Mahabir Singh Solutions for Chapter: SOF International Mathematics Olympiad, Exercise 2: MATHEMATICAL REASONING

Three identical spherical balls fit snugly into a cylindrical can. The radius of each of the spherical ball equals the radius of the can and the balls just touch the bottom and the top of the can. What fraction of the volume of the can is taken up by the balls?Enable GingerCannot connect to Ginger Check your internet connection
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The given bar graph represents the petrol and the diesel cars sold in a mega city from $2010$ to $2015$. Study the graph carefully and answer the following questions.

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Find the difference between the number of petrol cars sold in $2010$ and $2011$ together and the number of diesel cars sold in $2015$ ?

The given bar graph represents the petrol and the diesel cars sold in a mega city from $2010$ to $2015$. Study the graph carefully and answer the following questions.

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In which year, the total sale of petrol and diesel cars altogether was maximum ?

In the given figure (not drawn to scale), $AB$ and $BC$ are two chords of the circle with centre $O$. Find the value of $x$

Study the given statements carefully and select the correct option.

$\mathrm{P}.$ The congruent figures have equal areas and vice versa

$\mathrm{Q}.$ Parallelograms on the same base (or equal bases) and between the same parallels are equal in area.

$\mathrm{R}.$ If a parallelogram and a triangle are on the same base and between the same parallels, then area of the triangle is equal to the area of parallelogram.

$\mathrm{S}.$Triangles on the same base (or equal bases) and having equal areas lie between the same parallels.

In the given figure, $\mathrm{ABCD}$ is rhombus whose diagonals intersect at $O.$ If $E$ and $F$ are mid-points of $AO$ and $BO$ respectively, then find the length of $EF$.

The traffic police recorded the speed in $\left(\raisebox{1ex}{$\mathrm{km}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{hr}$}\right.\right)$ of $10$ motorcyclists as $32,45,46,55,65,38,45,41,39,62.$ Later on, an error in recording instrument was found. Find the correct average speed of the motorcyclists,if the instrument recorded $3 \raisebox{1ex}{$\mathrm{km}$}\!\left/ \!\raisebox{-1ex}{$\mathrm{hr}$}\right.$less in each case. 

In the given figure (not to drawn to scale), $ABCD$ is quadrilateral. If $AE\parallel DC$, $BE\parallel AD$ and $AE$ intersects $BC$ at $F$,then find $\angle EBF$