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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)

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

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 cm

Thus the closest option greater than 10.5

A bit more accurate measurement will be slightly more than half the rectangle.

Thus the shaded region >1/2 x 7x3 = 10.5 cm

^{2}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%

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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