In a $\Delta ABC$,$\angle A = 30^\circ$ and $a = 8 \text{ cm}$,then the distance of the orthocentre from vertex $A$ is equal to (with usual notations).

In a $\Delta ABC$,the expression $\frac{\cos C + \cos A}{c + a} + \frac{\cos B}{b}$ is equal to:

In a triangle $PQR$,let $\angle PQR = 30^{\circ}$ and the sides $PQ$ and $QR$ have lengths $10\sqrt{3}$ and $10$,respectively. Then,which of the following statement$(s)$ is (are) $TRUE$?
$(A)$ $\angle QPR = 45^{\circ}$
$(B)$ The area of the triangle $PQR$ is $25\sqrt{3}$ and $\angle QRP = 120^{\circ}$
$(C)$ The radius of the incircle of the triangle $PQR$ is $10\sqrt{3} - 15$
$(D)$ The area of the circumcircle of the triangle $PQR$ is $100\pi$

If $(a+b) \cos C + (b+c) \cos A + (c+a) \cos B = 72$ and if $a = 18, b = 24$,then the area of the triangle $ABC$ is

With usual notations,if in $\triangle ABC$,$s$ is the semi-perimeter and $(s-a)(s-b)=s(s-c)$,then $\triangle ABC$ is

In a $\Delta PQR$,if $3 \sin P + 4 \cos Q = 6$ and $4 \sin Q + 3 \cos P = 1$,then the angle $R$ is equal to: