1. There are two identical solid blocks of aluminium. When the first blocks is melted and recast into spheres of equal radii ‘a’, then 14 cc of iron was left, but when the second block was melted and recast into sphere each of equal radii ‘2a’, then 36 cc of iron was left. The volumes of the solid blocks and all the spheres are in integers. What is the volume (in cm^{2}) of each of the larger spheres of radius ‘2a’?
2. Initially the diameter of a balloon is 24 cm. It can explode when the diameter becomes 3/2 times of the initial diameter. Air is blown at 342 cc/s. It is known that the shape of balloon always remains spherical. In approximately how many seconds the balloon will explode?
a) 50 s
b) 68 s
c) 87 s
d) Can’t be determined
3. Ramit has 125 small cubes of 1 cm^{3}. He wants to arrange all of them in a cuboidal shape, such that the surface area will be minimum. What is the diagonal of this larger cuboid?
4. ABCDEF is a regular hexagon of side 8 cm. What is the area of triangle BDF? ABCDEF is a regular hexagon of side 8 cm. What is the area of triangle BDF?
5. A sector of the circle measures 19^{0} as shown in the figure below. Is it possible to split the circle into 360 equal sectors of 1^{0} central angle if measured by a scale and compass only?
a) Yes
b) No
c) Yes, only if radius is known
d) Can’t be determined
6. In the adjoining figure ABC is an equilateral triangle inscribing a square of maximum possible area. Again in this square there is an equilateral triangle whose side is same as that of the square. Further the smaller equilateral triangle inscribes a square of maximum possible area. What is the area of the innermost square if the each side of the outermost triangle be 0.01 m?
7. A square and rhombus have the same base. If the rhombus is inclined at 60, find the ratio of area of square to the area of the rhombus:
8. Four isosceles triangles are cut off from the corners of a square of area 900 m^{2}. Find the area of new smaller square (in m^{2}):
9. In the adjoining figure the cross-section of a swimming pool is shown. If the length of the swimming pool is 90 m then the amount of water it can hold is
a) 17280 m^{3}
b) 19600 m^{3}
c) 17200 m^{3}
d) None of these
10. A multipurpose hall has to be constructed in a big rectangular form that can accommodate 500 people with 20 m^{2} space for each person. The height of the wall has been fixed at 15 m and the total inner surface area of the walls must be 1950 m^{2}. What is the length and breadth of the hall (in metres)?
a) 30, 20
b) 45, 20
c) 40, 25
d) 35, 30
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Explanation:1 Let the volume of solid block be V and radius of the spheres formed from the first block be a_{1 }, then the volume of each sphere be V_{1 },
Similarly, let the radius of each sphere obtained from second block be a_{2 }(= 2a_{1}), then the volume of each sphere be
V_{2 }= (8V_{1})
V = kV_{1 }+ 14 ……(i)
and V = lV_{2} + 36
or V = 8lV_{1} + 36 …..(ii)
From equation (i) and (ii)
kV_{1} + 14 = 8lV_{1} + 36
_{ } V_{1} (k – 8l) = 22
The possible value of V_{1} = 22, 11, 2 or 1
But V_{1} can never be equal to less than 14 (since remainder is always less than divisior) So, the only possible value of V1 = 22.
Explanation:2 Initial radius = 12 cm
Radius at a time when the balloon explodes = 18 cm
Explanation:3 For the given volume, cube has minimum possible length of diagonal.
Explanation:4 You should know that
Thus we make equal 19 measurements each of 19^{0,} then we get (361 - 360) = 1^{0} angle at the centre. Thus, moving continuously in the similar fashion, we get all the 360^{0} angle i.e, 360 equal sectors of 1^{0}.
Explanation:6
Explanation: 7
Explanation: 8
Explanation: 9
Explanation: 10
Best way is to go through option. Given that height of room = 10 m.
Volume of room = 25 400 = 10000 m^{3}
and Surface area of walls = 2h (l + b) = 1950 m^{2}
Now, consider option (c) and verify it.
where l = 40 and b = 25
Explanation:1 Let the volume of solid block be V and radius of the spheres formed from the first block be a_{1 }, then the volume of each sphere be V_{1 },
Similarly, let the radius of each sphere obtained from second block be a_{2 }(= 2a_{1}), then the volume of each sphere be
V_{2 }= (8V_{1})
V = kV_{1 }+ 14 ……(i)
and V = lV_{2} + 36
or V = 8lV_{1} + 36 …..(ii)
From equation (i) and (ii)
kV_{1} + 14 = 8lV_{1} + 36
_{ } V_{1} (k – 8l) = 22
The possible value of V_{1} = 22, 11, 2 or 1
But V_{1} can never be equal to less than 14 (since remainder is always less than divisior) So, the only possible value of V1 = 22.
Explanation:2 Initial radius = 12 cm
Radius at a time when the balloon explodes = 18 cm
Explanation:3 For the given volume, cube has minimum possible length of diagonal.
Explanation:4 You should know that
Thus we make equal 19 measurements each of 19^{0,} then we get (361 - 360) = 1^{0} angle at the centre. Thus, moving continuously in the similar fashion, we get all the 360^{0} angle i.e, 360 equal sectors of 1^{0}.
Explanation:6
Explanation: 7
Explanation:
Explanation:
Explanation:
Best way is to go through option. Given that height of room = 10 m.
Volume of room = 25 400 = 10000 m^{3}
and Surface area of walls = 2h (l + b) = 1950 m^{2}
Now, consider option (c) and verify it.
where l = 40 and b = 25
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