If the equation $2 \sin^2 x + \frac{\sin 2x}{2} = k$ has at least one real solution,then the sum of all integral values of $k$ is

Which of the following is true?

Consider a triangle $PQR$ having sides of lengths $p, q$ and $r$ opposite to the angles $P, Q$ and $R$,respectively. Then which of the following statements is (are) $TRUE$?
$(A)$ $\cos P \geq 1-\frac{p^2}{2qr}$
$(B)$ $\cos R \geq \left(\frac{q-r}{p+q}\right) \cos P + \left(\frac{p-r}{p+q}\right) \cos Q$
$(C)$ $\frac{q+r}{p} < 2 \frac{\sqrt{\sin Q \sin R}}{\sin P}$
$(D)$ If $p < q$ and $p < r$,then $\cos Q > \frac{p}{r}$ and $\cos R > \frac{p}{q}$

In $\triangle ABC$, if $R = \frac{65}{8}$, $rr_1 = 42$ and $r_1 - r = 6.5$, then $s(s-a) = $

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

Let $P_1, P_2, P_3$ be the altitudes of a triangle $ABC$ from the vertices $A, B, C$ respectively. If $\Delta$ denotes the area of the triangle and $s$ is the semi-perimeter of the triangle, then $\frac{\cos A}{P_1} + \frac{\cos B}{P_2} + \frac{\cos C}{P_3} = $