If a person travels from a point $L$,towards east for $12 \ km$ and then travels $5 \ km$ towards north and reaches a point $M$,then the shortest distance from $L$ to $M$ is: (in $km$)

From the top of a $60 \ m$ high lighthouse with its base at sea level,the angle of depression of a boat is $15^{\circ}$. The distance of the boat from the lighthouse is

$A$ person of height $6 \ ft$ wants to pluck a fruit which is on a $\frac{26}{3} \ ft$ high tree. If the person is standing $\frac{8}{\sqrt{3}} \ ft$ away from the base of the tree,then at what angle should he throw a stone so that it hits the fruit (in $^{\circ}$)?

Two posts are $x \text{ m}$ apart and the height of one is double that of the other. If from the midpoint of the line joining their feet,an observer finds the angular elevations of their tops to be complementary,then the height (in $\text{m}$) of the shorter post is

The angle of elevation of an aeroplane from a point on the ground is $60^{\circ}$. After flying for $30$ seconds, the angle of elevation changes to $30^{\circ}$. If the aeroplane is flying at a constant height of $4500 \text{ m}$, what is the speed (in $\text{m/s}$) of the aeroplane (in $\sqrt{3}$)?

At a point,$15 \, m$ away from the base of a $15 \, m$ high house,the angle of elevation of the top is (in $^{\circ}$)