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Geometry and Mensuration: Level 1 Test 7
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Question 1 
The area of the largest circle that can be drawn inside a rectangle with side 18 cm and 14 cm is
49 cm^{2}  
154 cm^{2}  
378 cm^{2}  
1078 cm^{2} 
Question 2 
Three identical cones with base radius r are placed on their bases so that each in touching the other two. The radius of the circle drawn through their vertices is :
Smaller than r  
Equal to r  
Larger than r  
Depends on the height of the cones 
Question 2 Explanation:
$ \displaystyle \begin{array}{l}The\text{ }length\text{ }of\text{ }the\text{ }side\text{ }of\text{ }the\text{ }equilateral\text{ }triangle\\formed\text{ }by\text{ }joining\text{ }the\text{ }center\text{ }of\text{ }the\text{ }bases\text{ }of\text{ }the\text{ }cones\text{ }=\text{ }2r\text{ }units.\\The\text{ }radius\text{ }of\text{ }the\text{ }circle\text{ }drawn\text{ }through\text{ }the\text{ }vertices\\\text{ }is\text{ }actually\text{ }the\,\,2/3\text{ }of\text{ }the\text{ }altitude\text{ }of\text{ }the\text{ }triangle\text{ }\\=~\frac{\frac{2}{3}\times \sqrt{3}}{2}2r\,\,=\,\,\frac{2\times 1.7}{3}r>r\\Thus\text{ }correct\text{ }option\text{ }is\text{ }\left( c \right)\end{array}$
Question 3 
A conical vessel of base radius 2 cm and height 3 cm is filled with kerosene. This liquid leaks through a hole in the bottom and collects in a cylindrical jar of radius 2 cm. The kerosene level in the jar is
∏cm  
1.5 cm  
1 cm  
3 cm

Question 3 Explanation:
$ \displaystyle \begin{array}{l}The\text{ }kerosene\text{ }level\text{ }=\text{ }height\text{ }of\text{ }the\text{ }cylinder\text{ }\left( liquid\text{ }level \right)\\\frac{1}{3}\times \pi \times {{\left( 2 \right)}^{2}}\times 3=\pi \times {{\left( 2 \right)}^{2}}\times h\\\Rightarrow \,\,h=1\,\,cm\end{array}$
Question 4 
The diameter of hollow cone is equal to the diameter of a spherical ball. If the ball is placed at the base of the cone, what portion of the ball will be outside the cone?
50%  
Less than 50%  
More than 50%  
100% 
Question 4 Explanation:
The diameter of both the shapes is same and thus we can conclude that for a height of the radius of the sphere the volume of the cone is always less than the sphere.
Thus the sphere will remain outside the cone by more than 50% always.
Thus the sphere will remain outside the cone by more than 50% always.
Question 5 
The diagonals of a rhombus are 24 cm and 10 cm. Its area is
240 cm^{2}  
312 cm^{2}  
130 cm^{2}  
120 cm^{2} 
Question 5 Explanation:
$ \displaystyle The\text{ }area\,=\frac{24\times 10}{2}=120\,\,c{{m}^{2}}$
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