SECTION

In trapezium$ABCD,$$AB\parallel DC$, $DC=2AB$. $EF$ drawn parallel to $AB$ cuts $AD$ in $F$ and $BC$ in $E$ such that $\frac{BE}{EC}=\frac{3}{4}$. Diagonal$DB$ intersects $EF$ at $G$. Prove that $7FE=10AB$.

Draw a circle of radius $4 \mathrm{cm}$. Construct a pair of tangents to it, the angle between which is $60°$. Also justify the construction. Measure the distance between the centre of the circle and the point of intersection of tangents.

If $a\mathrm{sin\theta }+b\mathrm{cos\theta }=c$, then prove that $a\mathrm{cos\theta }-b\mathrm{sin\theta }=\pm \sqrt{a^2+b^2-c^2}$.

Prove that $\frac{\mathrm{tan\theta }}{1-\mathrm{tan\theta }}-\frac{\mathrm{cot\theta }}{1-\mathrm{cot\theta }}=\frac{\mathrm{cos\theta }{\displaystyle +}{\displaystyle \sin }\mathrm{\theta }}{\mathrm{cos\theta }{\displaystyle -}{\displaystyle \sin }\mathrm{\theta }}$.

A straight highway leads to the foot of the tower. A man standing at the top of the tower observes the car at an angle of depression of$30°$, which is approaching the foot of the tower with the uniform speed. Six seconds later, the angle of depression of the car found is to be$60°$. Find the time taken (in seconds) by the car to reach the foot of the tower from this point.

A container shaped like a right circular cylinder having diameter$12 \mathrm{cm}$ and height $15 \mathrm{cm}$ is full of ice cream. The ice cream is to be filled into cones of height $12 \mathrm{cm}$ and diameter $6 \mathrm{cm}$, having a hemispherical shape on the top. Find the number of such cones which can be filled with ice cream.

In Figure; $PSR$, $RTQ$ and $PAQ$ are three semicircles of diameter $10 \mathrm{cm}, 3 \mathrm{cm}$ and $7 \mathrm{cm}$ respectively. Find the perimeter of the shaded region.(Use $\mathrm{\pi }=3.14$)

In a class test, marks obtained by $120$ students are given in the following frequency distribution. If it is given that mean is $59$, find the missing frequencies $x$ and $y$.