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## Geometry and Mensuration: Level 3 Test 8

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Question 1 |

A car is being driven in a straight line and at a uniform speed towards the base of a vertical tower. The top of the tower is observed from the car and in the process, it takes 10 minutes for the angle of elevation to change from 45° to 60°. After how much more time will this car reach the base of the tower ?

$ \displaystyle 5\left( \sqrt{3}+1 \right)$ | |

$ \displaystyle 6\left( \sqrt{3}+\sqrt{2} \right)$ | |

$ \displaystyle 7\left( \sqrt{3}-1 \right)$ | |

$ \displaystyle 8\left( \sqrt{3}-2 \right)$ |

Question 1 Explanation:

$ \displaystyle \begin{array}{l}Let\text{ }the\text{ }height\text{ }of\text{ }the\text{ }tower\text{ }is\text{ }x\text{ }m\\Thus\text{ }when\text{ }the\text{ }angle\text{ }was\text{ }45\text{ }degrees\text{ }the\text{ }height\text{ }of\text{ }the\text{ }tower\text{ }=\text{ }x\text{ }m.\\When\text{ }the\text{ }inclination\text{ }was\text{ }60\text{ }degrees\text{ }the\text{ }height\text{ }of\text{ }the\text{ }tower\text{ }=\frac{x}{\sqrt{3}}~~~m\\Thus\text{ }the\text{ }speed\text{ }=\frac{x-\frac{x}{\sqrt{3}}}{10}\\Thus\text{ }the\text{ }required\text{ }time\text{ }=~\frac{\frac{x}{\sqrt{3}}}{\frac{x-\frac{x}{\sqrt{3}}}{10}}~=~5\left( \sqrt{3}+1 \right)\\Correct\text{ }option\text{ }is\text{ }\left( a \right)\end{array}$

Question 2 |

In the adjoining figure, chord ED is parallel to the diameter AC of the circle. If ∠CBE = 65

^{o}, then what is the value of ∠DEC?35 ^{o} | |

55 ^{o} | |

45 ^{o} | |

25 ^{o} |

Question 3 |

A circle with radius 2 is placed against a right angle. Another smaller circle is also placed as shown in the adjoining figure. What is the radius of the smaller circle?

$ \displaystyle 3-2\sqrt{2}$ | |

$ \displaystyle 4-2\sqrt{2}$ | |

$ \displaystyle 7-4\sqrt{2}$ | |

$ \displaystyle 6-4\sqrt{2}$ |

Question 3 Explanation:

The radius of big circle is 2.

So the diameter of the smaller circle must be less than ½ of that which is 1.

Thus the radius must be just less than 0.5

By the options only (d) is correct

So the diameter of the smaller circle must be less than ½ of that which is 1.

Thus the radius must be just less than 0.5

By the options only (d) is correct

Question 4 |

What is the distance in cm between two parallel chords of lengths 32 cm and 24 cm in a circle of radius 20 cm?

1 or 7 | |

2 or 14 | |

3 or 21 | |

4 or 28 |

Question 4 Explanation:

$ \displaystyle \begin{array}{l}The~distance\text{ }of\text{ }one\text{ }chord\text{ }of\text{ }length\text{ }32\text{ }cm\text{ }is\\\sqrt{{{20}^{2}}-{{16}^{2}}}=12\\and\,\,the\,\,distnce\,\,of\,\,the\,\,other\,\,chord\,\,is\,\,\sqrt{{{20}^{2}}-{{12}^{2}}}=16\\If\text{ }the\text{ }two\text{ }chords\text{ }are\text{ }on\text{ }the\text{ }same\text{ }side\text{ }of\text{ }the\text{ }center\text{ }\\then\text{ }the\text{ }distance\text{ }is\text{ }16-12\text{ }=4\text{ }and\text{ }if\text{ }on\text{ }different\text{ }sides\text{ }\\then\text{ }the\text{ }distance\text{ }is\text{ }16+12\text{ }=\text{ }28\text{ }cm.\\Correct\text{ }option\text{ }is\text{ }\left( d \right)\end{array}$

Question 5 |

Four points A, B, C and D lie on a straight line in the X-Y plane, such that AB =BC =CD and the length of AB is 1 metre. An ant at A wants to reach a sugar particle at D. But there are insect repellents kept at points B and C. The ant would not go within one metre of any insect repellent. The minimum distance in metres the ant must traverse to reach the sugar particle is

$ \displaystyle 3\sqrt{2}$ | |

$ \displaystyle 1+\pi $ | |

$ \displaystyle \frac{4\pi }{3}$ | |

$ \displaystyle 5$ |

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