Embibe Experts Solutions for Chapter: UPSC CDS (II) Previous Year Paper 2021, Exercise 1: Elementary Mathematics

$ABC$ is a triangle with sides $AB=6 \mathrm{cm}, BC=10 \mathrm{cm} \text{and} CA=8 \mathrm{cm}$. With vertices $A, B \text{and} C$ as centres, three circles are drawn each touching the other two externally.

If $P, Q \text{and} R$ are the areas of sectors at $A, B \text{and} C$ within the triangle respectively, then which of the following is/are correct?

$$\begin{gathered} 1. P={\mathrm{\pi cm}}^2 \\ 2. 9\mathrm{Q}+4\mathrm{R}=36\mathrm{\pi } {\mathrm{cm}}^2 \end{gathered}$$

Select the correct answer using the code given below:

$ABC$ is a triangle in which $AB=AC$ and $D$ is any point on $BC$.

Which one of the following is correct?

$ABC$ is a triangle in which $AB=AC$ and $D$ is any point on $BC$.

If $AD=5 \mathrm{cm}, BD=4 \mathrm{cm} \text{and} CD=6 \mathrm{cm}$, then what is $AB$ equal to?

$ABC$ is a triangle with $AB=1.6 \mathrm{cm}, BC=6.3 \mathrm{cm} \text{and} CA=6.5 \mathrm{cm}$. Let $P \text{and} Q$ be the mid-points of $AB and BC$ Respectively.

What is $A{B}^2+4B{Q}^2$ equal to?

$ABC$ is a triangle with $AB=1.6 \mathrm{cm}, BC=6.3 \mathrm{cm} \text{and} CA=6.5 \mathrm{cm}$. Let $P \text{and} Q$ be the mid-points of $AB and BC$ Respectively.

What is $A{Q}^2+C{P}^2$ equal to?

$ABC$ is a triangle with $AB=1.6 \mathrm{cm}, BC=6.3 \mathrm{cm} \text{and} CA=6.5 \mathrm{cm}$. Let $P \text{and} Q$ be the mid-points of $AB and BC$ Respectively.

What is $4\left(C{P}^2-A{Q}^2\right)$ equal to?

$AB$ is a diameter of a circle with centre $O$. Radius $OP$ is perpendicular to $AB$. Let $Q$ be any point on arc $PB$.

What is $\angle BAP$ equal to?

$AB$ is a diameter of a circle with centre $O$. Radius $OP$ is perpendicular to $AB$. Let $Q$ be any point on arc $PB$.

What is $\angle AQP$ equal to?