Embibe Experts Solutions for Chapter: UPSC CDS (II) Previous Year Paper 2017, Exercise 1: Elementary Mathematics

Two straight lines $AB$ and $AC$ include an angle. A circle is drawn in this angle which touches both these lines. One more circle is drawn which touches both these lines as well as the previous circle. If the area of the bigger circle is $9$ times the area of the smaller circle, then what must be the angle $A$?

In a triangle $PQR$, $X$ is a point on $PR, Y$ is a point on $QR$ such that $PR=10 \mathrm{cm}, \mathrm{RX}=4 \mathrm{cm}, \mathrm{YR}=2 \mathrm{cm}, \mathrm{QR}=5 \mathrm{cm}$. Which one the following is correct?

Consider the following statements in respect of three straight lines $A, B$ and $C$ on a plane:

$1.$ If $A$ and $C$ are parallel and $B$ and $C$ are parallel, then $A$ and $B$ are parallel.

$2.$ If $A$ is perpendicular to $C$ and $B$ is perpendicular to $C$, then $A$ and $B$ are parallel.

$3.$ If the acute angle between $A$ and $C$ is equal to the acute angle between $B$ and $C$, then $A$ and $B$ are parallel.

Which of the above statements are correct?

The diagonals of a rhombus are of length $20 \mathrm{cm}$ and$48 \mathrm{cm}$. What is the length of a side of the rhombus?
 

An arc of a circle subtends an angle $𝜋$ at the centre. If the length of the arc is $22 \mathrm{cm}$, then what is the radius of the circle? 

$(Take 𝜋=\frac{22}{ 7} )$

One-fifth of the area of a triangle $ABC$ is cut off by a line $DE$ drawn parallel to $BC$ such that$D$ is on $AB$ and $E$ is on $AC$. If $BC=10 \mathrm{cm}$, then what is $DE$ equal to?

There are$8$lines in a plane, no two of which are parallel. What is the maximum number of points at which they can intersect?

A closed polygon has six sides and one of its angles is $30°$ greater than each of the other five equal angles. What is the value of one of the equal angles?