{ are in } AP$
$(C) \ \frac{a}{c} = \frac{\sin(A-B)}{\sin(B-C)}$	$3. \ \Delta$
$(D) \ bc \cos^2 \frac{A}{2}$	$4. \ R r_1 r_2 r_3$
	$5. \ s(s-a)$