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## Geometry and Mensuration: Level 3 Test 5

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*Geometry and Mensuration: Level 3 Test 5*.You scored %%SCORE%% out of %%TOTAL%%.You correct answer percentage: %%PERCENTAGE%% .Your performance has been rated as %%RATING%% Your answers are highlighted below.

Question 1 |

AB is the diameter of the given circle, while points C and D lie on the circumference as shown. If AB is 15 cm, AC is 12 cm and BD is 9 cm, find the area of quadrilateral ACBD.

A | 54p sq. cm |

B | 216p sq. cm |

C | 162p sq. cm |

D | None of these |

Question 1 Explanation:

ACB and ADB are inscribed in a semicircle and

thus angle ACB and angle ADB = 90

Thus by Pythagoras theorem,

BC and AD = 9 and 12 respectively.

Thus the area of ACBD = ½ x 2 x 9 x 12 = 108

Thus correct option is (d)

thus angle ACB and angle ADB = 90

^{o}Thus by Pythagoras theorem,

BC and AD = 9 and 12 respectively.

Thus the area of ACBD = ½ x 2 x 9 x 12 = 108

Thus correct option is (d)

Question 2 |

The sum of the areas of two circles, which touch each other externally, is 153 ft. If the sum of their radii is 15, find the ratio of the larger to the smaller radius.

A | 4 |

B | 2 |

C | 3 |

D | None of these |

Question 2 Explanation:

$ \displaystyle \begin{array}{l}\pi {{r}_{1}}^{2}+\pi {{r}_{2}}^{2}=153................1\\{{r}_{1}}+{{r}_{2}}=15............................2\\from\,\,1\\\pi \left( {{r}_{1}}^{2}+{{r}_{2}}^{2} \right)=153.....................3\\from\,\,2\\{{r}_{1}}^{2}+{{r}_{2}}^{2}+2{{r}_{1}}{{r}_{2}}=225\\{{r}_{1}}^{2}+{{r}_{2}}^{2}=225-2{{r}_{1}}{{r}_{2}}\\Put\,\,this\,\,value\,\,in\,\,3\\\pi \left( 225-2{{r}_{1}}{{r}_{2}} \right)=153\\Now\,\,from\,\,2\\{{r}_{1}}=15-{{r}_{2}}\\\pi \left\{ 225-2\left( 15-{{r}_{2}} \right){{r}_{2}} \right\}=153\\225-30{{r}_{2}}+2{{r}_{2}}^{2}-153=0\\72+2{{r}_{2}}^{2}-30{{r}_{2}}=0\\{{r}_{2}}^{2}-15{{r}_{2}}+36=0\\By\,\,solving\,\,this\,\,equation\,\\we\,\,get\,two\,\,values\,\,for\,\,{{r}_{2}}\\i.e\,\,3,12\\so\,\,the\,\,required\,\,ratio\,\,is\,\,4:1\end{array}$

Question 3 |

In a rectangle, the difference between the sum of the adjacent sides and the diagonal is half the length of longer side. What is the ratio of the shorter to the longer side?

A | 13:2 |

B | 1: 13 |

C | 2: 5 |

D | 3: 4 |

Question 3 Explanation:

$ \displaystyle \begin{array}{l}Let\text{ }the\text{ }two\text{ }sides\text{ }be\text{ }x\text{ }and\text{ }y\text{ }and\text{ }y>xThus,\\x+y-\sqrt{{{x}^{2}}+{{y}^{2}}}=\frac{1}{2}x\\=>\frac{x}{y}+y=\sqrt{{{x}^{2}}+{{y}^{2}}}\\=>\frac{{{x}^{2}}}{4}+{{y}^{2}}+\frac{2xy}{2}={{x}^{2}}+{{y}^{2}}\\=>\frac{3x}{4}=y\\=>y:x=4:3\end{array}$

Question 4 |

Four identical coins are placed in a square. For each coin, the ratio of area to circumference is same as the ratio of circumference to area. Then, find the area of the square that is not covered by the coins.

A | 16(Π-1) |

B | 16(16Π-1) |

C | 16(4-Π) |

D | 16(4 – Π/2) |

Question 5 |

Consider a circle with unit radius. There are seven adjacent sectors, S1, S2, S3, …., S7 , in the circle such that their total area is 1/8 of the area of the circle. Further, the area of the jth sector is twice that of the (j - 1)

^{th}sector, for j = 2, .....,7 What is the angle, in radians, subtended by the arc of 51 at the centre of the circle?A | $ \displaystyle \frac{\pi }{508}$ |

B | $ \displaystyle \frac{\pi }{2040}$ |

C | $ \displaystyle \frac{\pi }{1016}$ |

D | $ \displaystyle \frac{\pi }{1524}$ |

Question 5 Explanation:

Let the area of S1 be a

Thus the total area =(2

The total angle = 127a x 8 = 1016a

Thus the angle of sector S1 = Π/1016

Correct option is (a)

Thus the total area =(2

^{7 }– 1)a = 127aThe total angle = 127a x 8 = 1016a

Thus the angle of sector S1 = Π/1016

Correct option is (a)

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