Geometry — CDS Elementary Mathematics PYQs

$C$ is the centre of a circle of radius 20 cm. $AB$ is a chord of length 32 cm. $E$ is a point on $AB$ such that $CE=13$ cm. What is $AE\times EB$ equal to?

In a triangle $ABC$, $AB=2$ cm, $BC=4$ cm and $AC=3$ cm. The bisector of angle $A$ meets $BC$ at $D$ and the bisector of angle $B$ meets $AD$ at $E$. What is $AE:ED$ equal to?

In a triangle $ABC$, the bisector of angle $A$ cuts $BC$ at $D$. If $AB+AC=10$ cm and $BD:DC=3:1$, then what is the length of $AC$?

The measure of an angle formed by the bisectors of the angles $A$ and $C$ of the triangle $ABC$ is 130°. What is the measure of the angle $B$?

Consider the following for the next two (02) items that follow:
$ABC$ is a triangle right-angled at $B$. The perimeter of the triangle is 24 cm and the difference between the sum of the perpendicular sides and the hypotenuse is 4 cm.

A circle is inscribed in the triangle. What is its radius?

Consider the following for the next two (02) items that follow:
A circle $M$ of radius 8 cm touches externally with another circle $N$ of radius 16 cm. Let $P$, $Q$ be the points where the common tangent touches the circles $M$ and $N$ respectively.

What is the length of the common tangent $PQ$?

Consider the following for the next two (02) items that follow:
A circle $M$ of radius 8 cm touches externally with another circle $N$ of radius 16 cm. Let $P$, $Q$ be the points where the common tangent touches the circles $M$ and $N$ respectively.

If $U$, $V$ are the centres of the circles $M$ and $N$ respectively, then what is the area of the quadrilateral formed by the points $P$, $Q$, $V$ and $U$?

Consider the following for the next two (02) items that follow:
The perimeter of a triangle $ABC$ is 105 cm. The altitudes $AD$, $BE$ and $CF$ are in the ratio $3:5:6$.

What is $AB:BC:CA$ equal to?

Consider the following for the next two (02) items that follow:
A pot is made from a hollow sphere of inner radius 20 cm by cutting its upper portion horizontally. The height of the pot is 30 cm.

What is the angle made by the line joining the centre of the sphere and any point on the rim of the circular opening with a vertical line passing through the centre?

Consider the following for the next two (02) items that follow:
A hall is of length $l$, breadth $b$ and height $h$. The maximum distance between any two points (say $P$ and $Q$) inside the hall is 14 m, whereas the maximum distance between two points (say $P$ and $R$) on the floor is $6\sqrt{5}$ m.

If $\alpha$ is the angle between the line joining $P$ and $Q$, and the line joining $P$ and $R$, then what is $\cos \alpha$ equal to?