FIRST SUMMATIVE TEST

The ratio of two positive integers is $2:3$. If $3$ is added to each number then the product of the numbers so obtained is $108$. Find the numbers.

The length of the hypotenuse of a right-angled triangle is $13 \mathrm{cm}$ and the difference of the other two sides is $7 \mathrm{cm}$. Find the length of those two sides.

If $q$ is the mean proportion of $p$ and $r$, then show that, ${\left(\frac{pq+qr+rp}{p+q+r}\right)}^3=pqr$.

Simplify: $\frac{3\sqrt{7}}{\sqrt{5}+\sqrt{2}}-\frac{5\sqrt{5}}{\sqrt{2}+\sqrt{7}}+\frac{2\sqrt{2}}{\sqrt{7}+\sqrt{5}}$

Prove that, two equal chords of a circle are equidistant from the centre.

Prove that, on the same arc of a circle, the angle at the centre is twice the angle on the circumference.

The inside length, breadth, and height of a water container are in the ratio $6:5:4$. The container is in the shape of a rectangular parallelepiped and its maximum capacity is $50625 \mathrm{litre}$. What is the depth (in $\mathrm{dm}$) of the container? [$1 \mathrm{cube} \mathrm{decimetre}=1 \mathrm{litre}$]

A right circular cylindrical tank $9 \mathrm{metres}$ high is full of water. If water is removed by a pipe of diameter $6 \mathrm{cm}$with a velocity of $225 \mathrm{metres} \mathrm{per} \mathrm{minute}$, the total water is removed in $36 \mathrm{minutes}$. Find the diameter of the tank.