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A tower \( PQ \) stands on a horizontal ground with base \( Q \) on the ground. The point \( R \) divides the tower in two parts such that \( QR = 15 \, \text{m} \). If from a point \( A \) on the ground the angle of elevation of \( R \) is \( 60^\circ \) and the part \( PR \) of the tower subtends an angle of \( 15^\circ \) at \( A \), then the height of the tower is :

Let \( \dfrac{\sin A}{\sin B} = \dfrac{\sin(A - C)}{\sin(C - B)} \) where \( A, B, C \) are angles of a triangle \( ABC \). If the lengths of the sides opposite these angles are \( a, b, c \) respectively, then:

For a triangle \( \triangle ABC \), the value of \( \cos 2A+\cos 2B+\cos 2C \) is least. If its inradius is \( 3 \) and incentre is \( M \), then which of the following is NOT correct?

In \( \triangle ABC \), the lengths of sides \( AC \) and \( AB \) are \( 12 \, \text{cm} \) and \( 5 \, \text{cm} \), respectively. If the area of \( \triangle ABC \) is \( 30 \, \text{cm}^{2} \) and \( R \) and \( r \) are respectively the radii of circumcircle and incircle of \( \triangle ABC \), then the value of \( 2R+r \) (in \( \text{cm} \)) is equal to ____________.

A spherical gas balloon of radius \( 16 \) meter subtends an angle \( 60^\circ \) at the eye of the observer \( A \) while the angle of elevation of its center from the eye of \( A \) is \( 75^\circ \). Then the height (in meter) of the top most point of the balloon from the level of the observer's eye is: