CDS Mathematics Mensuration Medium Questions

Let the area of the largest possible square inscribed in a circle of unit radius be $x$. Let the area of the largest possible circle inscribed in a square of unit side length be $y$. What is the relation between $x$ and $y$?

A Question is given followed by two Statements I and II. Consider the Question and the Statements.

Question :

$ABCD$ is a parallelogram with $\angle ABC=60^{\circ }$. If the area of the parallelogram is $7\sqrt{3}$ square units, then what is the perimeter of the parallelogram?

Statement-I :

The lengths of the sides $AB$ and $DA$ are prime numbers.

Statement-II :

The lengths of the sides are natural numbers each greater than 1 unit.

Which one of the following is correct in respect of the above Question and the Statements?

A wire is in the form of an equilateral triangle with an area of $36\sqrt{3}{\mathrm{cm}}^2$. If it is changed into a semicircle, then what is its radius?

The difference between an interior angle and an exterior angle of a regular polygon is $120^{\circ }$. What is the number of sides of the polygon?

A square and a rectangle have the same perimeter $p$ and their areas differ by $q$ units. What is the square of the difference between the length and breadth of the rectangle?

A sphere of diameter $14$ cm is dropped into a cylindrical vessel partly filled with water. The radius of the vessel is $14$ cm. If the sphere is completely submerged in water, then how much will the level of water rise?

The volume of a cube is $8{\text{cm}}^3$ and is equal to the volume of a cuboid of length $x$, breadth $y$ and height $z$, where $x$, $y$ and $z$ are natural numbers ($x>y>z$). Let $n$ be the number of such cuboids with different dimensions. What is the value of $n$?

Let $x,y,z$ (all prime) be the length, breadth and height respectively of a cuboid with $x>y>z$. The volume of the cuboid is $30k^3$ cubic units, where $k$ is a natural number. What is the total surface area of the cuboid?

The sum of the height and the radius of a right circular cylinder is 21 cm, and the radius is less than the height. If the curved surface area of the cylinder is 616 $c{m}^2$, then what is the volume of the cylinder? (Take $\pi =22/7$)

The length, breadth and height of a cuboid are consecutive integers. If the volume of the cuboid is 336 cubic units, then what is the total surface area of the cuboid?

A conical tent has an angle of $60^{\circ }$ at the vertex. If the curved surface area is $100{\mathrm{m}}^2$, then what is the volume of the tent?

Let $x$ be the length of a diagonal of a face of a cube and $y$ be the length of a diagonal of the cube. If $x+y=\left(5+2\sqrt{6}\right)$ units, then what is the total surface area of the cube ?

A right circular cone and a hemisphere have equal base and equal volume. What is the ratio of the height of the cone to the radius of the hemisphere?