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Geometry and Mensuration: Level 1 Test 6

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Question 1
A solid metallic sphere of radius r is converted into a solid right circular cylinder of radius R. If the height of the cylinder is twice the radius of the sphere, then
A
R = r
B
$ \displaystyle R=\text{ }r\sqrt{\frac{2}{3}}$
C
$ \displaystyle R=\sqrt{\frac{2r}{3}}$
D
$ \displaystyle R=\frac{2r}{3}$
Question 1 Explanation: 
$ \begin{array}{l}\frac{4}{3}\pi {{r}^{3}}=\pi {{R}^{2}}\times 2r\\=>\frac{2}{3}{{r}^{2}}={{R}^{2}}\\=>R=\sqrt{\frac{2}{3}r}\end{array}$
Question 2
The radius of a circle is twice the side of a square of area 196 sq.cm. Length of a rectangle is twice diameter of the circle. What is the perimeter of the rectangle if its breadth is haft the length of the rectangle?
A
244 cm
B
168 cm
C
336 cm
D
Cannot be determined
Question 2 Explanation: 
Side of the square = 14 cm.
Radius of the circle = 28 cm.
Diameter of the circle = 56 cm.
Length of the rectangle = 112 cm.
Breadth =1/2 x 112 = 56cm
Perimeter of the rectangle = 2(112+56) = 336 cm.
Correct option is (c)
Question 3
The smallest side of a right angled triangle is 6 cm, and second largest side is 8 cm. The side of a square is thrice the largest side of the triangle. What is the length of the diagonal of the square?
A
30√2 cm
B
60√2 cm
C
30 cm.
D
Cannot be determined
Question 3 Explanation: 
\[\begin{align} & The\,\,third\,\,side=\sqrt{({{8}^{2}}+{{6}^{2}})}=\sqrt{(64+36)}=10 \\ & The\,\,side\,\,of\,\,the\,\,square=3\times 10=30\,cm \\ & The\,\,length\,\,of\,\,the\,\,diagonal\,\,is\,\,30\sqrt{2}\,cm \\ & Correct\,\,option\,\,is\,\,(a) \\ \end{align}\]
Question 4
In the figure there are two rectangles ABCD and DEBG, each of length 7 cm and width 3 cm. The area of shaded region, in cm, is approximately 75
A
12
B
10
C
8
D
4
Question 4 Explanation: 
The shaded region covers roughly half of the rectangle.
A bit more accurate measurement will be slightly more than half the rectangle.
Thus the shaded region >1/2 x 7x3 = 10.5 cm2
Thus the closest option greater than 10.5
Question 5
Height of a cylindrical jar is decreased by 36%. By what percent must the radius be increased so that there is no change in its volume?
A
25
B
35
C
36
D
40
Question 5 Explanation: 
Let the height be h cm
The new height = .64h.
Thus for constant volume, the radius must be increased by
{(1/0.8) – 1}=1.25-1=(0.25/1)x100 = 25%
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