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Geometry and Mensuration: Level 3 Test 7

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Question 1
The figure shows the rectangle ABCD with a semicircle and a circle inscribed inside in it as shown. What is the ratio of the area of the circle to that of the semicircle?
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A
$ \displaystyle {{\left( \sqrt{2}-1 \right)}^{2}}:1$
B
$ \displaystyle 2{{\left( \sqrt{2}-1 \right)}^{2}}:1$
C
$ \displaystyle {{\left( \sqrt{2}-1 \right)}^{2}}:2$
D
None of these
Question 1 Explanation: 
$ \displaystyle \begin{array}{l}Let\,\,R\,\,be\,\,the\,\,radius\,\,of\,\,the\,\,semicircle\,\\Therefore\,\,OC=R=CB\\\sin ce\,\,OCB\,\,is\,a\,\,right\,\,angle\,\,triangle\,\,\\so\,\,OB=R\sqrt{2}\\Now\,\,area\,\,of\,semicircle=\frac{\pi {{r}^{2}}}{2}=\frac{\pi {{R}^{2}}}{2}\\Now\,\,FB=OB-OF=R\sqrt{2}-R=R\left( \sqrt{2}-1 \right)\\therfore\,\,the\,\,area\,\,of\,\,circle=\frac{\pi {{R}^{2}}{{\left( \sqrt{2}-1 \right)}^{2}}}{{{2}^{2}}}\\therfore\,\,the\,\,required\,\,ratio=\frac{\frac{\pi {{R}^{2}}{{\left( \sqrt{2}-1 \right)}^{2}}}{{{2}^{2}}}}{\frac{\pi {{R}^{2}}}{2}}={{\left( \sqrt{2}-1 \right)}^{2}}:2\end{array}$
Question 2
A wooden box (open at the top) of thickness 0•5 cm, length 21 cm, width 11 cm and height 6 cm is painted on the inside. The expenses of painting are Rs.70. What is the rate of painting per square centimeter?
A
Re. 0•7
B
Re 0•5
C
Re 0•1
D
Re 0•2
Question 2 Explanation: 
The actual surface area painted = 2{(11-1)(21-1)+20X5+10X5} = 700 sq. units.
Thus cost per sq. cm =70/700 =0.10 Rs
Correct option is (c)
Question 3
Based on the figure below, what is the value of x, if Y= 10.
123
A
10
B
11
C
12
D
None of these
Question 3 Explanation: 
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$ \begin{array}{l}Now\,\,AB=\sqrt{{{\left( x+3 \right)}^{2}}+{{(x-3)}^{2}}}=\sqrt{2({{x}^{2}}+9)}\\BD=\sqrt{100-{{(x-3)}^{2}}}=\sqrt{2\left( {{x}^{2}}+9 \right)}\\BD=\sqrt{100-{{(x-3)}^{2}}}\\thus\\\sqrt{100-(x-3){}^{2}}+x=\sqrt{2({{x}^{2}}+9)}\\Putting\text{ }the\text{ }values\text{ }of\text{ }x\text{ }none\text{ }of\text{ }them\text{ }satisfy\text{ }the\text{ }equation.\\Thus\text{ }the\text{ }correct\text{ }option\text{ }is\text{ }\left( d \right)\end{array}$
Question 4
A certain city has a circular wall around it, and this wall has four gates pointing north south, east and west. A house stands outside the city, three kms north of the north gate, and it can just be seen from a point nine kms east of the south gate. What is the diameter of the wall that surrounds the city?
A
6 km
B
9 km
C
12 km
D
None of these
Question 4 Explanation: 
$\displaystyle \begin{array}{l}Let\text{ }the\text{ }radius\text{ }be\text{ }r\text{ }and\text{ }the\text{ }center\text{ }be\text{ }0,0\\Now\text{ }since\text{ }between\text{ }the\text{ }point\text{ }from\text{ }where\text{ }it\text{ }can\text{ }be\text{ }just\text{ }seen\text{ }\left( -9,-r \right)\\\sqrt{6r+9}+9=\sqrt{{{(2r+3)}^{2}}+{{9}^{2}}}\\By\,\,putting\,the\,\,values\,\,only\,\,b\,\,satisfies\end{array}$
Question 5
Consider a cylinder of height h cm and radius r = r/2cm as shown in the figure (not drawn to scale).
A string of a certain length, when wound on its cylindrical surface, starting at point A and ending at
point B, gives a maximum of n turns (in other words, the string's length is the minimum length required
to wind n turns.
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What is the vertical spacing in ems between two consecutive turns
A
h/n
B
h/√n
C
h/h2
D
Cannot be determined with given information
Question 5 Explanation: 
Opening up the cylinder into a rectangle we can see that the
string is wound to a height of h with equal vertical spacing for n times.
Thus the distance = h/n
Correct option is (a) 
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