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Directions: Study the following questions carefully and choose the right answer.
1
The area of a sector of a circle of radius 36 cm is 72π cm2. The length of the corresponding arc of the sector is
» Explain it
D
Given that, radius (r) = 36 cm

And, Area of sector = 72π cm2

⇒   πr2Θ  = 72π
360°

∴  Θ =  72π × 360°  =  72 × 360°  = 20°
πr2 36 × 36

Now, length of arc =  πrΘ  =  π × 36 × 20°  = 4π cm
180° 180°

Hence, optiion D is correct.

2
A square is inscribed in a circle of diameter 2a and another square is circumscribing circle. The difference between the areas of outer and inner squares is
» Explain it
B

Given that, Diameter = 2a

For inscribed square,

Diameter of circle = Diagonal of inner square

For circumscribed square,

Diameter of circle = Side of outer square
 
∴  Area of inner square =  1 (diagonal)2 =  1  × (2a)2 = 2a2
2 2

And, Area of outer square = (side)2 = (2a)2 = 4a2

Now, Required difference = 4a2 – 2a2 = 2a2


Hence, option B is correct.

3
ABC is a triangle right angled at A. AB = 6 cm and AC = 8 cm. Semi-circles drawn (outside the triangle) on AB, AC and BC as diameters which enclose areas x, y and z square units, respectively. What is x + y – z equal to ?
» Explain it
C
 
Given that, AB = 6 cm and AC = 8 cm

In ΔABC, by Pythagoras theorem,

BC =  AB2 + AC2  =  62 + 82  =  100  = 10 cm

Now, Area of that semi-circle which diameter is AB =  π(3)2
2

∴  x =   cm2
2

Similarly, Area of that semi-circle which diameter is AC =  π(4)2
2

∴  y =  16π  cm2
2

Similarly, Area of that semi-circle which diameter is BC =  π(5)2
2

∴  z =  25π  cm2
2

Now, x + y – z =  (  +  16π )  –  25π  = 0
2 2 2

Hence, option C is correct.

4
Consider an equiateral triangle of a side of unit length. A new equilateral triangle is formed by joining the mid-points of one, then a third equilateral triangle is formed by joining the mid-points of second. The process is continued. The perimeter of all triangles, thus formed is
» Explain it
C

Perimeter of all triangles = (3 × 1) + (3 × 0.5) + (3 × 0.25) + (3 × 0.125)

= 3 + 1.5 + 0.75 + 0.375 = 5.625 ≈ 6 units

Hence, option C is correct.
 
5
If AB and CD are two diameters of a circle of radius r and they are mutually perpendicular, then what is the ratio of the area of the circle to the area of the ΔACD ?
» Explain it
B
 

Required ratio = Area of circle : Area of ΔACD
 
= πr2 :  1  × 2r × r   =   π
2

Hence, option B is correct.